Subgroups — C₁₀₉×#2410 #28359664

Order 13952.

1246 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#121×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₂×#120×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#120×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#394×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#392×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	#386×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#386×C₁₀₉	C₂	1	2	yes	maximal, complemented
10	#385×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#379×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂⁵×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂⁵×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₀₉	—	2	4	no	—
15	C₂²×D₄×C₁₀₉	—	2	8	no	—
16	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	—
18	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	—
19	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	complemented
21	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
22	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	complemented
24	#121×C₁₀₉	—	2	8	no	—
25	#120×C₁₀₉	—	2	4	no	—
26	#120×C₁₀₉	—	2	4	no	—
27	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
28	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
29	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
33	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
35	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
37	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
38	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
39	#95×C₁₀₉	C₂²	1	2	yes	complemented
40	#95×C₁₀₉	C₂²	1	2	yes	complemented
41	C₂⁴×C₁₀₉	—	2	2	no	—
42	C₂⁴×C₁₀₉	—	2	4	no	—
43	C₂⁴×C₁₀₉	—	2	4	no	—
44	C₂⁴×C₁₀₉	—	2	2	no	—
45	C₂⁴×C₁₀₉	—	2	2	no	—
46	C₂⁴×C₁₀₉	—	2	2	no	—
47	C₂⁴×C₁₀₉	—	2	2	no	—
48	C₂⁴×C₁₀₉	—	4	2	no	—
49	C₂⁴×C₁₀₉	—	4	2	no	—
50	C₂⁴×C₁₀₉	D₄	1	4	yes	complemented
51	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
54	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
55	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
56	C₂×D₄×C₁₀₉	—	2	32	no	—
57	C₂×D₄×C₁₀₉	—	2	4	no	—
58	C₂×D₄×C₁₀₉	—	2	4	no	—
59	C₂×D₄×C₁₀₉	—	2	4	no	—
60	C₂×D₄×C₁₀₉	—	2	4	no	—
61	C₂×D₄×C₁₀₉	—	2	2	no	—
62	C₂×D₄×C₁₀₉	—	2	2	no	—
63	C₂×D₄×C₁₀₉	—	2	2	no	—
64	C₂×D₄×C₁₀₉	—	2	2	no	—
65	C₂×D₄×C₁₀₉	—	2	4	no	—
66	C₂×D₄×C₁₀₉	—	2	4	no	—
67	C₂×D₄×C₁₀₉	—	2	2	no	—
68	C₂×D₄×C₁₀₉	—	2	4	no	—
69	C₂×D₄×C₁₀₉	—	2	4	no	—
70	C₂×D₄×C₁₀₉	—	4	2	no	—
71	C₂×D₄×C₁₀₉	—	4	2	no	—
72	C₂×D₄×C₁₀₉	D₄	1	8	yes	complemented
73	C₂²×C₄×C₁₀₉	—	2	2	no	—
74	C₂²×C₄×C₁₀₉	—	2	4	no	—
75	C₂²×C₄×C₁₀₉	—	2	4	no	—
76	C₂²×C₄×C₁₀₉	—	2	4	no	—
77	C₂²×C₄×C₁₀₉	—	4	2	no	—
78	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
81	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
83	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
84	C₄⋊C₄×C₁₀₉	—	2	4	no	—
85	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
86	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
87	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
88	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
89	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
90	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
91	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
92	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
93	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
94	C₂²⋊C₄×C₁₀₉	D₄	1	4	yes	complemented
95	C₂³×C₁₀₉	—	2	1	no	—
96	C₂³×C₁₀₉	—	2	2	no	—
97	C₂³×C₁₀₉	—	2	1	no	—
98	C₂³×C₁₀₉	—	2	2	no	—
99	C₂³×C₁₀₉	—	2	2	no	—
100	C₂³×C₁₀₉	—	2	8	no	—
101	C₂³×C₁₀₉	—	2	8	no	—
102	C₂³×C₁₀₉	—	2	8	no	—
103	C₂³×C₁₀₉	—	2	1	no	—
104	C₂³×C₁₀₉	—	2	2	no	—
105	C₂³×C₁₀₉	—	2	2	no	—
106	C₂³×C₁₀₉	—	2	2	no	—
107	C₂³×C₁₀₉	—	2	2	no	—
108	C₂³×C₁₀₉	—	2	1	no	—
109	C₂³×C₁₀₉	—	2	2	no	—
110	C₂³×C₁₀₉	—	2	2	no	—
111	C₂³×C₁₀₉	—	2	2	no	—
112	C₂³×C₁₀₉	—	2	2	no	—
113	C₂³×C₁₀₉	—	2	2	no	—
114	C₂³×C₁₀₉	—	2	1	no	—
115	C₂³×C₁₀₉	—	2	1	no	—
116	C₂³×C₁₀₉	—	2	1	no	—
117	C₂³×C₁₀₉	—	2	2	no	—
118	C₂³×C₁₀₉	—	4	4	no	—
119	C₂³×C₁₀₉	—	4	8	no	—
120	C₂³×C₁₀₉	—	4	8	no	—
121	C₂³×C₁₀₉	—	4	8	no	—
122	C₂³×C₁₀₉	—	4	8	no	—
123	C₂³×C₁₀₉	—	4	4	no	—
124	C₂³×C₁₀₉	—	4	4	no	—
125	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
126	C₂³×C₁₀₉	C₂×D₄	1	4	yes	—
127	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
128	C₂³×C₁₀₉	C₂×D₄	1	4	yes	—
129	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
130	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
131	C₂³×C₁₀₉	#41	1	2	yes	—
132	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
133	D₄×C₁₀₉	—	2	16	no	—
134	D₄×C₁₀₉	—	2	16	no	—
135	D₄×C₁₀₉	—	4	8	no	—
136	D₄×C₁₀₉	—	4	4	no	—
137	D₄×C₁₀₉	—	4	4	no	—
138	D₄×C₁₀₉	—	4	4	no	—
139	C₂×C₄×C₁₀₉	—	2	8	no	—
140	C₂×C₄×C₁₀₉	—	2	2	no	—
141	C₂×C₄×C₁₀₉	—	2	2	no	—
142	C₂×C₄×C₁₀₉	—	2	2	no	—
143	C₂×C₄×C₁₀₉	—	2	2	no	—
144	C₂×C₄×C₁₀₉	—	2	2	no	—
145	C₂×C₄×C₁₀₉	—	2	2	no	—
146	C₂×C₄×C₁₀₉	—	2	1	no	—
147	C₂×C₄×C₁₀₉	—	2	1	no	—
148	C₂×C₄×C₁₀₉	—	2	1	no	—
149	C₂×C₄×C₁₀₉	—	2	1	no	—
150	C₂×C₄×C₁₀₉	—	2	2	no	—
151	C₂×C₄×C₁₀₉	—	2	1	no	—
152	C₂×C₄×C₁₀₉	—	2	2	no	—
153	C₂×C₄×C₁₀₉	—	2	2	no	—
154	C₂×C₄×C₁₀₉	—	4	8	no	—
155	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
156	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	complemented
157	C₂²×C₁₀₉	—	2	2	no	—
158	C₂²×C₁₀₉	—	2	2	no	—
159	C₂²×C₁₀₉	—	2	2	no	—
160	C₂²×C₁₀₉	—	2	2	no	—
161	C₂²×C₁₀₉	—	2	4	no	—
162	C₂²×C₁₀₉	—	2	4	no	—
163	C₂²×C₁₀₉	—	2	4	no	—
164	C₂²×C₁₀₉	—	2	4	no	—
165	C₂²×C₁₀₉	—	2	2	no	—
166	C₂²×C₁₀₉	—	2	4	no	—
167	C₂²×C₁₀₉	—	2	2	no	—
168	C₂²×C₁₀₉	—	2	2	no	—
169	C₂²×C₁₀₉	—	2	4	no	—
170	C₂²×C₁₀₉	—	2	4	no	—
171	C₂²×C₁₀₉	—	4	16	no	—
172	C₂²×C₁₀₉	—	4	16	no	—
173	C₂²×C₁₀₉	—	4	2	no	—
174	C₂²×C₁₀₉	—	4	2	no	—
175	C₂²×C₁₀₉	—	4	2	no	—
176	C₂²×C₁₀₉	—	4	2	no	—
177	C₂²×C₁₀₉	—	4	2	no	—
178	C₂²×C₁₀₉	—	4	2	no	—
179	C₂²×C₁₀₉	—	4	1	no	—
180	C₂²×C₁₀₉	—	4	1	no	—
181	C₂²×C₁₀₉	—	4	1	no	—
182	C₂²×C₁₀₉	—	4	2	no	—
183	C₂²×C₁₀₉	—	4	2	no	—
184	C₂²×C₁₀₉	#120	1	4	yes	complemented
185	C₂²×C₁₀₉	#121	1	4	yes	complemented
186	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
187	C₂²×C₁₀₉	C₂²×D₄	1	2	yes	—
188	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
189	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
190	C₂²×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
191	C₄×C₁₀₉	—	2	4	no	—
192	C₄×C₁₀₉	—	2	4	no	—
193	C₄×C₁₀₉	—	4	1	no	—
194	C₄×C₁₀₉	—	4	1	no	—
195	C₄×C₁₀₉	—	4	1	no	—
196	C₄×C₁₀₉	—	4	2	no	—
197	C₂×C₁₀₉	—	2	4	no	—
198	C₂×C₁₀₉	—	2	4	no	—
199	C₂×C₁₀₉	—	4	2	no	—
200	C₂×C₁₀₉	—	4	4	no	—
201	C₂×C₁₀₉	—	8	1	no	—
202	C₂×C₁₀₉	C₂×#120	1	1	yes	characteristic
203	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
204	C₂×C₁₀₉	#534	1	1	yes	characteristic
205	C₂×C₁₀₉	D₄²	1	2	yes	—
206	C₂×C₁₀₉	#546	1	2	yes	—
207	#2410	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₁₀₉	#2410	1	1	yes	characteristic, direct product factor, Sylow
209	C₂³×D₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
210	C₂×#121	C₂×C₁₀₉	1	2	yes	complemented
211	C₂×#120	C₂×C₁₀₉	1	1	yes	characteristic, complemented
212	C₂×#120	C₂×C₁₀₉	1	1	yes	characteristic, complemented
213	C₂²×C₂²⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
214	#394	C₂×C₁₀₉	1	2	yes	complemented
215	#392	C₂×C₁₀₉	1	2	yes	complemented
216	#386	C₂×C₁₀₉	1	1	yes	characteristic, complemented
217	#386	C₂×C₁₀₉	1	2	yes	complemented
218	#385	C₂×C₁₀₉	1	1	yes	characteristic, complemented
219	#379	C₂×C₁₀₉	1	1	yes	characteristic, complemented
220	C₂⁵	C₂²×C₁₀₉	1	1	yes	characteristic
221	C₂⁵	C₂²×C₁₀₉	1	1	yes	characteristic
222	C₂²×D₄	—	2	4	no	—
223	C₂²×D₄	—	2	8	no	—
224	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
225	C₂²×D₄	C₂²×C₁₀₉	1	2	yes	—
226	C₂²×D₄	C₂²×C₁₀₉	1	2	yes	—
227	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
228	C₂²×D₄	C₂²×C₁₀₉	1	2	yes	complemented
229	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
230	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
231	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	complemented
232	#121	—	2	8	no	—
233	#120	—	2	4	no	—
234	#120	—	2	4	no	—
235	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	complemented
236	C₂×C₂²⋊C₄	—	2	4	no	—
237	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
238	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
239	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
240	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	complemented
241	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
242	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	complemented
243	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
244	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	complemented
245	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	complemented
246	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	complemented
247	#95	C₂²×C₁₀₉	1	2	yes	complemented
248	#95	C₂²×C₁₀₉	1	2	yes	complemented
249	C₂⁴	—	2	2	no	—
250	C₂⁴	—	2	4	no	—
251	C₂⁴	—	2	4	no	—
252	C₂⁴	—	2	2	no	—
253	C₂⁴	—	2	2	no	—
254	C₂⁴	—	2	2	no	—
255	C₂⁴	—	2	2	no	—
256	C₂⁴	—	4	2	no	—
257	C₂⁴	—	4	2	no	—
258	C₂⁴	D₄×C₁₀₉	1	4	yes	complemented
259	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
260	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
261	C₂⁴	C₂³×C₁₀₉	1	2	yes	—
262	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
263	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
264	C₂×D₄	—	2	32	no	—
265	C₂×D₄	—	2	4	no	—
266	C₂×D₄	—	2	4	no	—
267	C₂×D₄	—	2	4	no	—
268	C₂×D₄	—	2	4	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	4	no	—
274	C₂×D₄	—	2	4	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	2	4	no	—
277	C₂×D₄	—	2	4	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	—	4	2	no	—
280	C₂×D₄	D₄×C₁₀₉	1	8	yes	complemented
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	4	no	—
283	C₂²×C₄	—	2	4	no	—
284	C₂²×C₄	—	2	4	no	—
285	C₂²×C₄	—	4	2	no	—
286	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
287	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
288	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
289	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
290	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
291	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
292	C₄⋊C₄	—	2	4	no	—
293	C₂²⋊C₄	—	2	2	no	—
294	C₂²⋊C₄	—	2	2	no	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	2	4	no	—
298	C₂²⋊C₄	—	2	4	no	—
299	C₂²⋊C₄	—	2	4	no	—
300	C₂²⋊C₄	—	2	4	no	—
301	C₂²⋊C₄	—	4	2	no	—
302	C₂²⋊C₄	D₄×C₁₀₉	1	4	yes	complemented
303	C₂³	—	2	1	no	—
304	C₂³	—	2	2	no	—
305	C₂³	—	2	1	no	—
306	C₂³	—	2	2	no	—
307	C₂³	—	2	2	no	—
308	C₂³	—	2	8	no	—
309	C₂³	—	2	8	no	—
310	C₂³	—	2	8	no	—
311	C₂³	—	2	1	no	—
312	C₂³	—	2	2	no	—
313	C₂³	—	2	2	no	—
314	C₂³	—	2	2	no	—
315	C₂³	—	2	2	no	—
316	C₂³	—	2	1	no	—
317	C₂³	—	2	2	no	—
318	C₂³	—	2	2	no	—
319	C₂³	—	2	2	no	—
320	C₂³	—	2	2	no	—
321	C₂³	—	2	2	no	—
322	C₂³	—	2	1	no	—
323	C₂³	—	2	1	no	—
324	C₂³	—	2	1	no	—
325	C₂³	—	2	2	no	—
326	C₂³	—	4	4	no	—
327	C₂³	—	4	8	no	—
328	C₂³	—	4	8	no	—
329	C₂³	—	4	8	no	—
330	C₂³	—	4	8	no	—
331	C₂³	—	4	4	no	—
332	C₂³	—	4	4	no	—
333	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
334	C₂³	C₂×D₄×C₁₀₉	1	4	yes	—
335	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
336	C₂³	C₂×D₄×C₁₀₉	1	4	yes	—
337	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
338	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
339	C₂³	#41×C₁₀₉	1	2	yes	—
340	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
341	D₄	—	2	16	no	—
342	D₄	—	2	16	no	—
343	D₄	—	4	8	no	—
344	D₄	—	4	4	no	—
345	D₄	—	4	4	no	—
346	D₄	—	4	4	no	—
347	C₂×C₄	—	2	8	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	4	8	no	—
363	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
364	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	complemented
365	C₂²	—	2	2	no	—
366	C₂²	—	2	2	no	—
367	C₂²	—	2	2	no	—
368	C₂²	—	2	2	no	—
369	C₂²	—	2	4	no	—
370	C₂²	—	2	4	no	—
371	C₂²	—	2	4	no	—
372	C₂²	—	2	4	no	—
373	C₂²	—	2	2	no	—
374	C₂²	—	2	4	no	—
375	C₂²	—	2	2	no	—
376	C₂²	—	2	2	no	—
377	C₂²	—	2	4	no	—
378	C₂²	—	2	4	no	—
379	C₂²	—	4	16	no	—
380	C₂²	—	4	16	no	—
381	C₂²	—	4	2	no	—
382	C₂²	—	4	2	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	4	2	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	4	1	no	—
388	C₂²	—	4	1	no	—
389	C₂²	—	4	1	no	—
390	C₂²	—	4	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	#120×C₁₀₉	1	4	yes	complemented
393	C₂²	#121×C₁₀₉	1	4	yes	complemented
394	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
395	C₂²	C₂²×D₄×C₁₀₉	1	2	yes	—
396	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
397	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
398	C₂²	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
399	C₄	—	2	4	no	—
400	C₄	—	2	4	no	—
401	C₄	—	4	1	no	—
402	C₄	—	4	1	no	—
403	C₄	—	4	1	no	—
404	C₄	—	4	2	no	—
405	C₂	—	2	4	no	—
406	C₂	—	2	4	no	—
407	C₂	—	4	2	no	—
408	C₂	—	4	4	no	—
409	C₂	—	8	1	no	—
410	C₂	C₂×#120×C₁₀₉	1	1	yes	characteristic
411	C₂	C₂×#121×C₁₀₉	1	1	yes	characteristic
412	C₂	#534×C₁₀₉	1	1	yes	characteristic
413	C₂	D₄²×C₁₀₉	1	2	yes	—
414	C₂	#546×C₁₀₉	1	2	yes	—

	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. C₂³×D₄×C₁₀₉	C₂	semidirect
2. C₂×#121×C₁₀₉	C₂	semidirect
3. C₂×#120×C₁₀₉	C₂	semidirect
4. C₂×#120×C₁₀₉	C₂	semidirect
5. C₂²×C₂²⋊C₄×C₁₀₉	C₂	semidirect
6. #394×C₁₀₉	C₂	semidirect
7. #392×C₁₀₉	C₂	semidirect
8. #386×C₁₀₉	C₂	semidirect
9. #386×C₁₀₉	C₂	semidirect
10. #385×C₁₀₉	C₂	semidirect
11. #379×C₁₀₉	C₂	semidirect
19. C₂²×D₄×C₁₀₉	C₂²	semidirect
20. C₂²×D₄×C₁₀₉	C₂²	semidirect
21. C₂²×D₄×C₁₀₉	C₂²	semidirect
23. C₂³×C₄×C₁₀₉	C₂²	semidirect
27. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
32. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
33. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
34. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
35. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
36. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
37. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
38. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
39. #95×C₁₀₉	C₂²	semidirect
40. #95×C₁₀₉	C₂²	semidirect
50. C₂⁴×C₁₀₉	D₄	semidirect
72. C₂×D₄×C₁₀₉	D₄	semidirect
94. C₂²⋊C₄×C₁₀₉	D₄	semidirect
156. C₂×C₄×C₁₀₉	C₂×D₄	semidirect
184. C₂²×C₁₀₉	#120	semidirect
185. C₂²×C₁₀₉	#121	semidirect
207. #2410	C₁₀₉	direct (N × H)
208. C₁₀₉	#2410	direct (N × H)
209. C₂³×D₄	C₂×C₁₀₉	semidirect
210. C₂×#121	C₂×C₁₀₉	semidirect
211. C₂×#120	C₂×C₁₀₉	semidirect
212. C₂×#120	C₂×C₁₀₉	semidirect
213. C₂²×C₂²⋊C₄	C₂×C₁₀₉	semidirect
214. #394	C₂×C₁₀₉	semidirect
215. #392	C₂×C₁₀₉	semidirect
216. #386	C₂×C₁₀₉	semidirect
217. #386	C₂×C₁₀₉	semidirect
218. #385	C₂×C₁₀₉	semidirect
219. #379	C₂×C₁₀₉	semidirect
227. C₂²×D₄	C₂²×C₁₀₉	semidirect
228. C₂²×D₄	C₂²×C₁₀₉	semidirect
229. C₂²×D₄	C₂²×C₁₀₉	semidirect
231. C₂³×C₄	C₂²×C₁₀₉	semidirect
235. C₂×C₄⋊C₄	C₂²×C₁₀₉	semidirect
240. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
241. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
242. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
243. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
244. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
245. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
246. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
247. #95	C₂²×C₁₀₉	semidirect
248. #95	C₂²×C₁₀₉	semidirect
258. C₂⁴	D₄×C₁₀₉	semidirect
280. C₂×D₄	D₄×C₁₀₉	semidirect
302. C₂²⋊C₄	D₄×C₁₀₉	semidirect
364. C₂×C₄	C₂×D₄×C₁₀₉	semidirect
392. C₂²	#120×C₁₀₉	semidirect
393. 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
C₂³×D₄×C₁₀₉	2
C₂×#121×C₁₀₉	2
C₂×#120×C₁₀₉	2
C₂²×C₂²⋊C₄×C₁₀₉	2
#394×C₁₀₉	2
#392×C₁₀₉	2
#386×C₁₀₉	2
#385×C₁₀₉	2
#379×C₁₀₉	2
#2410	109

Subgroups — C109×#2410 #28359664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2410 #28359664