Subgroups — C₁₀₉×#2359 #28359613

Order 13952.

652 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#123×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#389×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#388×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#388×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#387×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#387×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#386×C₁₀₉	C₂	1	1	yes	maximal, characteristic
10	#385×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#385×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#95×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	#123×C₁₀₉	C₄	1	8	yes	complemented
22	C₄×D₄×C₁₀₉	—	2	4	no	—
23	#117×C₁₀₉	—	2	4	no	—
24	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
40	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
42	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
43	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
44	#95×C₁₀₉	C₂²	1	1	yes	characteristic
45	#95×C₁₀₉	C₂²	1	1	yes	characteristic
46	#95×C₁₀₉	C₂²	1	1	yes	characteristic
47	#95×C₁₀₉	C₂²	1	1	yes	characteristic
48	#95×C₁₀₉	C₂²	1	1	yes	characteristic
49	#95×C₁₀₉	C₂²	1	1	yes	characteristic
50	#95×C₁₀₉	C₂²	1	1	yes	characteristic
51	#95×C₁₀₉	C₂²	1	1	yes	characteristic
52	#95×C₁₀₉	C₂²	1	1	yes	characteristic
53	#95×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
55	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
56	C₂×D₄×C₁₀₉	—	2	2	no	—
57	C₂×D₄×C₁₀₉	—	2	2	no	—
58	C₂×D₄×C₁₀₉	C₂×C₄	1	4	yes	—
59	C₂²×C₄×C₁₀₉	—	2	2	no	—
60	C₂²×C₄×C₁₀₉	—	2	2	no	—
61	C₂²×C₄×C₁₀₉	—	2	2	no	—
62	C₂²×C₄×C₁₀₉	—	2	2	no	—
63	C₂²×C₄×C₁₀₉	—	2	2	no	—
64	C₂²×C₄×C₁₀₉	—	2	2	no	—
65	C₂²×C₄×C₁₀₉	—	2	2	no	—
66	C₂²×C₄×C₁₀₉	—	2	2	no	—
67	C₂²×C₄×C₁₀₉	—	4	1	no	—
68	C₂²×C₄×C₁₀₉	—	4	1	no	—
69	C₂²×C₄×C₁₀₉	—	4	1	no	—
70	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	complemented
71	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₁₀₉	—	2	2	no	—
85	C₄⋊C₄×C₁₀₉	—	2	2	no	—
86	C₄⋊C₄×C₁₀₉	—	2	2	no	—
87	C₄⋊C₄×C₁₀₉	—	2	2	no	—
88	C₄⋊C₄×C₁₀₉	—	4	1	no	—
89	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	complemented
90	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	complemented
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	—	4	1	no	—
95	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	complemented
96	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	complemented
97	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	complemented
98	C₂²⋊C₄×C₁₀₉	D₄	1	4	yes	—
99	C₄²×C₁₀₉	—	2	2	no	—
100	C₄²×C₁₀₉	—	2	2	no	—
101	C₄²×C₁₀₉	—	2	2	no	—
102	C₄²×C₁₀₉	—	2	2	no	—
103	C₄²×C₁₀₉	—	4	1	no	—
104	C₂³×C₁₀₉	—	2	1	no	—
105	C₂³×C₁₀₉	—	2	1	no	—
106	C₂³×C₁₀₉	—	2	1	no	—
107	C₂³×C₁₀₉	—	2	1	no	—
108	C₂³×C₁₀₉	—	2	1	no	—
109	C₂³×C₁₀₉	—	2	1	no	—
110	C₂³×C₁₀₉	—	2	1	no	—
111	C₂³×C₁₀₉	—	2	1	no	—
112	C₂³×C₁₀₉	—	2	1	no	—
113	C₂³×C₁₀₉	—	2	1	no	—
114	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
115	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
116	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
117	C₂³×C₁₀₉	#41	1	2	yes	—
118	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
119	D₄×C₁₀₉	—	2	8	no	—
120	C₂×C₄×C₁₀₉	—	2	8	no	—
121	C₂×C₄×C₁₀₉	—	2	1	no	—
122	C₂×C₄×C₁₀₉	—	2	1	no	—
123	C₂×C₄×C₁₀₉	—	2	1	no	—
124	C₂×C₄×C₁₀₉	—	2	1	no	—
125	C₂×C₄×C₁₀₉	—	2	1	no	—
126	C₂×C₄×C₁₀₉	—	2	1	no	—
127	C₂×C₄×C₁₀₉	—	2	1	no	—
128	C₂×C₄×C₁₀₉	—	2	1	no	—
129	C₂×C₄×C₁₀₉	—	2	1	no	—
130	C₂×C₄×C₁₀₉	—	2	1	no	—
131	C₂×C₄×C₁₀₉	—	2	1	no	—
132	C₂×C₄×C₁₀₉	—	2	1	no	—
133	C₂×C₄×C₁₀₉	—	2	1	no	—
134	C₂×C₄×C₁₀₉	—	2	1	no	—
135	C₂×C₄×C₁₀₉	—	2	1	no	—
136	C₂×C₄×C₁₀₉	—	2	1	no	—
137	C₂×C₄×C₁₀₉	—	2	1	no	—
138	C₂×C₄×C₁₀₉	—	2	1	no	—
139	C₂×C₄×C₁₀₉	—	2	1	no	—
140	C₂×C₄×C₁₀₉	—	2	1	no	—
141	C₂×C₄×C₁₀₉	—	2	1	no	—
142	C₂×C₄×C₁₀₉	—	2	1	no	—
143	C₂×C₄×C₁₀₉	—	2	1	no	—
144	C₂×C₄×C₁₀₉	—	2	1	no	—
145	C₂×C₄×C₁₀₉	—	2	1	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	1	no	—
151	C₂×C₄×C₁₀₉	—	4	4	no	—
152	C₂×C₄×C₁₀₉	—	4	4	no	—
153	C₂×C₄×C₁₀₉	—	4	4	no	—
154	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
155	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
156	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
157	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
158	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
159	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
160	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
161	C₂×C₄×C₁₀₉	#41	1	2	yes	—
162	C₂×C₄×C₁₀₉	#41	1	2	yes	—
163	C₂²×C₁₀₉	—	2	2	no	—
164	C₂²×C₁₀₉	—	2	2	no	—
165	C₂²×C₁₀₉	—	2	2	no	—
166	C₂²×C₁₀₉	—	2	2	no	—
167	C₂²×C₁₀₉	—	2	2	no	—
168	C₂²×C₁₀₉	—	2	2	no	—
169	C₂²×C₁₀₉	—	2	2	no	—
170	C₂²×C₁₀₉	—	2	2	no	—
171	C₂²×C₁₀₉	—	2	2	no	—
172	C₂²×C₁₀₉	—	4	1	no	—
173	C₂²×C₁₀₉	—	4	1	no	—
174	C₂²×C₁₀₉	—	4	1	no	—
175	C₂²×C₁₀₉	—	4	1	no	—
176	C₂²×C₁₀₉	C₄×D₄	1	4	yes	—
177	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
178	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
179	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
180	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
181	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
182	C₂²×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂²×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
184	C₄×C₁₀₉	—	2	2	no	—
185	C₄×C₁₀₉	—	2	2	no	—
186	C₄×C₁₀₉	—	2	2	no	—
187	C₄×C₁₀₉	—	2	2	no	—
188	C₄×C₁₀₉	—	2	2	no	—
189	C₄×C₁₀₉	—	2	2	no	—
190	C₄×C₁₀₉	—	2	2	no	—
191	C₄×C₁₀₉	—	2	2	no	—
192	C₄×C₁₀₉	—	2	2	no	—
193	C₄×C₁₀₉	—	4	1	no	—
194	C₄×C₁₀₉	—	4	1	no	—
195	C₄×C₁₀₉	—	4	1	no	—
196	C₄×C₁₀₉	—	4	1	no	—
197	C₂×C₁₀₉	—	2	4	no	—
198	C₂×C₁₀₉	—	4	2	no	—
199	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
200	C₂×C₁₀₉	C₄×#41	1	1	yes	characteristic
201	C₂×C₁₀₉	#520	1	1	yes	characteristic
202	C₂×C₁₀₉	#546	1	1	yes	characteristic
203	C₂×C₁₀₉	#547	1	1	yes	characteristic
204	C₂×C₁₀₉	#551	1	1	yes	characteristic
205	C₂×C₁₀₉	#552	1	1	yes	characteristic
206	#2359	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₁₀₉	#2359	1	1	yes	characteristic, direct product factor, Sylow
208	C₂×#123	C₂×C₁₀₉	1	1	yes	characteristic
209	C₂×C₄×D₄	C₂×C₁₀₉	1	1	yes	characteristic
210	C₂×#117	C₂×C₁₀₉	1	1	yes	characteristic, complemented
211	#389	C₂×C₁₀₉	1	1	yes	characteristic, complemented
212	#388	C₂×C₁₀₉	1	1	yes	characteristic, complemented
213	#388	C₂×C₁₀₉	1	1	yes	characteristic, complemented
214	#387	C₂×C₁₀₉	1	1	yes	characteristic, complemented
215	#387	C₂×C₁₀₉	1	1	yes	characteristic, complemented
216	#386	C₂×C₁₀₉	1	1	yes	characteristic
217	#385	C₂×C₁₀₉	1	1	yes	characteristic, complemented
218	#385	C₂×C₁₀₉	1	1	yes	characteristic, complemented
219	C₄×C₄⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
220	C₄×C₂²⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
221	C₄×C₂²⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
222	C₂×#95	C₂×C₁₀₉	1	1	yes	characteristic, complemented
223	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
224	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
225	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
226	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
227	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
228	#123	C₄×C₁₀₉	1	8	yes	complemented
229	C₄×D₄	—	2	4	no	—
230	#117	—	2	4	no	—
231	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
232	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
233	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
234	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
235	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
236	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
237	C₂×C₄⋊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	1	yes	characteristic
241	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
242	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
243	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
244	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
245	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
246	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
247	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
248	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
249	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
250	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
251	#95	C₂²×C₁₀₉	1	1	yes	characteristic
252	#95	C₂²×C₁₀₉	1	1	yes	characteristic
253	#95	C₂²×C₁₀₉	1	1	yes	characteristic
254	#95	C₂²×C₁₀₉	1	1	yes	characteristic
255	#95	C₂²×C₁₀₉	1	1	yes	characteristic
256	#95	C₂²×C₁₀₉	1	1	yes	characteristic
257	#95	C₂²×C₁₀₉	1	1	yes	characteristic
258	#95	C₂²×C₁₀₉	1	1	yes	characteristic
259	#95	C₂²×C₁₀₉	1	1	yes	characteristic
260	#95	C₂²×C₁₀₉	1	1	yes	characteristic
261	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
262	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	C₂×C₄×C₁₀₉	1	4	yes	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	—	4	1	no	—
276	C₂²×C₄	—	4	1	no	—
277	C₂²×C₄	C₂×C₄×C₁₀₉	1	4	yes	complemented
278	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
279	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
280	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
281	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
282	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
283	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
284	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
285	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
286	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
287	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
288	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
289	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
290	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	complemented
297	C₄⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	complemented
298	C₂²⋊C₄	—	2	2	no	—
299	C₂²⋊C₄	—	2	2	no	—
300	C₂²⋊C₄	—	2	2	no	—
301	C₂²⋊C₄	—	4	1	no	—
302	C₂²⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	complemented
303	C₂²⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	complemented
304	C₂²⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	complemented
305	C₂²⋊C₄	D₄×C₁₀₉	1	4	yes	—
306	C₄²	—	2	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	2	2	no	—
310	C₄²	—	4	1	no	—
311	C₂³	—	2	1	no	—
312	C₂³	—	2	1	no	—
313	C₂³	—	2	1	no	—
314	C₂³	—	2	1	no	—
315	C₂³	—	2	1	no	—
316	C₂³	—	2	1	no	—
317	C₂³	—	2	1	no	—
318	C₂³	—	2	1	no	—
319	C₂³	—	2	1	no	—
320	C₂³	—	2	1	no	—
321	C₂³	C₂²×C₄×C₁₀₉	1	2	yes	—
322	C₂³	C₂²×C₄×C₁₀₉	1	2	yes	—
323	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
324	C₂³	#41×C₁₀₉	1	2	yes	—
325	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
326	D₄	—	2	8	no	—
327	C₂×C₄	—	2	8	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	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	—	4	4	no	—
359	C₂×C₄	—	4	4	no	—
360	C₂×C₄	—	4	4	no	—
361	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
362	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
363	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
364	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
365	C₂×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
366	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
367	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
368	C₂×C₄	#41×C₁₀₉	1	2	yes	—
369	C₂×C₄	#41×C₁₀₉	1	2	yes	—
370	C₂²	—	2	2	no	—
371	C₂²	—	2	2	no	—
372	C₂²	—	2	2	no	—
373	C₂²	—	2	2	no	—
374	C₂²	—	2	2	no	—
375	C₂²	—	2	2	no	—
376	C₂²	—	2	2	no	—
377	C₂²	—	2	2	no	—
378	C₂²	—	2	2	no	—
379	C₂²	—	4	1	no	—
380	C₂²	—	4	1	no	—
381	C₂²	—	4	1	no	—
382	C₂²	—	4	1	no	—
383	C₂²	C₄×D₄×C₁₀₉	1	4	yes	—
384	C₂²	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
385	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
386	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
387	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
388	C₂²	C₂×#41×C₁₀₉	1	1	yes	characteristic
389	C₂²	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
390	C₂²	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
391	C₄	—	2	2	no	—
392	C₄	—	2	2	no	—
393	C₄	—	2	2	no	—
394	C₄	—	2	2	no	—
395	C₄	—	2	2	no	—
396	C₄	—	2	2	no	—
397	C₄	—	2	2	no	—
398	C₄	—	2	2	no	—
399	C₄	—	2	2	no	—
400	C₄	—	4	1	no	—
401	C₄	—	4	1	no	—
402	C₄	—	4	1	no	—
403	C₄	—	4	1	no	—
404	C₂	—	2	4	no	—
405	C₂	—	4	2	no	—
406	C₂	C₂×C₄×D₄×C₁₀₉	1	1	yes	characteristic
407	C₂	C₄×#41×C₁₀₉	1	1	yes	characteristic
408	C₂	#520×C₁₀₉	1	1	yes	characteristic
409	C₂	#546×C₁₀₉	1	1	yes	characteristic
410	C₂	#547×C₁₀₉	1	1	yes	characteristic
411	C₂	#551×C₁₀₉	1	1	yes	characteristic
412	C₂	#552×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₂³	C₂⁴×C₁₀₉
prime radical	C₂²×D₄×C₁₀₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#117×C₁₀₉	C₂	semidirect
4. #389×C₁₀₉	C₂	semidirect
5. #388×C₁₀₉	C₂	semidirect
6. #388×C₁₀₉	C₂	semidirect
7. #387×C₁₀₉	C₂	semidirect
8. #387×C₁₀₉	C₂	semidirect
10. #385×C₁₀₉	C₂	semidirect
11. #385×C₁₀₉	C₂	semidirect
12. C₄×C₄⋊C₄×C₁₀₉	C₂	semidirect
13. C₄×C₂²⋊C₄×C₁₀₉	C₂	semidirect
14. C₄×C₂²⋊C₄×C₁₀₉	C₂	semidirect
15. C₂×#95×C₁₀₉	C₂	semidirect
21. #123×C₁₀₉	C₄	semidirect
70. C₂²×C₄×C₁₀₉	C₂×C₄	semidirect
89. C₄⋊C₄×C₁₀₉	C₂×C₄	semidirect
90. C₄⋊C₄×C₁₀₉	C₂×C₄	semidirect
95. C₂²⋊C₄×C₁₀₉	C₂×C₄	semidirect
96. C₂²⋊C₄×C₁₀₉	C₂×C₄	semidirect
97. C₂²⋊C₄×C₁₀₉	C₂×C₄	semidirect
206. #2359	C₁₀₉	direct (N × H)
207. C₁₀₉	#2359	direct (N × H)
210. C₂×#117	C₂×C₁₀₉	semidirect
211. #389	C₂×C₁₀₉	semidirect
212. #388	C₂×C₁₀₉	semidirect
213. #388	C₂×C₁₀₉	semidirect
214. #387	C₂×C₁₀₉	semidirect
215. #387	C₂×C₁₀₉	semidirect
217. #385	C₂×C₁₀₉	semidirect
218. #385	C₂×C₁₀₉	semidirect
219. C₄×C₄⋊C₄	C₂×C₁₀₉	semidirect
220. C₄×C₂²⋊C₄	C₂×C₁₀₉	semidirect
221. C₄×C₂²⋊C₄	C₂×C₁₀₉	semidirect
222. C₂×#95	C₂×C₁₀₉	semidirect
228. #123	C₄×C₁₀₉	semidirect
277. C₂²×C₄	C₂×C₄×C₁₀₉	semidirect
296. C₄⋊C₄	C₂×C₄×C₁₀₉	semidirect
297. C₄⋊C₄	C₂×C₄×C₁₀₉	semidirect
302. C₂²⋊C₄	C₂×C₄×C₁₀₉	semidirect
303. C₂²⋊C₄	C₂×C₄×C₁₀₉	semidirect
304. C₂²⋊C₄	C₂×C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#123×C₁₀₉	2
C₂×C₄×D₄×C₁₀₉	2
C₂×#117×C₁₀₉	2
#389×C₁₀₉	2
#388×C₁₀₉	2
#387×C₁₀₉	2
#386×C₁₀₉	2
#385×C₁₀₉	2
C₄×C₄⋊C₄×C₁₀₉	2
C₄×C₂²⋊C₄×C₁₀₉	2
C₂×#95×C₁₀₉	2
#2359	109

Subgroups — C109×#2359 #28359613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2359 #28359613