Subgroups — #26029971

Order 11392.

452 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152398	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152396	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15668032	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15668020	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151872	C₂	1	2	yes	maximal, complemented
6	#19151867	C₂	1	2	yes	maximal, complemented
7	#19151864	C₂	1	2	yes	maximal, complemented
8	#19151860	C₂	1	2	yes	maximal, complemented
9	C₂×#15667947	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151580	C₂	1	1	yes	maximal, characteristic, complemented
11	#413×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13083574	C₂²	1	1	yes	characteristic
13	C₂×#13083574	C₂²	1	1	yes	characteristic
14	#15668045	—	2	1	no	—
15	#15668045	C₂²	1	2	yes	—
16	#15668043	C₂²	1	2	yes	—
17	#15668042	—	2	1	no	—
18	#15668040	—	2	1	no	—
19	#15668040	C₂²	1	2	yes	—
20	#15668038	C₂²	1	2	yes	—
21	#15668037	—	2	1	no	—
22	#15668034	—	2	2	no	—
23	#15668032	—	2	2	no	—
24	#15668032	—	2	1	no	—
25	#15668032	—	2	1	no	—
26	#15668032	—	2	1	no	—
27	#15668032	—	2	1	no	—
28	C₂×#13083564	C₂²	1	1	yes	characteristic, complemented
29	C₂×#13083562	C₂²	1	1	yes	characteristic, complemented
30	C₂×#13083561	C₂²	1	1	yes	characteristic, complemented
31	C₂×#13083559	C₂²	1	1	yes	characteristic, complemented
32	#15668020	—	2	1	no	—
33	#15668020	—	2	1	no	—
34	#15668020	C₂²	1	2	yes	—
35	#15668020	C₂²	1	2	yes	—
36	C₂×#13083558	C₂²	1	1	yes	characteristic
37	C₂×#13083557	—	2	1	no	—
38	C₂²×#10623494	C₂²	1	1	yes	characteristic
39	C₂×C₈×D₈₉	—	2	1	no	—
40	#15667947	C₂²	1	2	yes	complemented
41	#15667947	C₂²	1	2	yes	complemented
42	#15667930	C₂²	1	2	yes	complemented
43	#15667928	C₂²	1	2	yes	complemented
44	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
46	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
47	#101×C₈₉	C₂²	1	2	yes	complemented
48	#100×C₈₉	C₂²	1	2	yes	complemented
49	#13083574	—	2	1	no	—
50	#13083574	—	2	1	no	—
51	#13083574	—	2	1	no	—
52	#13083574	—	2	1	no	—
53	#13083574	—	2	1	no	—
54	#13083574	—	2	1	no	—
55	#13083574	D₄	1	2	yes	—
56	#13083574	C₂³	1	2	yes	—
57	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
58	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
59	C₂×D₃₅₆	D₄	1	1	yes	characteristic
60	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₈₉	—	2	1	no	—
62	C₂×C₂²⋊D₈₉	—	2	1	no	—
63	C₂×C₄×D₈₉	—	2	1	no	—
64	C₂×C₄×D₈₉	—	2	1	no	—
65	Dic₃₅₆	—	2	2	no	—
66	#13083564	—	2	2	no	—
67	#13083564	—	2	1	no	—
68	#13083564	—	2	1	no	—
69	#13083563	—	2	2	no	—
70	#13083563	—	2	2	no	—
71	#13083562	—	2	2	no	—
72	#13083562	—	2	1	no	—
73	#13083562	—	2	1	no	—
74	#13083561	—	2	2	no	—
75	#13083561	—	2	1	no	—
76	#13083561	—	2	1	no	—
77	D₇₁₂	—	2	2	no	—
78	#13083559	—	2	2	no	—
79	#13083559	—	2	1	no	—
80	#13083559	—	2	1	no	—
81	#13083558	—	2	1	no	—
82	#13083558	C₂³	1	2	yes	—
83	#13083557	—	2	2	no	—
84	#13083557	—	2	2	no	—
85	C₂×#10623494	—	2	1	no	—
86	C₂×#10623494	D₄	1	1	yes	characteristic
87	C₂×#10623494	D₄	1	1	yes	characteristic
88	C₂×#10623494	D₄	1	1	yes	characteristic
89	C₂×#10623494	D₄	1	1	yes	characteristic
90	C₂×#10623494	C₂³	1	1	yes	characteristic
91	C₂×#10623494	C₂³	1	1	yes	characteristic
92	C₈×D₈₉	—	2	2	no	—
93	C₈×D₈₉	—	2	2	no	—
94	#41×C₈₉	—	2	1	no	—
95	#41×C₈₉	—	2	1	no	—
96	#41×C₈₉	—	2	1	no	—
97	#41×C₈₉	—	2	1	no	—
98	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₈₉	—	2	1	no	—
100	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	—	2	1	no	—
102	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
103	M₁₆×C₈₉	—	2	1	no	—
104	M₁₆×C₈₉	—	2	1	no	—
105	M₁₆×C₈₉	C₂³	1	2	yes	—
106	M₁₆×C₈₉	C₂³	1	2	yes	—
107	C₂×C₈×C₈₉	—	2	1	no	—
108	C₂×C₈×C₈₉	—	2	1	no	—
109	C₂³×C₈₉	—	2	1	no	—
110	C₂³×C₈₉	#41	1	1	yes	characteristic
111	C₂²×D₈₉	—	2	1	no	—
112	C₂²×D₈₉	—	2	1	no	—
113	Q₈×C₈₉	—	2	1	no	—
114	Q₈×C₈₉	—	2	1	no	—
115	D₄×C₈₉	—	2	1	no	—
116	D₄×C₈₉	—	2	1	no	—
117	D₄×C₈₉	—	4	1	no	—
118	D₄×C₈₉	—	4	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	4	1	no	—
121	C₂×C₄×C₈₉	—	4	1	no	—
122	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₈₉	—	2	2	no	—
129	C₂²⋊D₈₉	—	2	2	no	—
130	C₂²⋊D₈₉	—	4	1	no	—
131	C₂²⋊D₈₉	—	4	1	no	—
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	—	2	1	no	—
134	D₃₅₆	—	2	1	no	—
135	D₃₅₆	—	2	1	no	—
136	D₃₅₆	—	2	1	no	—
137	D₃₅₆	C₂×D₄	1	2	yes	—
138	C₄×D₈₉	—	2	2	no	—
139	C₄×D₈₉	—	2	2	no	—
140	C₄×D₈₉	—	4	1	no	—
141	C₄×D₈₉	—	4	1	no	—
142	Dic₁₇₈	—	2	1	no	—
143	Dic₁₇₈	—	2	1	no	—
144	Dic₁₇₈	—	2	1	no	—
145	Dic₁₇₈	C₂×D₄	1	2	yes	—
146	C₈×C₈₉	—	2	2	no	—
147	C₈×C₈₉	—	2	2	no	—
148	#10623494	—	2	2	no	—
149	#10623494	C₂×D₄	1	1	yes	characteristic
150	#10623494	C₂×D₄	1	1	yes	characteristic
151	#10623494	C₂×D₄	1	1	yes	characteristic
152	#10623494	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₈₉	—	4	1	no	—
154	C₂²×C₈₉	—	4	1	no	—
155	C₂²×C₈₉	—	8	1	no	—
156	C₂²×C₈₉	#121	1	1	yes	characteristic
157	C₂²×C₈₉	#121	1	1	yes	characteristic
158	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
159	C₂×D₈₉	—	2	2	no	—
160	C₂×D₈₉	—	4	2	no	—
161	C₂×D₈₉	—	4	1	no	—
162	C₂×D₈₉	—	8	1	no	—
163	C₄×C₈₉	—	4	1	no	—
164	C₄×C₈₉	#121	1	1	yes	characteristic
165	C₄×C₈₉	#121	1	1	yes	characteristic
166	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
167	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
168	Dic₈₉	—	2	2	no	—
169	Dic₈₉	—	4	1	no	—
170	C₂×C₈₉	—	2	1	no	—
171	C₂×C₈₉	—	2	1	no	—
172	C₂×C₈₉	—	2	1	no	—
173	C₂×C₈₉	—	8	1	no	—
174	C₂×C₈₉	C₂×#121	1	1	yes	characteristic
175	D₈₉	—	4	2	no	—
176	D₈₉	—	8	1	no	—
177	#3067	—	89	1	no	maximal, Sylow
178	C₈₉	#3067	1	1	yes	characteristic, complemented, Sylow
179	#575	—	89	1	no	—
180	#575	—	89	1	no	—
181	C₂×#135	—	89	1	no	—
182	C₂×#131	—	89	1	no	—
183	#473	—	89	2	no	—
184	#472	—	89	2	no	—
185	#471	—	89	2	no	—
186	#471	—	89	2	no	—
187	C₂×#108	—	89	1	no	—
188	#413	—	89	1	no	—
189	#413	D₈₉	1	1	yes	characteristic, complemented
190	C₂×#41	—	89	1	no	—
191	C₂×#41	—	89	1	no	—
192	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
193	#137	—	89	2	no	—
194	#137	—	89	2	no	—
195	#137	—	178	1	no	—
196	#137	—	178	1	no	—
197	AGL(1,ℤ/8)	—	89	2	no	—
198	AGL(1,ℤ/8)	—	89	2	no	—
199	AGL(1,ℤ/8)	—	178	1	no	—
200	AGL(1,ℤ/8)	—	178	1	no	—
201	#135	—	178	2	no	—
202	#135	—	178	1	no	—
203	#135	—	178	1	no	—
204	#135	—	178	1	no	—
205	#135	—	178	1	no	—
206	#135	—	178	2	no	—
207	C₂×Q₁₆	—	89	1	no	—
208	C₂×SD₁₆	—	89	1	no	—
209	C₂×SD₁₆	—	89	1	no	—
210	C₂×D₈	—	89	1	no	—
211	#131	—	89	2	no	—
212	#131	—	89	2	no	—
213	#131	—	178	1	no	—
214	#131	—	178	1	no	—
215	C₂×M₁₆	—	89	1	no	—
216	C₂×M₁₆	—	178	1	no	—
217	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
218	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
219	C₂²×C₈	—	89	1	no	—
220	C₂²×C₈	—	178	1	no	—
221	#108	—	89	2	no	—
222	#108	—	89	2	no	—
223	#101	—	89	2	no	—
224	#101	C₂×D₈₉	1	2	yes	complemented
225	#100	—	89	2	no	—
226	#100	C₂×D₈₉	1	2	yes	complemented
227	#41	—	2	1	no	—
228	#41	—	2	1	no	—
229	#41	—	2	1	no	—
230	#41	—	2	1	no	—
231	#41	—	89	2	no	—
232	#41	—	89	2	no	—
233	#41	—	178	1	no	—
234	#41	—	178	1	no	—
235	#41	—	178	1	no	—
236	#41	—	178	1	no	—
237	#41	—	178	1	no	—
238	#41	—	178	1	no	—
239	C₂×Q₈	—	89	1	no	—
240	C₂×Q₈	—	89	1	no	—
241	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	89	1	no	—
244	C₂×D₄	—	89	1	no	—
245	C₂×D₄	—	178	1	no	—
246	C₂×D₄	—	178	1	no	—
247	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	178	1	no	—
250	C₂²×C₄	—	178	1	no	—
251	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
252	Q₁₆	—	178	2	no	—
253	Q₁₆	—	178	1	no	—
254	Q₁₆	—	178	1	no	—
255	Q₁₆	—	178	2	no	—
256	SD₁₆	—	178	2	no	—
257	SD₁₆	—	178	2	no	—
258	SD₁₆	—	178	1	no	—
259	SD₁₆	—	178	1	no	—
260	SD₁₆	—	178	1	no	—
261	SD₁₆	—	178	1	no	—
262	SD₁₆	—	178	2	no	—
263	SD₁₆	—	178	2	no	—
264	D₈	—	178	2	no	—
265	D₈	—	178	1	no	—
266	D₈	—	178	1	no	—
267	D₈	—	178	2	no	—
268	M₁₆	—	2	1	no	—
269	M₁₆	—	2	1	no	—
270	M₁₆	—	89	2	no	—
271	M₁₆	—	178	2	no	—
272	M₁₆	—	178	2	no	—
273	M₁₆	—	178	1	no	—
274	M₁₆	C₂²×D₈₉	1	2	yes	—
275	M₁₆	C₂²×D₈₉	1	2	yes	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	89	1	no	—
279	C₂×C₈	—	89	1	no	—
280	C₂×C₈	—	89	1	no	—
281	C₂×C₈	—	89	1	no	—
282	C₂×C₈	—	89	1	no	—
283	C₂×C₈	—	89	1	no	—
284	C₂×C₈	—	178	2	no	—
285	C₂×C₈	—	178	2	no	—
286	C₂×C₈	—	178	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	178	1	no	—
289	C₂³	—	178	1	no	—
290	C₂³	#13083574	1	1	yes	characteristic
291	Q₈	—	2	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	89	2	no	—
294	Q₈	—	178	1	no	—
295	Q₈	—	178	1	no	—
296	Q₈	—	178	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	2	1	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	89	2	no	—
302	D₄	—	356	1	no	—
303	D₄	—	356	1	no	—
304	D₄	—	178	2	no	—
305	D₄	—	178	2	no	—
306	D₄	—	178	1	no	—
307	D₄	—	178	1	no	—
308	D₄	—	178	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	356	1	no	—
313	C₂×C₄	—	356	1	no	—
314	C₂×C₄	—	178	1	no	—
315	C₂×C₄	—	178	1	no	—
316	C₂×C₄	—	178	2	no	—
317	C₂×C₄	—	178	2	no	—
318	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
319	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
320	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
321	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
322	C₂×C₄	#13083574	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	2	no	—
326	C₈	—	89	1	no	—
327	C₈	—	89	1	no	—
328	C₈	—	89	1	no	—
329	C₈	—	89	1	no	—
330	C₈	—	178	2	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	8	1	no	—
334	C₂²	—	356	2	no	—
335	C₂²	—	356	1	no	—
336	C₂²	—	178	2	no	—
337	C₂²	—	712	1	no	—
338	C₂²	#15667985	1	1	yes	characteristic
339	C₂²	#15667985	1	1	yes	characteristic
340	C₂²	C₂×#13083574	1	1	yes	characteristic
341	C₄	—	4	1	no	—
342	C₄	—	356	1	no	—
343	C₄	—	178	2	no	—
344	C₄	#15667985	1	1	yes	characteristic
345	C₄	#15667985	1	1	yes	characteristic
346	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
347	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
348	C₂	—	2	1	no	—
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	8	1	no	—
352	C₂	—	356	2	no	—
353	C₂	—	712	1	no	—
354	C₂	C₂×#15667985	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15667985
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#413×C₈₉	C₂
socle	C₂×C₈₉	C₂×#121
hypercenter	#413	D₈₉
hypocenter	C₈₉	#3067
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

Complements

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

N	H ≅ G/N	type
1. #19152398	C₂	semidirect
2. #19152396	C₂	semidirect
3. C₂×#15668032	C₂	semidirect
4. C₂×#15668020	C₂	semidirect
5. #19151872	C₂	semidirect
6. #19151867	C₂	semidirect
7. #19151864	C₂	semidirect
8. #19151860	C₂	semidirect
9. C₂×#15667947	C₂	semidirect
10. #19151580	C₂	semidirect
11. #413×C₈₉	C₂	semidirect
28. C₂×#13083564	C₂²	semidirect
29. C₂×#13083562	C₂²	semidirect
30. C₂×#13083561	C₂²	semidirect
31. C₂×#13083559	C₂²	semidirect
40. #15667947	C₂²	semidirect
41. #15667947	C₂²	semidirect
42. #15667930	C₂²	semidirect
43. #15667928	C₂²	semidirect
47. #101×C₈₉	C₂²	semidirect
48. #100×C₈₉	C₂²	semidirect
178. C₈₉	#3067	semidirect
189. #413	D₈₉	semidirect
224. #101	C₂×D₈₉	semidirect
226. #100	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152398	2
#19152396	2
C₂×#15668032	2
C₂×#15668020	2
#19151872	2
#19151867	2
#19151864	2
#19151860	2
C₂×#15667947	2
#19151580	2
#413×C₈₉	2
#3067	89