Subgroups — C₂×C₉×#397 #11767111

Order 1152.

1230 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₉×C₂²⋊C₄	C₂	1	2	yes	maximal
3	C₂²×C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₉×#397	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₉×#95	C₂	1	2	yes	maximal, complemented
6	C₂×C₉×#95	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#397	C₃	1	1	yes	maximal, characteristic
8	C₂⁵×C₉	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₉	C₂²	1	2	yes	—
10	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₉	C₂²	1	2	yes	—
12	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₉×C₄⋊C₄	—	2	4	no	—
14	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	complemented
15	C₂×C₉×C₂²⋊C₄	—	2	8	no	—
16	C₂×C₉×C₂²⋊C₄	—	2	4	no	—
17	C₂×C₉×C₂²⋊C₄	C₂²	1	8	yes	—
18	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
19	C₉×#95	C₂²	1	8	yes	complemented
20	C₉×#95	C₂²	1	4	yes	complemented
21	C₂²×C₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
22	C₂²×C₃×C₂²⋊C₄	C₂×C₃	1	2	yes	—
23	C₂²×C₃×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
24	C₃×#397	C₂×C₃	1	8	yes	—
25	C₂×C₃×#95	C₂×C₃	1	2	yes	—
26	C₂×C₃×#95	C₂×C₃	1	1	yes	characteristic
27	C₂⁴×C₉	—	2	2	no	—
28	C₂⁴×C₉	—	2	2	no	—
29	C₂⁴×C₉	—	2	4	no	—
30	C₂⁴×C₉	—	2	4	no	—
31	C₂⁴×C₉	D₄	1	4	yes	—
32	C₂⁴×C₉	C₂³	1	2	yes	—
33	C₂⁴×C₉	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₉	—	2	4	no	—
35	C₂²×C₄×C₉	—	2	4	no	—
36	C₂²×C₄×C₉	—	2	4	no	—
37	C₂²×C₄×C₉	—	2	2	no	—
38	C₂²×C₄×C₉	—	2	4	no	—
39	C₂²×C₄×C₉	—	2	4	no	—
40	C₂²×C₄×C₉	—	2	4	no	—
41	C₂²×C₄×C₉	—	2	4	no	—
42	C₂²×C₄×C₉	—	2	4	no	—
43	C₂²×C₄×C₉	D₄	1	4	yes	—
44	C₂²×C₄×C₉	C₂³	1	4	yes	—
45	C₂²×C₄×C₉	C₂³	1	4	yes	—
46	C₂²×C₄×C₉	C₂³	1	2	yes	—
47	C₂²×C₄×C₉	C₂³	1	2	yes	—
48	C₉×C₄⋊C₄	—	2	8	no	—
49	C₉×C₂²⋊C₄	—	2	16	no	—
50	C₉×C₂²⋊C₄	—	2	8	no	—
51	C₂×#397	C₉	1	1	yes	characteristic, direct product factor, Sylow
52	C₂⁵×C₃	C₂²×C₃	1	1	yes	characteristic
53	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
54	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
55	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
56	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃×C₄⋊C₄	—	2	4	no	—
58	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
59	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
60	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
61	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
62	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
63	C₃×#95	C₂²×C₃	1	8	yes	—
64	C₃×#95	C₂²×C₃	1	4	yes	—
65	C₂³×C₉	—	2	8	no	—
66	C₂³×C₉	—	2	16	no	—
67	C₂³×C₉	—	2	2	no	—
68	C₂³×C₉	—	2	2	no	—
69	C₂³×C₉	—	2	4	no	—
70	C₂³×C₉	—	2	4	no	—
71	C₂³×C₉	—	4	4	no	—
72	C₂³×C₉	—	4	4	no	—
73	C₂³×C₉	—	4	8	no	—
74	C₂³×C₉	C₂×D₄	1	4	yes	—
75	C₂³×C₉	C₂×D₄	1	2	yes	—
76	C₂³×C₉	C₂×D₄	1	2	yes	—
77	C₂³×C₉	#41	1	2	yes	—
78	C₂³×C₉	#41	1	4	yes	—
79	C₂³×C₉	#41	1	4	yes	—
80	C₂³×C₉	C₂⁴	1	1	yes	characteristic
81	C₂×C₄×C₉	—	2	8	no	—
82	C₂×C₄×C₉	—	2	4	no	—
83	C₂×C₄×C₉	—	2	4	no	—
84	C₂×C₄×C₉	—	2	4	no	—
85	C₂×C₄×C₉	—	2	2	no	—
86	C₂×C₄×C₉	—	2	4	no	—
87	C₂×C₄×C₉	—	2	4	no	—
88	C₂×C₄×C₉	—	2	4	no	—
89	C₂×C₄×C₉	—	2	4	no	—
90	C₂×C₄×C₉	—	2	4	no	—
91	C₂×C₄×C₉	—	4	8	no	—
92	C₂×C₄×C₉	—	4	8	no	—
93	C₂×C₄×C₉	—	4	4	no	—
94	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
95	C₂²×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
96	C₂²×C₂²⋊C₄	C₂×C₉	1	2	yes	—
97	C₂²×C₂²⋊C₄	C₂×C₉	1	1	yes	characteristic
98	#397	C₂×C₉	1	8	yes	direct product factor
99	C₂×#95	C₂×C₉	1	2	yes	complemented
100	C₂×#95	C₂×C₉	1	1	yes	characteristic, complemented
101	C₂⁴×C₃	—	2	2	no	—
102	C₂⁴×C₃	—	2	2	no	—
103	C₂⁴×C₃	—	2	4	no	—
104	C₂⁴×C₃	—	2	4	no	—
105	C₂⁴×C₃	C₃×D₄	1	4	yes	—
106	C₂⁴×C₃	C₂³×C₃	1	2	yes	—
107	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
108	C₂²×C₃×C₄	—	2	4	no	—
109	C₂²×C₃×C₄	—	2	4	no	—
110	C₂²×C₃×C₄	—	2	4	no	—
111	C₂²×C₃×C₄	—	2	2	no	—
112	C₂²×C₃×C₄	—	2	4	no	—
113	C₂²×C₃×C₄	—	2	4	no	—
114	C₂²×C₃×C₄	—	2	4	no	—
115	C₂²×C₃×C₄	—	2	4	no	—
116	C₂²×C₃×C₄	—	2	4	no	—
117	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
118	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
119	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
120	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
121	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
122	C₃×C₄⋊C₄	—	2	8	no	—
123	C₃×C₂²⋊C₄	—	2	16	no	—
124	C₃×C₂²⋊C₄	—	2	8	no	—
125	C₂²×C₉	—	2	4	no	—
126	C₂²×C₉	—	2	8	no	—
127	C₂²×C₉	—	4	2	no	—
128	C₂²×C₉	—	4	2	no	—
129	C₂²×C₉	—	4	4	no	—
130	C₂²×C₉	—	4	16	no	—
131	C₂²×C₉	#121	1	4	yes	—
132	C₂²×C₉	#123	1	4	yes	—
133	C₂²×C₉	#123	1	8	yes	—
134	C₂²×C₉	#124	1	4	yes	—
135	C₂²×C₉	#126	1	8	yes	—
136	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
137	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
138	C₂²×C₉	C₂×#41	1	1	yes	characteristic
139	C₂²×C₉	C₂×#41	1	2	yes	—
140	C₂²×C₉	C₂×#41	1	2	yes	—
141	C₄×C₉	—	2	4	no	—
142	C₄×C₉	—	4	4	no	—
143	C₄×C₉	—	4	4	no	—
144	C₄×C₉	—	4	2	no	—
145	C₂⁵	C₂²×C₉	1	1	yes	characteristic
146	C₂³×C₄	C₂²×C₉	1	2	yes	—
147	C₂³×C₄	C₂²×C₉	1	2	yes	—
148	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
149	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
150	C₂×C₄⋊C₄	—	2	4	no	—
151	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	complemented
152	C₂×C₂²⋊C₄	—	2	8	no	—
153	C₂×C₂²⋊C₄	—	2	4	no	—
154	C₂×C₂²⋊C₄	C₂²×C₉	1	8	yes	—
155	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
156	#95	C₂²×C₉	1	8	yes	complemented
157	#95	C₂²×C₉	1	4	yes	complemented
158	C₂³×C₃	—	2	8	no	—
159	C₂³×C₃	—	2	16	no	—
160	C₂³×C₃	—	2	2	no	—
161	C₂³×C₃	—	2	2	no	—
162	C₂³×C₃	—	2	4	no	—
163	C₂³×C₃	—	2	4	no	—
164	C₂³×C₃	—	4	4	no	—
165	C₂³×C₃	—	4	4	no	—
166	C₂³×C₃	—	4	8	no	—
167	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
168	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
169	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
170	C₂³×C₃	C₃×#41	1	2	yes	—
171	C₂³×C₃	C₃×#41	1	4	yes	—
172	C₂³×C₃	C₃×#41	1	4	yes	—
173	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
174	C₂×C₃×C₄	—	2	8	no	—
175	C₂×C₃×C₄	—	2	4	no	—
176	C₂×C₃×C₄	—	2	4	no	—
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	4	no	—
180	C₂×C₃×C₄	—	2	4	no	—
181	C₂×C₃×C₄	—	2	4	no	—
182	C₂×C₃×C₄	—	2	4	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	—	4	8	no	—
185	C₂×C₃×C₄	—	4	8	no	—
186	C₂×C₃×C₄	—	4	4	no	—
187	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
188	C₂×C₉	—	4	4	no	—
189	C₂×C₉	#397	1	8	yes	direct product factor
190	C₂×C₉	C₂×#121	1	1	yes	characteristic
191	C₂×C₉	C₂×#123	1	1	yes	characteristic
192	C₂×C₉	C₂×#123	1	2	yes	—
193	C₂×C₉	C₂×#124	1	1	yes	characteristic
194	C₂×C₉	C₂×#126	1	2	yes	—
195	C₂⁴	—	2	2	no	—
196	C₂⁴	—	2	2	no	—
197	C₂⁴	—	2	4	no	—
198	C₂⁴	—	2	4	no	—
199	C₂⁴	D₄×C₉	1	4	yes	—
200	C₂⁴	C₂³×C₉	1	2	yes	—
201	C₂⁴	C₂³×C₉	1	1	yes	characteristic
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	D₄×C₉	1	4	yes	—
212	C₂²×C₄	C₂³×C₉	1	4	yes	—
213	C₂²×C₄	C₂³×C₉	1	2	yes	—
214	C₂²×C₄	C₂³×C₉	1	4	yes	—
215	C₂²×C₄	C₂³×C₉	1	2	yes	—
216	C₄⋊C₄	—	2	8	no	—
217	C₂²⋊C₄	—	2	16	no	—
218	C₂²⋊C₄	—	2	8	no	—
219	C₂²×C₃	—	2	4	no	—
220	C₂²×C₃	—	2	8	no	—
221	C₂²×C₃	—	4	16	no	—
222	C₂²×C₃	—	4	2	no	—
223	C₂²×C₃	—	4	2	no	—
224	C₂²×C₃	—	4	4	no	—
225	C₂²×C₃	C₃×#121	1	4	yes	—
226	C₂²×C₃	C₃×#123	1	4	yes	—
227	C₂²×C₃	C₃×#123	1	8	yes	—
228	C₂²×C₃	C₃×#124	1	4	yes	—
229	C₂²×C₃	C₃×#126	1	8	yes	—
230	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
231	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
232	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
233	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
234	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
235	C₃×C₄	—	2	4	no	—
236	C₃×C₄	—	4	4	no	—
237	C₃×C₄	—	4	4	no	—
238	C₃×C₄	—	4	2	no	—
239	C₉	C₂×#397	1	1	yes	characteristic, direct product factor, Sylow
240	C₂³	—	2	8	no	—
241	C₂³	—	2	16	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	4	no	—
245	C₂³	—	2	4	no	—
246	C₂³	—	4	4	no	—
247	C₂³	—	4	4	no	—
248	C₂³	—	4	8	no	—
249	C₂³	C₂×D₄×C₉	1	4	yes	—
250	C₂³	C₂×D₄×C₉	1	2	yes	—
251	C₂³	C₂×D₄×C₉	1	2	yes	—
252	C₂³	C₉×#41	1	2	yes	—
253	C₂³	C₉×#41	1	4	yes	—
254	C₂³	C₉×#41	1	4	yes	—
255	C₂³	C₂⁴×C₉	1	1	yes	characteristic
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	8	no	—
267	C₂×C₄	—	4	8	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
270	C₂×C₃	—	4	4	no	—
271	C₂×C₃	C₃×#397	1	8	yes	—
272	C₂×C₃	C₂×C₃×#121	1	1	yes	characteristic
273	C₂×C₃	C₂×C₃×#123	1	1	yes	characteristic
274	C₂×C₃	C₂×C₃×#123	1	2	yes	—
275	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
276	C₂×C₃	C₂×C₃×#126	1	2	yes	—
277	C₂²	—	2	4	no	—
278	C₂²	—	2	8	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	4	16	no	—
283	C₂²	C₉×#121	1	4	yes	—
284	C₂²	C₉×#123	1	4	yes	—
285	C₂²	C₉×#123	1	8	yes	—
286	C₂²	C₉×#124	1	4	yes	—
287	C₂²	C₉×#126	1	8	yes	—
288	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
289	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
290	C₂²	C₂×C₉×#41	1	1	yes	characteristic
291	C₂²	C₂×C₉×#41	1	2	yes	—
292	C₂²	C₂×C₉×#41	1	2	yes	—
293	C₄	—	2	4	no	—
294	C₄	—	4	4	no	—
295	C₄	—	4	4	no	—
296	C₄	—	4	2	no	—
297	C₃	C₂×C₃×#397	1	1	yes	characteristic
298	C₂	—	4	4	no	—
299	C₂	C₉×#397	1	8	yes	direct product factor
300	C₂	C₂×C₉×#121	1	1	yes	characteristic
301	C₂	C₂×C₉×#123	1	1	yes	characteristic
302	C₂	C₂×C₉×#123	1	2	yes	—
303	C₂	C₂×C₉×#124	1	1	yes	characteristic
304	C₂	C₂×C₉×#126	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₉	C₂³
commutator	C₂³	C₂⁴×C₉
Frattini, cosocle	C₂³×C₃	C₂⁴×C₃
socle	C₂⁴×C₃	C₂³×C₃
prime radical	C₂⁵×C₃	C₂²×C₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₉×C₄⋊C₄	C₂	semidirect
4. C₉×#397	C₂	direct (N × H)
5. C₂×C₉×#95	C₂	semidirect
6. C₂×C₉×#95	C₂	semidirect
14. C₂×C₉×C₄⋊C₄	C₂²	semidirect
19. C₉×#95	C₂²	semidirect
20. C₉×#95	C₂²	semidirect
51. C₂×#397	C₉	direct (N × H)
95. C₂²×C₄⋊C₄	C₂×C₉	semidirect
98. #397	C₂×C₉	direct (N × H)
99. C₂×#95	C₂×C₉	semidirect
100. C₂×#95	C₂×C₉	semidirect
151. C₂×C₄⋊C₄	C₂²×C₉	semidirect
156. #95	C₂²×C₉	semidirect
157. #95	C₂²×C₉	semidirect
189. C₂×C₉	#397	direct (N × H)
239. C₉	C₂×#397	direct (N × H)
299. C₂	C₉×#397	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×C₄⋊C₄	2
C₂²×C₉×C₂²⋊C₄	2
C₉×#397	2
C₂×C₉×#95	2
C₂×C₃×#397	3

Subgroups — C2×C9×#397 #11767111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#397 #11767111