Subgroups — #26032716

Order 11392.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152252	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152008	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151922	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151916	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151914	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151909	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151895	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151888	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151849	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151832	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151708	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151670	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151641	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#15667943	C₂	1	1	yes	maximal, characteristic, complemented
15	#553×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083573	—	2	1	no	—
17	C₂×#13083559	C₂²	1	1	yes	characteristic
18	C₂×#13083558	C₂²	1	1	yes	characteristic
19	C₂²×#10623494	C₂²	1	1	yes	characteristic
20	#15668012	C₂²	1	1	yes	characteristic
21	#15667996	—	2	1	no	—
22	#15667996	—	2	1	no	—
23	#15667992	—	2	1	no	—
24	#15667986	C₂²	1	1	yes	characteristic
25	#15667986	C₂²	1	1	yes	characteristic
26	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
29	#15667945	C₂²	1	1	yes	characteristic
30	#15667945	C₂²	1	1	yes	characteristic
31	#15667945	C₂²	1	1	yes	characteristic
32	#15667943	C₂²	1	1	yes	characteristic
33	#15667943	C₂²	1	1	yes	characteristic
34	#15667943	C₂²	1	1	yes	characteristic
35	#15667943	C₂²	1	1	yes	characteristic
36	#15667943	C₂²	1	1	yes	characteristic
37	#15667943	C₂²	1	1	yes	characteristic
38	C₄⋊#10623494	C₂²	1	1	yes	characteristic
39	#15667933	C₂²	1	1	yes	characteristic
40	#15667933	C₂²	1	1	yes	characteristic
41	#15667933	C₂²	1	1	yes	characteristic
42	#15667932	C₂²	1	1	yes	characteristic
43	#15667932	C₂²	1	1	yes	characteristic
44	#15667932	C₂²	1	1	yes	characteristic
45	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
46	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
47	#15667923	C₂²	1	1	yes	characteristic
48	#126×C₈₉	—	2	1	no	—
49	#125×C₈₉	C₂²	1	1	yes	characteristic
50	#123×C₈₉	—	2	1	no	—
51	#121×C₈₉	—	2	1	no	—
52	#121×C₈₉	C₂²	1	1	yes	characteristic
53	#121×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₉	—	2	1	no	—
55	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
56	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
57	#117×C₈₉	C₂²	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
59	#13083573	—	4	4	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
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	C₂×C₄×D₈₉	—	2	1	no	—
67	#13083559	—	2	2	no	—
68	#13083558	—	2	2	no	—
69	C₂×#10623494	—	2	2	no	—
70	C₂×#10623494	C₂³	1	1	yes	characteristic
71	C₂×#10623494	C₂³	1	1	yes	characteristic
72	C₂×#10623494	C₂³	1	1	yes	characteristic
73	C₂×#10623494	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₉	—	2	1	no	—
75	C₄⋊Dic₈₉	—	2	1	no	—
76	C₄⋊Dic₈₉	—	2	1	no	—
77	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
80	#13083551	—	2	1	no	—
81	#13083551	—	2	1	no	—
82	#13083551	—	2	1	no	—
83	C₂×D₄×C₈₉	—	2	1	no	—
84	C₂×D₄×C₈₉	—	2	1	no	—
85	C₂×D₄×C₈₉	D₄	1	1	yes	characteristic
86	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	—	2	1	no	—
88	C₂²×C₄×C₈₉	—	2	1	no	—
89	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
91	C₄⋊C₄×C₈₉	—	2	1	no	—
92	C₄⋊C₄×C₈₉	—	2	1	no	—
93	C₄⋊C₄×C₈₉	—	2	1	no	—
94	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
95	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₂²⋊C₄×C₈₉	—	2	1	no	—
99	C₂²⋊C₄×C₈₉	—	2	1	no	—
100	C₂²⋊C₄×C₈₉	—	2	1	no	—
101	C₂²⋊C₄×C₈₉	—	2	1	no	—
102	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
103	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
104	C₄²×C₈₉	—	2	1	no	—
105	C₄²×C₈₉	C₂³	1	1	yes	characteristic
106	C₂³×C₈₉	—	2	1	no	—
107	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₈₉	—	2	1	no	—
110	D₄×C₈₉	—	2	2	no	—
111	D₄×C₈₉	—	2	1	no	—
112	D₄×C₈₉	—	2	1	no	—
113	D₄×C₈₉	—	4	2	no	—
114	D₄×C₈₉	—	8	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	8	1	no	—
123	C₂×C₄×C₈₉	—	8	1	no	—
124	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
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	—	4	2	no	—
131	C₂×Dic₈₉	—	2	1	no	—
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	—	2	1	no	—
134	C₂×Dic₈₉	—	4	2	no	—
135	C₂×Dic₈₉	—	4	2	no	—
136	D₃₅₆	—	2	1	no	—
137	D₃₅₆	—	2	1	no	—
138	C₄×D₈₉	—	4	2	no	—
139	Dic₁₇₈	—	4	2	no	—
140	#10623494	—	2	1	no	—
141	#10623494	—	2	1	no	—
142	#10623494	—	2	1	no	—
143	#10623494	#41	1	2	yes	complemented
144	C₂²×C₈₉	—	2	1	no	—
145	C₂²×C₈₉	—	2	1	no	—
146	C₂²×C₈₉	—	2	1	no	—
147	C₂²×C₈₉	—	2	1	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	4	1	no	—
150	C₂²×C₈₉	—	4	1	no	—
151	C₂²×C₈₉	—	4	1	no	—
152	C₂²×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
153	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₈₉	—	4	1	no	—
155	C₂×D₈₉	—	4	1	no	—
156	C₂×D₈₉	—	4	1	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	—	4	1	no	—
162	C₄×C₈₉	—	4	1	no	—
163	C₄×C₈₉	—	4	1	no	—
164	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
165	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
166	Dic₈₉	—	4	1	no	—
167	Dic₈₉	—	4	1	no	—
168	Dic₈₉	—	4	1	no	—
169	C₂×C₈₉	—	2	2	no	—
170	C₂×C₈₉	—	4	1	no	—
171	C₂×C₈₉	—	8	1	no	—
172	C₂×C₈₉	#547	1	1	yes	characteristic
173	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₂×C₈₉	#577	1	1	yes	characteristic
175	D₈₉	—	8	1	no	—
176	#3345	—	89	1	no	maximal, Sylow
177	C₈₉	#3345	1	1	yes	characteristic, complemented, Sylow
178	#553	D₈₉	1	1	yes	characteristic, complemented
179	#547	—	89	1	no	—
180	#501	—	89	1	no	—
181	#482	—	89	1	no	—
182	#481	—	89	1	no	—
183	#481	—	89	1	no	—
184	#480	—	89	1	no	—
185	#478	—	89	1	no	—
186	#476	—	89	1	no	—
187	#469	—	89	1	no	—
188	#466	—	89	1	no	—
189	#442	—	89	1	no	—
190	#435	—	89	1	no	—
191	#428	—	89	1	no	—
192	C₂×#106	—	89	1	no	—
193	C₂×#41	—	178	1	no	—
194	C₂×D₈	—	89	1	no	—
195	C₂×M₁₆	—	89	1	no	—
196	C₂²×C₈	—	89	1	no	—
197	C₄⋊Q₈	—	89	1	no	—
198	#126	—	2	1	no	—
199	#125	C₂×D₈₉	1	1	yes	characteristic
200	#123	—	2	1	no	—
201	#123	—	178	1	no	—
202	#123	—	178	1	no	—
203	C₂²⋊Q₈	—	178	1	no	—
204	#121	—	2	1	no	—
205	#121	—	89	1	no	—
206	#121	—	89	1	no	—
207	#121	C₂×D₈₉	1	1	yes	characteristic
208	#121	C₂×D₈₉	1	1	yes	characteristic
209	C₄×D₄	—	2	1	no	—
210	C₄×D₄	—	89	1	no	—
211	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
212	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
213	#117	C₂×D₈₉	1	1	yes	characteristic
214	C₂×C₄⋊C₄	—	89	1	no	—
215	C₂×C₄⋊C₄	—	89	1	no	—
216	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
217	#107	—	89	1	no	—
218	#107	—	89	1	no	—
219	#107	—	89	1	no	—
220	#106	—	89	1	no	—
221	#106	—	89	1	no	—
222	#106	—	89	1	no	—
223	#106	—	89	1	no	—
224	#106	—	89	1	no	—
225	#106	—	89	1	no	—
226	C₄⋊C₈	—	89	1	no	—
227	#102	—	89	1	no	—
228	#102	—	89	1	no	—
229	#102	—	89	1	no	—
230	#102	—	89	1	no	—
231	#102	—	89	1	no	—
232	#102	—	89	1	no	—
233	C₂²⋊C₈	—	89	1	no	—
234	C₂²⋊C₈	—	89	1	no	—
235	#97	—	89	1	no	—
236	#41	—	356	4	no	—
237	C₂×Q₈	—	178	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	89	1	no	—
241	C₂×D₄	—	178	1	no	—
242	C₂×D₄	—	178	1	no	—
243	C₂×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
244	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	178	1	no	—
248	C₂²×C₄	—	178	1	no	—
249	C₂²×C₄	—	178	1	no	—
250	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
251	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
252	D₈	—	178	2	no	—
253	M₁₆	—	178	2	no	—
254	C₂×C₈	—	89	1	no	—
255	C₂×C₈	—	89	1	no	—
256	C₂×C₈	—	89	1	no	—
257	C₂×C₈	—	89	1	no	—
258	C₂×C₈	—	178	2	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	89	1	no	—
263	C₄⋊C₄	—	89	1	no	—
264	C₄⋊C₄	—	89	1	no	—
265	C₄⋊C₄	—	178	1	no	—
266	C₄⋊C₄	—	178	1	no	—
267	C₄⋊C₄	—	178	1	no	—
268	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	178	1	no	—
277	C₂²⋊C₄	—	178	1	no	—
278	C₂²⋊C₄	—	178	1	no	—
279	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
280	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
281	C₄²	—	2	1	no	—
282	C₄²	C₂²×D₈₉	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	178	1	no	—
285	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
286	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
287	Q₈	—	356	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	4	2	no	—
292	D₄	—	8	1	no	—
293	D₄	—	356	2	no	—
294	D₄	—	356	2	no	—
295	D₄	—	178	1	no	—
296	D₄	—	178	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	356	2	no	—
307	C₂×C₄	—	356	2	no	—
308	C₂×C₄	—	356	2	no	—
309	C₂×C₄	—	178	1	no	—
310	C₂×C₄	—	178	1	no	—
311	C₂×C₄	—	178	1	no	—
312	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
313	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
314	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
315	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
317	C₈	—	89	2	no	—
318	C₈	—	178	1	no	—
319	C₈	—	178	1	no	—
320	C₈	—	178	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	356	1	no	—
330	C₂²	—	356	1	no	—
331	C₂²	—	356	1	no	—
332	C₂²	#15668037	1	2	yes	—
333	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	356	1	no	—
342	C₄	—	356	1	no	—
343	C₄	—	356	1	no	—
344	C₄	#41×D₈₉	1	1	yes	characteristic
345	C₄	#15668059	1	1	yes	characteristic
346	C₂	—	2	2	no	—
347	C₂	—	4	1	no	—
348	C₂	—	8	1	no	—
349	C₂	—	712	1	no	—
350	C₂	#19152247	1	1	yes	characteristic
351	C₂	C₂×#15668037	1	1	yes	characteristic
352	C₂	#19152409	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#553×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#553	D₈₉
hypocenter	C₈₉	#3345
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #553

Complements

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

N	H ≅ G/N	type
1. #19152252	C₂	semidirect
2. #19152008	C₂	semidirect
3. #19151922	C₂	semidirect
4. #19151916	C₂	semidirect
5. #19151914	C₂	semidirect
6. #19151909	C₂	semidirect
7. #19151895	C₂	semidirect
8. #19151888	C₂	semidirect
9. #19151849	C₂	semidirect
10. #19151832	C₂	semidirect
11. #19151708	C₂	semidirect
12. #19151670	C₂	semidirect
13. #19151641	C₂	semidirect
14. C₂×#15667943	C₂	semidirect
15. #553×C₈₉	C₂	semidirect
143. #10623494	#41	semidirect
177. C₈₉	#3345	semidirect
178. #553	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152252	2
#19152008	2
#19151922	2
#19151916	2
#19151914	2
#19151909	2
#19151895	2
#19151888	2
#19151849	2
#19151832	2
#19151708	2
#19151670	2
#19151641	2
C₂×#15667943	2
#553×C₈₉	2
#3345	89