Subgroups — C₁₀₉×#2424 #28359678

Order 13952.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#121×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#121×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#120×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#120×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#396×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#394×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#394×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#392×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#390×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#388×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#386×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#386×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#385×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂⁵×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
24	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
27	#121×C₁₀₉	—	2	4	no	—
28	#121×C₁₀₉	—	2	4	no	—
29	#120×C₁₀₉	—	2	4	no	—
30	#120×C₁₀₉	—	2	4	no	—
31	C₄×D₄×C₁₀₉	—	2	4	no	—
32	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
33	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
37	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
43	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
44	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
45	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
48	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
49	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
50	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
51	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
52	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
53	#95×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
54	#95×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
55	#95×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
56	#95×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
57	C₂⁴×C₁₀₉	—	2	2	no	—
58	C₂⁴×C₁₀₉	—	2	2	no	—
59	C₂⁴×C₁₀₉	—	2	2	no	—
60	C₂⁴×C₁₀₉	—	2	2	no	—
61	C₂⁴×C₁₀₉	—	2	2	no	—
62	C₂⁴×C₁₀₉	—	4	1	no	—
63	C₂⁴×C₁₀₉	—	4	1	no	—
64	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
66	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₂×D₄×C₁₀₉	—	2	2	no	—
70	C₂×D₄×C₁₀₉	—	2	2	no	—
71	C₂×D₄×C₁₀₉	—	2	2	no	—
72	C₂×D₄×C₁₀₉	—	2	2	no	—
73	C₂×D₄×C₁₀₉	—	2	2	no	—
74	C₂×D₄×C₁₀₉	—	2	2	no	—
75	C₂×D₄×C₁₀₉	—	2	2	no	—
76	C₂×D₄×C₁₀₉	—	2	2	no	—
77	C₂×D₄×C₁₀₉	—	2	2	no	—
78	C₂×D₄×C₁₀₉	—	2	2	no	—
79	C₂×D₄×C₁₀₉	—	2	2	no	—
80	C₂×D₄×C₁₀₉	—	2	2	no	—
81	C₂×D₄×C₁₀₉	—	2	2	no	—
82	C₂×D₄×C₁₀₉	—	2	2	no	—
83	C₂×D₄×C₁₀₉	—	2	2	no	—
84	C₂×D₄×C₁₀₉	—	4	1	no	—
85	C₂×D₄×C₁₀₉	—	4	1	no	—
86	C₂×D₄×C₁₀₉	—	4	1	no	—
87	C₂×D₄×C₁₀₉	—	4	1	no	—
88	C₂×D₄×C₁₀₉	—	4	1	no	—
89	C₂×D₄×C₁₀₉	D₄	1	4	yes	complemented
90	C₂²×C₄×C₁₀₉	—	2	2	no	—
91	C₂²×C₄×C₁₀₉	—	2	2	no	—
92	C₂²×C₄×C₁₀₉	—	2	2	no	—
93	C₂²×C₄×C₁₀₉	—	2	2	no	—
94	C₂²×C₄×C₁₀₉	—	2	2	no	—
95	C₂²×C₄×C₁₀₉	—	2	2	no	—
96	C₂²×C₄×C₁₀₉	—	4	1	no	—
97	C₂²×C₄×C₁₀₉	—	4	1	no	—
98	C₂²×C₄×C₁₀₉	—	4	1	no	—
99	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₁₀₉	—	2	2	no	—
110	C₄⋊C₄×C₁₀₉	—	2	2	no	—
111	C₄⋊C₄×C₁₀₉	—	2	2	no	—
112	C₄⋊C₄×C₁₀₉	—	4	1	no	—
113	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
114	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
115	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
116	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
117	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
118	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
119	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
120	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
121	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
122	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
123	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
124	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
125	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
126	C₂²⋊C₄×C₁₀₉	D₄	1	4	yes	complemented
127	C₂²⋊C₄×C₁₀₉	D₄	1	4	yes	complemented
128	C₄²×C₁₀₉	—	2	2	no	—
129	C₂³×C₁₀₉	—	2	1	no	—
130	C₂³×C₁₀₉	—	2	1	no	—
131	C₂³×C₁₀₉	—	2	1	no	—
132	C₂³×C₁₀₉	—	2	1	no	—
133	C₂³×C₁₀₉	—	2	8	no	—
134	C₂³×C₁₀₉	—	2	1	no	—
135	C₂³×C₁₀₉	—	2	1	no	—
136	C₂³×C₁₀₉	—	2	1	no	—
137	C₂³×C₁₀₉	—	2	1	no	—
138	C₂³×C₁₀₉	—	2	1	no	—
139	C₂³×C₁₀₉	—	2	1	no	—
140	C₂³×C₁₀₉	—	2	1	no	—
141	C₂³×C₁₀₉	—	2	1	no	—
142	C₂³×C₁₀₉	—	2	1	no	—
143	C₂³×C₁₀₉	—	2	1	no	—
144	C₂³×C₁₀₉	—	2	1	no	—
145	C₂³×C₁₀₉	—	2	1	no	—
146	C₂³×C₁₀₉	—	2	1	no	—
147	C₂³×C₁₀₉	—	2	1	no	—
148	C₂³×C₁₀₉	—	2	1	no	—
149	C₂³×C₁₀₉	—	2	1	no	—
150	C₂³×C₁₀₉	—	2	1	no	—
151	C₂³×C₁₀₉	—	2	1	no	—
152	C₂³×C₁₀₉	—	2	1	no	—
153	C₂³×C₁₀₉	—	2	1	no	—
154	C₂³×C₁₀₉	—	2	1	no	—
155	C₂³×C₁₀₉	—	2	1	no	—
156	C₂³×C₁₀₉	—	2	1	no	—
157	C₂³×C₁₀₉	—	2	1	no	—
158	C₂³×C₁₀₉	—	2	1	no	—
159	C₂³×C₁₀₉	—	4	4	no	—
160	C₂³×C₁₀₉	—	4	4	no	—
161	C₂³×C₁₀₉	—	4	4	no	—
162	C₂³×C₁₀₉	—	4	4	no	—
163	C₂³×C₁₀₉	—	4	4	no	—
164	C₂³×C₁₀₉	—	4	4	no	—
165	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
166	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
167	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
168	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
169	C₂³×C₁₀₉	#41	1	2	yes	—
170	C₂³×C₁₀₉	#41	1	2	yes	—
171	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
172	D₄×C₁₀₉	—	2	8	no	—
173	D₄×C₁₀₉	—	4	4	no	—
174	D₄×C₁₀₉	—	4	4	no	—
175	D₄×C₁₀₉	—	4	4	no	—
176	D₄×C₁₀₉	—	4	4	no	—
177	D₄×C₁₀₉	—	4	4	no	—
178	D₄×C₁₀₉	—	4	4	no	—
179	C₂×C₄×C₁₀₉	—	2	1	no	—
180	C₂×C₄×C₁₀₉	—	2	1	no	—
181	C₂×C₄×C₁₀₉	—	2	1	no	—
182	C₂×C₄×C₁₀₉	—	2	1	no	—
183	C₂×C₄×C₁₀₉	—	2	1	no	—
184	C₂×C₄×C₁₀₉	—	2	1	no	—
185	C₂×C₄×C₁₀₉	—	2	1	no	—
186	C₂×C₄×C₁₀₉	—	2	1	no	—
187	C₂×C₄×C₁₀₉	—	2	1	no	—
188	C₂×C₄×C₁₀₉	—	2	1	no	—
189	C₂×C₄×C₁₀₉	—	2	1	no	—
190	C₂×C₄×C₁₀₉	—	2	1	no	—
191	C₂×C₄×C₁₀₉	—	2	1	no	—
192	C₂×C₄×C₁₀₉	—	2	1	no	—
193	C₂×C₄×C₁₀₉	—	2	1	no	—
194	C₂×C₄×C₁₀₉	—	2	1	no	—
195	C₂×C₄×C₁₀₉	—	2	1	no	—
196	C₂×C₄×C₁₀₉	—	2	1	no	—
197	C₂×C₄×C₁₀₉	—	2	1	no	—
198	C₂×C₄×C₁₀₉	—	2	1	no	—
199	C₂×C₄×C₁₀₉	—	2	1	no	—
200	C₂×C₄×C₁₀₉	—	2	1	no	—
201	C₂×C₄×C₁₀₉	—	2	1	no	—
202	C₂×C₄×C₁₀₉	—	2	1	no	—
203	C₂×C₄×C₁₀₉	—	2	1	no	—
204	C₂×C₄×C₁₀₉	—	4	4	no	—
205	C₂×C₄×C₁₀₉	—	4	4	no	—
206	C₂×C₄×C₁₀₉	—	4	4	no	—
207	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
208	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
209	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
210	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
211	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
212	C₂²×C₁₀₉	—	2	2	no	—
213	C₂²×C₁₀₉	—	2	2	no	—
214	C₂²×C₁₀₉	—	2	2	no	—
215	C₂²×C₁₀₉	—	2	2	no	—
216	C₂²×C₁₀₉	—	2	2	no	—
217	C₂²×C₁₀₉	—	2	2	no	—
218	C₂²×C₁₀₉	—	2	2	no	—
219	C₂²×C₁₀₉	—	2	2	no	—
220	C₂²×C₁₀₉	—	2	2	no	—
221	C₂²×C₁₀₉	—	2	2	no	—
222	C₂²×C₁₀₉	—	2	2	no	—
223	C₂²×C₁₀₉	—	2	2	no	—
224	C₂²×C₁₀₉	—	2	2	no	—
225	C₂²×C₁₀₉	—	2	2	no	—
226	C₂²×C₁₀₉	—	2	2	no	—
227	C₂²×C₁₀₉	—	4	16	no	—
228	C₂²×C₁₀₉	—	4	1	no	—
229	C₂²×C₁₀₉	—	4	1	no	—
230	C₂²×C₁₀₉	—	4	1	no	—
231	C₂²×C₁₀₉	—	4	1	no	—
232	C₂²×C₁₀₉	—	4	1	no	—
233	C₂²×C₁₀₉	—	4	1	no	—
234	C₂²×C₁₀₉	—	4	1	no	—
235	C₂²×C₁₀₉	—	4	1	no	—
236	C₂²×C₁₀₉	—	4	1	no	—
237	C₂²×C₁₀₉	—	4	1	no	—
238	C₂²×C₁₀₉	—	4	1	no	—
239	C₂²×C₁₀₉	—	4	1	no	—
240	C₂²×C₁₀₉	—	4	1	no	—
241	C₂²×C₁₀₉	—	4	1	no	—
242	C₂²×C₁₀₉	—	4	1	no	—
243	C₂²×C₁₀₉	—	4	1	no	—
244	C₂²×C₁₀₉	—	4	1	no	—
245	C₂²×C₁₀₉	—	4	1	no	—
246	C₂²×C₁₀₉	—	4	1	no	—
247	C₂²×C₁₀₉	#121	1	4	yes	complemented
248	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
249	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
250	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
251	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
252	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
253	C₂²×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
254	C₂²×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
255	C₄×C₁₀₉	—	2	2	no	—
256	C₄×C₁₀₉	—	2	2	no	—
257	C₄×C₁₀₉	—	2	2	no	—
258	C₄×C₁₀₉	—	2	2	no	—
259	C₄×C₁₀₉	—	2	2	no	—
260	C₄×C₁₀₉	—	4	1	no	—
261	C₄×C₁₀₉	—	4	1	no	—
262	C₄×C₁₀₉	—	4	1	no	—
263	C₄×C₁₀₉	—	4	1	no	—
264	C₄×C₁₀₉	—	4	1	no	—
265	C₂×C₁₀₉	—	2	4	no	—
266	C₂×C₁₀₉	—	4	2	no	—
267	C₂×C₁₀₉	—	4	2	no	—
268	C₂×C₁₀₉	—	4	2	no	—
269	C₂×C₁₀₉	—	8	1	no	—
270	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
271	C₂×C₁₀₉	#525	1	1	yes	characteristic
272	C₂×C₁₀₉	#538	1	1	yes	characteristic
273	C₂×C₁₀₉	D₄²	1	1	yes	characteristic
274	C₂×C₁₀₉	D₄²	1	1	yes	characteristic
275	C₂×C₁₀₉	#546	1	1	yes	characteristic
276	C₂×C₁₀₉	#546	1	1	yes	characteristic
277	#2424	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
278	C₁₀₉	#2424	1	1	yes	characteristic, direct product factor, Sylow
279	C₂×#121	C₂×C₁₀₉	1	1	yes	characteristic, complemented
280	C₂×#121	C₂×C₁₀₉	1	1	yes	characteristic, complemented
281	C₂×#120	C₂×C₁₀₉	1	1	yes	characteristic, complemented
282	C₂×#120	C₂×C₁₀₉	1	1	yes	characteristic, complemented
283	C₂×C₄×D₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
284	C₂²×C₂²⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
285	#396	C₂×C₁₀₉	1	1	yes	characteristic, complemented
286	#394	C₂×C₁₀₉	1	1	yes	characteristic, complemented
287	#394	C₂×C₁₀₉	1	1	yes	characteristic, complemented
288	#392	C₂×C₁₀₉	1	1	yes	characteristic, complemented
289	#390	C₂×C₁₀₉	1	1	yes	characteristic, complemented
290	#388	C₂×C₁₀₉	1	1	yes	characteristic, complemented
291	#386	C₂×C₁₀₉	1	1	yes	characteristic, complemented
292	#386	C₂×C₁₀₉	1	1	yes	characteristic, complemented
293	#385	C₂×C₁₀₉	1	1	yes	characteristic, complemented
294	C₂⁵	C₂²×C₁₀₉	1	1	yes	characteristic
295	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
296	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
297	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
298	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
299	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
300	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
301	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
302	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
303	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
304	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
305	#121	—	2	4	no	—
306	#121	—	2	4	no	—
307	#120	—	2	4	no	—
308	#120	—	2	4	no	—
309	C₄×D₄	—	2	4	no	—
310	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
311	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
312	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
313	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
314	C₂×C₂²⋊C₄	—	2	4	no	—
315	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
316	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
317	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
318	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
319	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
320	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
321	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
322	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
323	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
324	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
325	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
326	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
327	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
328	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
329	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
330	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
331	#95	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
332	#95	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
333	#95	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
334	#95	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
335	C₂⁴	—	2	2	no	—
336	C₂⁴	—	2	2	no	—
337	C₂⁴	—	2	2	no	—
338	C₂⁴	—	2	2	no	—
339	C₂⁴	—	2	2	no	—
340	C₂⁴	—	4	1	no	—
341	C₂⁴	—	4	1	no	—
342	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
343	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
344	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
345	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
346	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	2	2	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	2	no	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	—	4	1	no	—
363	C₂×D₄	—	4	1	no	—
364	C₂×D₄	—	4	1	no	—
365	C₂×D₄	—	4	1	no	—
366	C₂×D₄	—	4	1	no	—
367	C₂×D₄	D₄×C₁₀₉	1	4	yes	complemented
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	4	1	no	—
375	C₂²×C₄	—	4	1	no	—
376	C₂²×C₄	—	4	1	no	—
377	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
378	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
379	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
380	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
381	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
382	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
383	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
384	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
385	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
386	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	4	1	no	—
391	C₂²⋊C₄	—	2	2	no	—
392	C₂²⋊C₄	—	2	2	no	—
393	C₂²⋊C₄	—	2	2	no	—
394	C₂²⋊C₄	—	2	2	no	—
395	C₂²⋊C₄	—	2	2	no	—
396	C₂²⋊C₄	—	2	2	no	—
397	C₂²⋊C₄	—	2	2	no	—
398	C₂²⋊C₄	—	2	2	no	—
399	C₂²⋊C₄	—	2	2	no	—
400	C₂²⋊C₄	—	2	2	no	—
401	C₂²⋊C₄	—	2	2	no	—
402	C₂²⋊C₄	—	2	2	no	—
403	C₂²⋊C₄	—	4	1	no	—
404	C₂²⋊C₄	D₄×C₁₀₉	1	4	yes	complemented
405	C₂²⋊C₄	D₄×C₁₀₉	1	4	yes	complemented
406	C₄²	—	2	2	no	—
407	C₂³	—	2	1	no	—
408	C₂³	—	2	1	no	—
409	C₂³	—	2	1	no	—
410	C₂³	—	2	1	no	—
411	C₂³	—	2	8	no	—
412	C₂³	—	2	1	no	—
413	C₂³	—	2	1	no	—
414	C₂³	—	2	1	no	—
415	C₂³	—	2	1	no	—
416	C₂³	—	2	1	no	—
417	C₂³	—	2	1	no	—
418	C₂³	—	2	1	no	—
419	C₂³	—	2	1	no	—
420	C₂³	—	2	1	no	—
421	C₂³	—	2	1	no	—
422	C₂³	—	2	1	no	—
423	C₂³	—	2	1	no	—
424	C₂³	—	2	1	no	—
425	C₂³	—	2	1	no	—
426	C₂³	—	2	1	no	—
427	C₂³	—	2	1	no	—
428	C₂³	—	2	1	no	—
429	C₂³	—	2	1	no	—
430	C₂³	—	2	1	no	—
431	C₂³	—	2	1	no	—
432	C₂³	—	2	1	no	—
433	C₂³	—	2	1	no	—
434	C₂³	—	2	1	no	—
435	C₂³	—	2	1	no	—
436	C₂³	—	2	1	no	—
437	C₂³	—	4	4	no	—
438	C₂³	—	4	4	no	—
439	C₂³	—	4	4	no	—
440	C₂³	—	4	4	no	—
441	C₂³	—	4	4	no	—
442	C₂³	—	4	4	no	—
443	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
444	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
445	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
446	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
447	C₂³	#41×C₁₀₉	1	2	yes	—
448	C₂³	#41×C₁₀₉	1	2	yes	—
449	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
450	D₄	—	2	8	no	—
451	D₄	—	4	4	no	—
452	D₄	—	4	4	no	—
453	D₄	—	4	4	no	—
454	D₄	—	4	4	no	—
455	D₄	—	4	4	no	—
456	D₄	—	4	4	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	4	4	no	—
483	C₂×C₄	—	4	4	no	—
484	C₂×C₄	—	4	4	no	—
485	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
486	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
487	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
488	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
489	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	2	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	2	2	no	—
494	C₂²	—	2	2	no	—
495	C₂²	—	2	2	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	2	2	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	2	2	no	—
500	C₂²	—	2	2	no	—
501	C₂²	—	2	2	no	—
502	C₂²	—	2	2	no	—
503	C₂²	—	2	2	no	—
504	C₂²	—	2	2	no	—
505	C₂²	—	4	16	no	—
506	C₂²	—	4	1	no	—
507	C₂²	—	4	1	no	—
508	C₂²	—	4	1	no	—
509	C₂²	—	4	1	no	—
510	C₂²	—	4	1	no	—
511	C₂²	—	4	1	no	—
512	C₂²	—	4	1	no	—
513	C₂²	—	4	1	no	—
514	C₂²	—	4	1	no	—
515	C₂²	—	4	1	no	—
516	C₂²	—	4	1	no	—
517	C₂²	—	4	1	no	—
518	C₂²	—	4	1	no	—
519	C₂²	—	4	1	no	—
520	C₂²	—	4	1	no	—
521	C₂²	—	4	1	no	—
522	C₂²	—	4	1	no	—
523	C₂²	—	4	1	no	—
524	C₂²	—	4	1	no	—
525	C₂²	#121×C₁₀₉	1	4	yes	complemented
526	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
527	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
528	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
529	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
530	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
531	C₂²	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
532	C₂²	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
533	C₄	—	2	2	no	—
534	C₄	—	2	2	no	—
535	C₄	—	2	2	no	—
536	C₄	—	2	2	no	—
537	C₄	—	2	2	no	—
538	C₄	—	4	1	no	—
539	C₄	—	4	1	no	—
540	C₄	—	4	1	no	—
541	C₄	—	4	1	no	—
542	C₄	—	4	1	no	—
543	C₂	—	2	4	no	—
544	C₂	—	4	2	no	—
545	C₂	—	4	2	no	—
546	C₂	—	4	2	no	—
547	C₂	—	8	1	no	—
548	C₂	C₂×#121×C₁₀₉	1	1	yes	characteristic
549	C₂	#525×C₁₀₉	1	1	yes	characteristic
550	C₂	#538×C₁₀₉	1	1	yes	characteristic
551	C₂	D₄²×C₁₀₉	1	1	yes	characteristic
552	C₂	D₄²×C₁₀₉	1	1	yes	characteristic
553	C₂	#546×C₁₀₉	1	1	yes	characteristic
554	C₂	#546×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₀₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#121×C₁₀₉	C₂	semidirect
2. C₂×#121×C₁₀₉	C₂	semidirect
3. C₂×#120×C₁₀₉	C₂	semidirect
4. C₂×#120×C₁₀₉	C₂	semidirect
5. C₂×C₄×D₄×C₁₀₉	C₂	semidirect
6. C₂²×C₂²⋊C₄×C₁₀₉	C₂	semidirect
7. #396×C₁₀₉	C₂	semidirect
8. #394×C₁₀₉	C₂	semidirect
9. #394×C₁₀₉	C₂	semidirect
10. #392×C₁₀₉	C₂	semidirect
11. #390×C₁₀₉	C₂	semidirect
12. #388×C₁₀₉	C₂	semidirect
13. #386×C₁₀₉	C₂	semidirect
14. #386×C₁₀₉	C₂	semidirect
15. #385×C₁₀₉	C₂	semidirect
23. C₂²×D₄×C₁₀₉	C₂²	semidirect
26. C₂³×C₄×C₁₀₉	C₂²	semidirect
32. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
33. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
34. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
35. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
47. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
48. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
49. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
50. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
51. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
52. C₂×C₄²×C₁₀₉	C₂²	semidirect
53. #95×C₁₀₉	C₂²	semidirect
54. #95×C₁₀₉	C₂²	semidirect
55. #95×C₁₀₉	C₂²	semidirect
56. #95×C₁₀₉	C₂²	semidirect
89. C₂×D₄×C₁₀₉	D₄	semidirect
126. C₂²⋊C₄×C₁₀₉	D₄	semidirect
127. C₂²⋊C₄×C₁₀₉	D₄	semidirect
247. C₂²×C₁₀₉	#121	semidirect
277. #2424	C₁₀₉	direct (N × H)
278. C₁₀₉	#2424	direct (N × H)
279. C₂×#121	C₂×C₁₀₉	semidirect
280. C₂×#121	C₂×C₁₀₉	semidirect
281. C₂×#120	C₂×C₁₀₉	semidirect
282. C₂×#120	C₂×C₁₀₉	semidirect
283. C₂×C₄×D₄	C₂×C₁₀₉	semidirect
284. C₂²×C₂²⋊C₄	C₂×C₁₀₉	semidirect
285. #396	C₂×C₁₀₉	semidirect
286. #394	C₂×C₁₀₉	semidirect
287. #394	C₂×C₁₀₉	semidirect
288. #392	C₂×C₁₀₉	semidirect
289. #390	C₂×C₁₀₉	semidirect
290. #388	C₂×C₁₀₉	semidirect
291. #386	C₂×C₁₀₉	semidirect
292. #386	C₂×C₁₀₉	semidirect
293. #385	C₂×C₁₀₉	semidirect
301. C₂²×D₄	C₂²×C₁₀₉	semidirect
304. C₂³×C₄	C₂²×C₁₀₉	semidirect
310. C₂×C₄⋊C₄	C₂²×C₁₀₉	semidirect
311. C₂×C₄⋊C₄	C₂²×C₁₀₉	semidirect
312. C₂×C₄⋊C₄	C₂²×C₁₀₉	semidirect
313. C₂×C₄⋊C₄	C₂²×C₁₀₉	semidirect
325. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
326. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
327. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
328. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
329. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
330. C₂×C₄²	C₂²×C₁₀₉	semidirect
331. #95	C₂²×C₁₀₉	semidirect
332. #95	C₂²×C₁₀₉	semidirect
333. #95	C₂²×C₁₀₉	semidirect
334. #95	C₂²×C₁₀₉	semidirect
367. C₂×D₄	D₄×C₁₀₉	semidirect
404. C₂²⋊C₄	D₄×C₁₀₉	semidirect
405. C₂²⋊C₄	D₄×C₁₀₉	semidirect
525. C₂²	#121×C₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#121×C₁₀₉	2
C₂×#120×C₁₀₉	2
C₂×C₄×D₄×C₁₀₉	2
C₂²×C₂²⋊C₄×C₁₀₉	2
#396×C₁₀₉	2
#394×C₁₀₉	2
#392×C₁₀₉	2
#390×C₁₀₉	2
#388×C₁₀₉	2
#386×C₁₀₉	2
#385×C₁₀₉	2
#2424	109

Subgroups — C109×#2424 #28359678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2424 #28359678