Subgroups — #26033337

Order 11392.

380 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152334	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152330	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151986	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151979	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151818	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151817	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151815	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151811	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151791	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151790	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151726	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151700	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151699	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#496×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083564	C₂²	1	1	yes	characteristic
17	C₂×#13083563	C₂²	1	1	yes	characteristic
18	C₂×#13083562	C₂²	1	1	yes	characteristic
19	C₂×#13083559	C₂²	1	1	yes	characteristic
20	C₂×#13083559	C₂²	1	1	yes	characteristic
21	#15668008	C₂²	1	1	yes	characteristic
22	#15668001	C₂²	1	1	yes	characteristic
23	#15667999	—	2	1	no	—
24	#15667999	C₂²	1	1	yes	characteristic
25	#15667999	C₂²	1	1	yes	characteristic
26	#15667998	—	2	1	no	—
27	#15667996	—	2	1	no	—
28	#15667994	—	2	1	no	—
29	#15667993	—	2	1	no	—
30	#15667993	—	2	1	no	—
31	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
33	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
34	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
35	D₄×Dic₈₉	—	2	1	no	—
36	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
37	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
38	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
39	#15667972	—	2	1	no	—
40	#15667946	C₂²	1	1	yes	characteristic
41	C₄⋊#10623494	C₂²	1	1	yes	characteristic
42	#15667941	C₂²	1	1	yes	characteristic
43	#15667941	C₂²	1	1	yes	characteristic
44	#15667937	C₂²	1	1	yes	characteristic
45	#15667936	C₂²	1	1	yes	characteristic
46	#15667934	C₂²	1	1	yes	characteristic
47	#15667933	C₂²	1	1	yes	characteristic
48	#15667933	C₂²	1	1	yes	characteristic
49	#15667933	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
55	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
56	#127×C₈₉	C₂²	1	1	yes	characteristic
57	#124×C₈₉	C₂²	1	1	yes	characteristic
58	#97×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
60	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
61	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₈₉	—	2	1	no	—
63	C₂×C₂²⋊D₈₉	—	2	1	no	—
64	C₂²×Dic₈₉	—	2	1	no	—
65	C₂²×Dic₈₉	—	2	1	no	—
66	C₂²×Dic₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	#13083564	—	4	1	no	—
69	#13083563	—	2	2	no	—
70	#13083562	—	4	1	no	—
71	#13083559	—	4	1	no	—
72	#13083559	—	4	1	no	—
73	C₂×#10623494	C₂³	1	1	yes	characteristic
74	C₂×#10623494	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₉	—	2	1	no	—
76	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
77	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
79	#13083553	—	2	1	no	—
80	#13083553	D₄	1	1	yes	characteristic
81	#13083553	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₈₉	—	2	1	no	—
83	C₂²⋊Dic₈₉	—	2	1	no	—
84	C₂²⋊Dic₈₉	—	2	1	no	—
85	C₂²⋊Dic₈₉	—	2	1	no	—
86	C₂²⋊Dic₈₉	—	2	1	no	—
87	C₂²⋊Dic₈₉	—	2	1	no	—
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	#13083551	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	C₄×Dic₈₉	—	2	1	no	—
93	C₄×Dic₈₉	—	2	1	no	—
94	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
95	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₈₉	—	2	1	no	—
98	C₂×D₄×C₈₉	—	2	1	no	—
99	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
102	SD₁₆×C₈₉	—	2	2	no	—
103	SD₁₆×C₈₉	—	4	1	no	—
104	D₈×C₈₉	—	2	2	no	—
105	D₈×C₈₉	—	4	1	no	—
106	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
108	C₂²⋊C₄×C₈₉	—	2	1	no	—
109	C₂²⋊C₄×C₈₉	—	2	1	no	—
110	C₄²×C₈₉	C₂³	1	1	yes	characteristic
111	C₂³×C₈₉	—	2	1	no	—
112	C₂³×C₈₉	—	2	1	no	—
113	C₂³×C₈₉	—	2	1	no	—
114	C₂²×D₈₉	—	2	1	no	—
115	Q₈×C₈₉	—	2	1	no	—
116	Q₈×C₈₉	—	2	1	no	—
117	D₄×C₈₉	—	2	1	no	—
118	D₄×C₈₉	—	2	1	no	—
119	D₄×C₈₉	—	2	1	no	—
120	D₄×C₈₉	—	2	1	no	—
121	D₄×C₈₉	—	2	1	no	—
122	D₄×C₈₉	—	2	1	no	—
123	D₄×C₈₉	—	4	2	no	—
124	D₄×C₈₉	—	8	1	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₈₉	—	4	2	no	—
130	C₂²⋊D₈₉	—	8	1	no	—
131	C₂×Dic₈₉	—	2	1	no	—
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	—	2	1	no	—
134	C₂×Dic₈₉	—	8	1	no	—
135	C₂×Dic₈₉	—	8	1	no	—
136	C₂×Dic₈₉	—	8	1	no	—
137	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
141	D₃₅₆	—	2	1	no	—
142	D₃₅₆	—	2	1	no	—
143	C₄×D₈₉	—	8	1	no	—
144	Dic₁₇₈	—	2	1	no	—
145	Dic₁₇₈	—	2	1	no	—
146	Dic₁₇₈	—	4	1	no	—
147	C₈×C₈₉	—	2	1	no	—
148	C₈×C₈₉	#41	1	2	yes	—
149	#10623494	—	2	1	no	—
150	#10623494	—	2	1	no	—
151	C₂²×C₈₉	—	4	1	no	—
152	C₂²×C₈₉	—	4	1	no	—
153	C₂²×C₈₉	—	4	1	no	—
154	C₂²×C₈₉	—	4	1	no	—
155	C₂²×C₈₉	—	4	1	no	—
156	C₂²×C₈₉	—	4	1	no	—
157	C₂²×C₈₉	—	4	1	no	—
158	C₂²×C₈₉	—	4	1	no	—
159	C₂²×C₈₉	—	4	1	no	—
160	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
161	C₂×D₈₉	—	4	1	no	—
162	C₂×D₈₉	—	4	1	no	—
163	C₂×D₈₉	—	4	1	no	—
164	C₄×C₈₉	—	2	1	no	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	4	1	no	—
167	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
168	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
169	Dic₈₉	—	2	1	no	—
170	Dic₈₉	—	2	1	no	—
171	Dic₈₉	—	2	1	no	—
172	Dic₈₉	—	2	1	no	—
173	Dic₈₉	—	4	1	no	—
174	Dic₈₉	—	4	1	no	—
175	Dic₈₉	—	4	1	no	—
176	C₂×C₈₉	—	8	1	no	—
177	C₂×C₈₉	—	8	1	no	—
178	C₂×C₈₉	—	8	1	no	—
179	C₂×C₈₉	#550	1	1	yes	characteristic
180	C₂×C₈₉	#575	1	1	yes	characteristic
181	C₂×C₈₉	#576	1	1	yes	characteristic
182	D₈₉	—	8	1	no	—
183	#3403	—	89	1	no	maximal, Sylow
184	C₈₉	#3403	1	1	yes	characteristic, complemented, Sylow
185	#559	—	89	1	no	—
186	#559	—	89	1	no	—
187	#496	—	89	1	no	—
188	#496	D₈₉	1	1	yes	characteristic, complemented
189	#495	—	89	1	no	—
190	#464	—	89	1	no	—
191	#464	—	89	1	no	—
192	#463	—	89	1	no	—
193	#463	—	89	1	no	—
194	#459	—	89	1	no	—
195	#459	—	89	1	no	—
196	#446	—	89	1	no	—
197	#440	—	89	1	no	—
198	#440	—	89	1	no	—
199	C₄×D₈	—	89	1	no	—
200	C₂×Q₁₆	—	89	1	no	—
201	C₂×SD₁₆	—	89	1	no	—
202	C₂×SD₁₆	—	89	1	no	—
203	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
204	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
205	C₂×D₈	—	89	1	no	—
206	C₂×D₈	—	89	1	no	—
207	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
208	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
209	#127	—	89	1	no	—
210	#127	C₂×D₈₉	1	1	yes	characteristic
211	#124	—	89	1	no	—
212	#124	—	89	1	no	—
213	#124	—	89	1	no	—
214	#124	—	178	1	no	—
215	#124	—	178	1	no	—
216	#124	C₂×D₈₉	1	1	yes	characteristic
217	#123	—	178	1	no	—
218	#123	—	178	1	no	—
219	#123	—	178	1	no	—
220	#123	—	178	1	no	—
221	C₄×Q₈	—	89	1	no	—
222	C₄×Q₈	—	89	1	no	—
223	C₄×Q₈	—	89	1	no	—
224	C₄×D₄	—	89	1	no	—
225	C₄×D₄	—	89	1	no	—
226	C₄×D₄	—	89	1	no	—
227	C₄×D₄	—	89	1	no	—
228	C₄×D₄	—	178	1	no	—
229	C₄×D₄	—	178	1	no	—
230	#107	—	89	1	no	—
231	C₄⋊C₈	—	89	1	no	—
232	C₄⋊C₈	—	89	1	no	—
233	C₄⋊C₈	—	89	1	no	—
234	#103	—	89	1	no	—
235	#103	—	89	1	no	—
236	#102	—	89	1	no	—
237	#102	—	89	1	no	—
238	#102	—	89	1	no	—
239	#102	—	89	1	no	—
240	#97	—	89	1	no	—
241	#97	C₂×D₈₉	1	1	yes	characteristic
242	C₄×C₈	—	89	1	no	—
243	C₂×Q₈	—	89	1	no	—
244	C₂×Q₈	—	89	1	no	—
245	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	89	1	no	—
249	C₂×D₄	—	178	1	no	—
250	C₂×D₄	—	178	1	no	—
251	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
252	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
253	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
254	C₂²×C₄	—	178	1	no	—
255	C₂²×C₄	—	178	1	no	—
256	C₂²×C₄	—	178	1	no	—
257	C₂²×C₄	—	178	1	no	—
258	Q₁₆	—	356	1	no	—
259	SD₁₆	—	2	2	no	—
260	SD₁₆	—	4	1	no	—
261	SD₁₆	—	356	1	no	—
262	SD₁₆	—	178	2	no	—
263	D₈	—	2	2	no	—
264	D₈	—	4	1	no	—
265	D₈	—	356	1	no	—
266	D₈	—	356	1	no	—
267	C₂×C₈	—	89	1	no	—
268	C₂×C₈	—	89	1	no	—
269	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
271	C₄⋊C₄	—	89	1	no	—
272	C₄⋊C₄	—	89	1	no	—
273	C₄⋊C₄	—	89	1	no	—
274	C₄⋊C₄	—	89	1	no	—
275	C₄⋊C₄	—	89	1	no	—
276	C₄⋊C₄	—	178	1	no	—
277	C₄⋊C₄	—	178	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	178	1	no	—
281	C₂²⋊C₄	—	178	1	no	—
282	C₂²⋊C₄	—	178	1	no	—
283	C₂²⋊C₄	—	178	1	no	—
284	C₂²⋊C₄	—	178	1	no	—
285	C₂²⋊C₄	—	178	1	no	—
286	C₂²⋊C₄	—	178	1	no	—
287	C₂²⋊C₄	—	178	1	no	—
288	C₂²⋊C₄	—	178	1	no	—
289	C₂²⋊C₄	—	178	1	no	—
290	C₄²	—	89	1	no	—
291	C₄²	—	89	1	no	—
292	C₄²	—	178	1	no	—
293	C₄²	—	178	1	no	—
294	C₄²	C₂²×D₈₉	1	1	yes	characteristic
295	C₂³	—	2	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	178	1	no	—
299	Q₈	—	2	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	356	1	no	—
302	Q₈	—	178	1	no	—
303	Q₈	—	178	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	4	2	no	—
311	D₄	—	8	1	no	—
312	D₄	—	356	2	no	—
313	D₄	—	178	1	no	—
314	D₄	—	178	1	no	—
315	D₄	—	712	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	89	1	no	—
318	C₂×C₄	—	89	1	no	—
319	C₂×C₄	—	89	1	no	—
320	C₂×C₄	—	89	1	no	—
321	C₂×C₄	—	178	1	no	—
322	C₂×C₄	—	178	1	no	—
323	C₂×C₄	—	178	1	no	—
324	C₂×C₄	—	712	1	no	—
325	C₂×C₄	—	712	1	no	—
326	C₂×C₄	—	712	1	no	—
327	C₂×C₄	—	712	1	no	—
328	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
329	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	178	1	no	—
333	C₈	—	178	1	no	—
334	C₈	#13083573	1	2	yes	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	356	1	no	—
345	C₂²	—	356	1	no	—
346	C₂²	—	356	1	no	—
347	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	356	1	no	—
352	C₄	—	356	1	no	—
353	C₄	—	356	1	no	—
354	C₄	—	178	1	no	—
355	C₄	—	178	1	no	—
356	C₄	—	178	1	no	—
357	C₄	—	178	1	no	—
358	C₄	C₂×#13083573	1	1	yes	characteristic
359	C₄	#15668057	1	1	yes	characteristic
360	C₂	—	8	1	no	—
361	C₂	—	8	1	no	—
362	C₂	—	8	1	no	—
363	C₂	—	712	1	no	—
364	C₂	#19152268	1	1	yes	characteristic
365	C₂	#19152394	1	1	yes	characteristic
366	C₂	#19152402	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#496×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#496	D₈₉
hypocenter	C₈₉	#3403
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #496

Complements

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

N	H ≅ G/N	type
1. #19152334	C₂	semidirect
2. #19152330	C₂	semidirect
3. #19151986	C₂	semidirect
4. #19151979	C₂	semidirect
5. #19151818	C₂	semidirect
6. #19151817	C₂	semidirect
7. #19151815	C₂	semidirect
8. #19151811	C₂	semidirect
9. #19151791	C₂	semidirect
10. #19151790	C₂	semidirect
11. #19151726	C₂	semidirect
12. #19151700	C₂	semidirect
13. #19151699	C₂	semidirect
14. D₈×Dic₈₉	C₂	semidirect
15. #496×C₈₉	C₂	semidirect
184. C₈₉	#3403	semidirect
188. #496	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152334	2
#19152330	2
#19151986	2
#19151979	2
#19151818	2
#19151817	2
#19151815	2
#19151811	2
#19151791	2
#19151790	2
#19151726	2
#19151700	2
#19151699	2
D₈×Dic₈₉	2
#496×C₈₉	2
#3403	89