Subgroups — #11784122

Order 1152.

438 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63464	C₂	1	1	yes	maximal, characteristic
2	C₉×#405	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#63384	C₂	1	1	yes	maximal, characteristic
4	C₂×#63375	C₂	1	1	yes	maximal, characteristic, complemented
5	#10588683	C₂	1	1	yes	maximal, characteristic
6	#10588231	C₂	1	1	yes	maximal, characteristic, complemented
7	#10588208	C₂	1	1	yes	maximal, characteristic, complemented
8	#69888	—	3	1	no	maximal
9	C₂×#3697	C₂²	1	1	yes	characteristic
10	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₉×#117	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
14	C₂×#3667	C₂²	1	1	yes	characteristic
15	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
16	#63464	—	2	2	no	—
17	#63464	C₄	1	4	yes	complemented
18	C₂×#3663	—	2	1	no	—
19	C₂×C₈×D₉	—	2	1	no	—
20	C₉×#97	—	2	1	no	—
21	C₄×C₈×C₉	—	2	1	no	—
22	#63384	C₄	1	4	yes	complemented
23	#63375	—	2	2	no	—
24	#63369	C₄	1	2	yes	complemented
25	#63368	C₄	1	2	yes	complemented
26	C₂×#901	—	3	1	no	—
27	C₃×#405	D₃	1	1	yes	characteristic
28	C₂×#825	—	3	1	no	—
29	C₂×#815	—	3	1	no	—
30	#5002	—	3	1	no	—
31	#4558	—	3	1	no	—
32	#4532	—	3	1	no	—
33	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₂×#611	—	2	1	no	—
35	C₂²×Dic₉	—	2	1	no	—
36	#3697	—	2	1	no	—
37	#3697	—	2	1	no	—
38	#3697	C₂×C₄	1	2	yes	—
39	#3697	C₂×C₄	1	2	yes	—
40	C₂×D₃₆	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₉	—	2	1	no	—
42	C₂×Dic₁₈	C₂×C₄	1	1	yes	characteristic
43	C₉×M₁₆	—	2	1	no	—
44	C₉×M₁₆	C₂×C₄	1	2	yes	—
45	C₂×C₈×C₉	—	2	1	no	—
46	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
47	C₂×C₈×C₉	D₄	1	2	yes	—
48	C₂×C₈×C₉	D₄	1	1	yes	characteristic
49	C₂×C₈×C₉	D₄	1	1	yes	characteristic
50	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
51	C₉×C₂²⋊C₄	—	2	1	no	—
52	C₄²×C₉	—	2	1	no	—
53	C₄×Dic₉	—	2	1	no	—
54	C₄×Dic₉	C₂×C₄	1	2	yes	—
55	#3667	—	2	1	no	—
56	#3667	C₂×C₄	1	2	yes	—
57	C₂×C₉⋊C₈	—	2	1	no	—
58	C₂×C₉⋊C₈	—	2	1	no	—
59	C₂×C₉⋊C₈	—	2	1	no	—
60	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
61	#3663	—	2	4	no	—
62	C₈×D₉	—	2	4	no	—
63	#1765	—	9	1	no	Sylow
64	C₂×#235	—	3	1	no	—
65	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
66	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
67	C₃×#117	C₂×D₃	1	1	yes	characteristic
68	C₂×C₄×Dic₃	—	3	1	no	—
69	C₂×#208	—	3	1	no	—
70	C₂²×C₃⋊C₈	—	3	1	no	—
71	#901	—	3	4	no	—
72	#901	—	6	2	no	—
73	C₂×ΓL(1,𝔽₂₅)	—	6	1	no	—
74	C₂×D₃×C₈	—	6	1	no	—
75	C₃×#97	—	2	1	no	—
76	C₃×C₄×C₈	—	2	1	no	—
77	#825	—	3	4	no	—
78	#815	—	6	2	no	—
79	#810	—	3	2	no	—
80	#809	—	3	2	no	—
81	C₂³×C₉	#41	1	1	yes	characteristic
82	C₂²×D₉	—	2	1	no	—
83	C₂×C₄×C₉	—	2	2	no	—
84	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₉	#41	1	1	yes	characteristic
90	#611	—	2	2	no	—
91	#611	—	2	2	no	—
92	C₂×Dic₉	—	2	1	no	—
93	C₂×Dic₉	—	2	2	no	—
94	C₂×Dic₉	—	4	2	no	—
95	D₃₆	—	2	1	no	—
96	D₃₆	C₄²	1	2	yes	—
97	C₄×D₉	—	2	2	no	—
98	C₄×D₉	—	2	2	no	—
99	Dic₁₈	—	2	1	no	—
100	Dic₁₈	C₄²	1	2	yes	—
101	C₈×C₉	—	2	2	no	—
102	C₈×C₉	C₂²⋊C₄	1	2	yes	—
103	C₈×C₉	C₂²⋊C₄	1	2	yes	—
104	C₉⋊C₈	—	2	2	no	—
105	C₉⋊C₈	—	2	2	no	—
106	C₂×#131	—	9	1	no	—
107	C₂×#104	—	9	1	no	—
108	#416	—	9	1	no	—
109	#405	D₉	1	1	yes	characteristic, complemented
110	C₂×#97	—	9	1	no	—
111	#341	—	9	1	no	—
112	#339	—	9	1	no	—
113	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
114	C₂×#67	—	6	1	no	—
115	C₂²×Dic₃	—	6	1	no	—
116	#235	—	3	2	no	—
117	#235	—	3	2	no	—
118	#235	—	6	1	no	—
119	#235	—	6	1	no	—
120	C₂×D₁₂	—	3	1	no	—
121	C₂×C₄×D₃	—	6	1	no	—
122	C₂×Dic₆	—	3	1	no	—
123	C₃×M₁₆	—	2	1	no	—
124	C₃×M₁₆	C₄×D₃	1	2	yes	—
125	C₂×C₃×C₈	—	2	1	no	—
126	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
127	C₂×C₃×C₈	D₁₂	1	2	yes	—
128	C₂×C₃×C₈	#67	1	1	yes	characteristic
129	C₂×C₃×C₈	#67	1	1	yes	characteristic
130	C₃×C₄⋊C₄	C₄×D₃	1	2	yes	—
131	C₃×C₂²⋊C₄	—	2	1	no	—
132	C₃×C₄²	—	2	1	no	—
133	C₄×Dic₃	—	3	2	no	—
134	C₄×Dic₃	—	6	1	no	—
135	#208	—	3	2	no	—
136	#208	—	6	1	no	—
137	C₂×C₃⋊C₈	—	3	2	no	—
138	C₂×C₃⋊C₈	—	6	1	no	—
139	C₂×C₃⋊C₈	—	6	1	no	—
140	C₂×C₃⋊C₈	—	6	1	no	—
141	ΓL(1,𝔽₂₅)	—	6	4	no	—
142	D₃×C₈	—	6	4	no	—
143	C₂²×C₉	—	2	2	no	—
144	C₂²×C₉	#117	1	1	yes	characteristic
145	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
146	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
147	C₂×D₉	—	2	2	no	—
148	C₂×D₉	—	4	2	no	—
149	C₄×C₉	—	4	2	no	—
150	C₄×C₉	C₂×C₄²	1	1	yes	characteristic
151	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₄×C₉	C₄×D₄	1	1	yes	characteristic
153	C₄×C₉	C₄×D₄	1	1	yes	characteristic
154	Dic₉	—	2	2	no	—
155	Dic₉	—	4	2	no	—
156	C₂×#41	—	9	1	no	—
157	#131	—	9	4	no	—
158	#131	—	18	2	no	—
159	C₂×M₁₆	—	9	1	no	—
160	C₂×M₁₆	—	18	1	no	—
161	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
162	C₂²×C₈	—	9	1	no	—
163	C₂²×C₈	—	18	1	no	—
164	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
165	#117	C₂×D₉	1	1	yes	characteristic
166	C₂×C₄²	—	9	1	no	—
167	#104	—	9	4	no	—
168	#103	—	9	2	no	—
169	#102	—	9	2	no	—
170	#97	—	2	1	no	—
171	#97	—	18	2	no	—
172	C₄×C₈	—	2	1	no	—
173	C₂³×C₃	#235	1	1	yes	characteristic
174	C₂²×D₃	—	6	1	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
180	C₂×C₃×C₄	#235	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
182	#67	—	6	2	no	—
183	#67	—	6	2	no	—
184	C₂×Dic₃	—	6	1	no	—
185	C₂×Dic₃	—	6	2	no	—
186	C₂×Dic₃	—	12	2	no	—
187	D₁₂	—	3	2	no	—
188	D₁₂	—	6	1	no	—
189	C₄×D₃	—	6	2	no	—
190	C₄×D₃	—	6	2	no	—
191	Dic₆	—	3	2	no	—
192	Dic₆	—	6	1	no	—
193	C₃×C₈	—	2	2	no	—
194	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
195	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
196	C₃⋊C₈	—	6	2	no	—
197	C₃⋊C₈	—	6	2	no	—
198	C₂×C₉	—	2	1	no	—
199	C₂×C₉	—	2	1	no	—
200	C₂×C₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₉	#444	1	2	yes	—
202	D₉	—	4	2	no	—
203	#41	—	9	2	no	—
204	#41	—	9	2	no	—
205	#41	—	18	1	no	—
206	#41	—	18	1	no	—
207	C₂×Q₈	—	9	1	no	—
208	C₂×D₄	—	9	1	no	—
209	C₂×D₄	—	18	1	no	—
210	C₂²×C₄	—	18	1	no	—
211	C₂²×C₄	—	18	1	no	—
212	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
213	M₁₆	—	2	1	no	—
214	M₁₆	—	9	2	no	—
215	M₁₆	—	18	1	no	—
216	M₁₆	—	18	4	no	—
217	M₁₆	C₄×D₉	1	2	yes	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	9	2	no	—
220	C₂×C₈	—	18	1	no	—
221	C₂×C₈	—	18	1	no	—
222	C₂×C₈	—	18	1	no	—
223	C₂×C₈	—	18	4	no	—
224	C₂×C₈	C₄×D₉	1	2	yes	—
225	C₂×C₈	D₃₆	1	2	yes	—
226	C₂×C₈	#611	1	1	yes	characteristic
227	C₂×C₈	#611	1	1	yes	characteristic
228	C₄⋊C₄	C₄×D₉	1	2	yes	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₄²	—	2	1	no	—
231	C₄²	—	9	2	no	—
232	C₄²	—	18	1	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	#872	1	1	yes	characteristic
235	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
236	C₂²×C₃	C₄×#67	1	1	yes	characteristic
237	C₂×D₃	—	6	2	no	—
238	C₂×D₃	—	12	2	no	—
239	C₃×C₄	—	4	2	no	—
240	C₃×C₄	C₄²×D₃	1	1	yes	characteristic
241	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
242	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
243	C₃×C₄	C₄×#67	1	1	yes	characteristic
244	Dic₃	—	6	2	no	—
245	Dic₃	—	12	2	no	—
246	C₉	#1765	1	1	yes	characteristic, complemented, Sylow
247	C₂³	—	18	1	no	—
248	C₂³	#3697	1	1	yes	characteristic
249	Q₈	—	9	2	no	—
250	Q₈	—	18	1	no	—
251	D₄	—	9	2	no	—
252	D₄	—	18	2	no	—
253	D₄	—	18	2	no	—
254	D₄	—	18	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	18	1	no	—
257	C₂×C₄	—	18	2	no	—
258	C₂×C₄	—	18	2	no	—
259	C₂×C₄	—	18	2	no	—
260	C₂×C₄	—	36	2	no	—
261	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
264	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
265	C₂×C₄	#3697	1	1	yes	characteristic
266	C₂×C₄	C₂×#611	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	18	2	no	—
269	C₈	—	18	2	no	—
270	C₈	(C₂×D₉)⋊C₄	1	2	yes	—
271	C₈	(C₂×D₉)⋊C₄	1	2	yes	—
272	C₂×C₃	—	2	1	no	—
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	C₄×(C₂×D₃)⋊C₄	1	1	yes	characteristic
275	C₂×C₃	#5145	1	2	yes	—
276	D₃	—	12	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	18	2	no	—
279	C₂²	—	36	2	no	—
280	C₂²	#63434	1	1	yes	characteristic
281	C₂²	C₄×D₃₆	1	1	yes	characteristic
282	C₂²	C₄×#611	1	1	yes	characteristic
283	C₄	—	4	2	no	—
284	C₄	—	18	2	no	—
285	C₄	—	36	2	no	—
286	C₄	C₄²×D₉	1	1	yes	characteristic
287	C₄	C₄×D₃₆	1	1	yes	characteristic
288	C₄	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
289	C₄	C₄×#611	1	1	yes	characteristic
290	C₃	#69888	1	1	yes	characteristic
291	C₂	—	2	1	no	—
292	C₂	—	2	1	no	—
293	C₂	—	36	2	no	—
294	C₂	C₄×(C₂×D₉)⋊C₄	1	1	yes	characteristic
295	C₂	#10588825	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#611
commutator	C₄×C₉	C₂×C₄²
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#405	C₂
socle	C₂²×C₃	C₄×#67
prime radical	C₂×#3697	C₂²
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#405	D₉
hypocenter	C₉	#1765
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. C₉×#405	C₂	semidirect
4. C₂×#63375	C₂	semidirect
6. #10588231	C₂	semidirect
7. #10588208	C₂	semidirect
17. #63464	C₄	semidirect
22. #63384	C₄	semidirect
24. #63369	C₄	semidirect
25. #63368	C₄	semidirect
109. #405	D₉	semidirect
246. C₉	#1765	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63464	2
C₉×#405	2
C₂×#63384	2
C₂×#63375	2
#10588683	2
#10588231	2
#10588208	2
#69888	3