Subgroups — #26032421

Order 11392.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152296	C₂	1	1	yes	maximal, characteristic
2	#19152261	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151965	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151941	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151904	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151845	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151829	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151802	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151739	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151738	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151697	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151666	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151605	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#15667935	C₂	1	1	yes	maximal, characteristic, complemented
15	#484×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	C₂×#13083563	C₂²	1	1	yes	characteristic
20	C₂×#13083561	C₂²	1	1	yes	characteristic
21	C₂×#13083557	C₂²	1	1	yes	characteristic
22	C₂²×#10623494	C₂²	1	1	yes	characteristic
23	#15668011	—	2	1	no	—
24	#15668010	C₂²	1	1	yes	characteristic
25	#15668007	—	2	1	no	—
26	#15668003	C₂²	1	1	yes	characteristic
27	#15667997	—	2	1	no	—
28	#15667992	C₂²	1	1	yes	characteristic
29	#15667990	—	2	1	no	—
30	#15667989	—	2	1	no	—
31	#15667987	C₂²	1	1	yes	characteristic
32	#15667986	—	2	1	no	—
33	#15667986	C₂²	1	1	yes	characteristic
34	#15667977	C₂²	1	1	yes	characteristic
35	#15667974	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊D₈₉	—	2	1	no	—
37	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
38	#15667972	—	2	1	no	—
39	#15667970	C₂²	1	1	yes	characteristic
40	#15667943	C₂²	1	1	yes	characteristic
41	#15667941	C₂²	1	1	yes	characteristic
42	#15667937	C₂²	1	1	yes	characteristic
43	#15667937	C₂²	1	1	yes	characteristic
44	#15667936	C₂²	1	1	yes	characteristic
45	#15667935	C₂²	1	1	yes	characteristic
46	#15667935	C₂²	1	1	yes	characteristic
47	#15667935	C₂²	1	1	yes	characteristic
48	#15667935	C₂²	1	1	yes	characteristic
49	#15667932	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	#15667922	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic
55	#107×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	—	2	2	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	—	2	1	no	—
62	C₂×Dic₁₇₈	—	4	1	no	—
63	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
64	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
65	Q₈×D₈₉	—	2	2	no	—
66	Q₈×D₈₉	—	4	1	no	—
67	C₂×D₃₅₆	—	2	1	no	—
68	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊D₈₉	—	2	1	no	—
70	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
71	C₂²×Dic₈₉	—	2	1	no	—
72	C₂×C₄×D₈₉	—	2	1	no	—
73	C₂×C₄×D₈₉	—	2	1	no	—
74	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
75	#13083563	—	2	2	no	—
76	#13083563	—	4	1	no	—
77	#13083561	—	4	1	no	—
78	#13083557	—	4	1	no	—
79	C₂×#10623494	—	4	1	no	—
80	C₂×#10623494	C₂³	1	1	yes	characteristic
81	C₂×#10623494	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₉	—	2	1	no	—
83	C₄⋊Dic₈₉	—	2	1	no	—
84	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
85	#13083553	—	2	1	no	—
86	#13083553	—	2	1	no	—
87	#13083553	—	2	1	no	—
88	#13083553	—	2	1	no	—
89	#13083553	D₄	1	1	yes	characteristic
90	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
91	#13083551	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	#13083551	—	2	1	no	—
94	#13083551	—	2	1	no	—
95	#13083551	—	2	1	no	—
96	#13083551	D₄	1	1	yes	characteristic
97	C₄×Dic₈₉	—	2	1	no	—
98	C₄×Dic₈₉	—	2	1	no	—
99	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
100	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	—	2	1	no	—
102	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	—	2	1	no	—
106	C₄⋊C₄×C₈₉	—	2	1	no	—
107	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₈₉	—	2	1	no	—
111	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₈₉	—	2	1	no	—
113	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₈₉	—	2	1	no	—
115	Q₈×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	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₈₉	—	2	2	no	—
126	C₂²⋊D₈₉	—	4	2	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	C₂×Dic₈₉	—	2	1	no	—
130	C₂×Dic₈₉	—	4	2	no	—
131	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
134	D₃₅₆	—	2	1	no	—
135	D₃₅₆	—	2	1	no	—
136	D₃₅₆	—	8	1	no	—
137	C₄×D₈₉	—	2	1	no	—
138	C₄×D₈₉	—	2	1	no	—
139	C₄×D₈₉	—	4	2	no	—
140	C₄×D₈₉	—	8	1	no	—
141	Dic₁₇₈	—	2	1	no	—
142	Dic₁₇₈	—	2	1	no	—
143	Dic₁₇₈	—	2	1	no	—
144	Dic₁₇₈	—	4	2	no	—
145	Dic₁₇₈	—	4	2	no	—
146	Dic₁₇₈	#41	1	2	yes	—
147	C₈×C₈₉	—	2	1	no	—
148	C₈×C₈₉	—	2	1	no	—
149	#10623494	—	2	1	no	—
150	#10623494	—	2	1	no	—
151	C₂²×C₈₉	—	2	1	no	—
152	C₂²×C₈₉	—	2	1	no	—
153	C₂²×C₈₉	#137	1	2	yes	—
154	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₈₉	—	2	1	no	—
156	C₂×D₈₉	—	2	1	no	—
157	C₂×D₈₉	—	2	1	no	—
158	C₂×D₈₉	—	4	1	no	—
159	C₂×D₈₉	—	4	1	no	—
160	C₂×D₈₉	—	4	1	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	—	4	1	no	—
163	C₄×C₈₉	—	4	1	no	—
164	C₄×C₈₉	—	4	1	no	—
165	C₄×C₈₉	—	4	1	no	—
166	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
167	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
168	Dic₈₉	—	2	2	no	—
169	Dic₈₉	—	2	1	no	—
170	Dic₈₉	—	2	1	no	—
171	Dic₈₉	—	2	1	no	—
172	Dic₈₉	—	4	1	no	—
173	Dic₈₉	—	4	1	no	—
174	C₂×C₈₉	—	2	2	no	—
175	C₂×C₈₉	#546	1	1	yes	characteristic
176	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
177	C₂×C₈₉	#577	1	1	yes	characteristic
178	D₈₉	—	4	1	no	—
179	D₈₉	—	8	1	no	—
180	#3320	—	89	1	no	maximal, Sylow
181	C₈₉	#3320	1	1	yes	characteristic, complemented, Sylow
182	#553	—	89	1	no	—
183	D₄×Q₈	—	89	1	no	—
184	#491	—	89	1	no	—
185	#485	—	89	1	no	—
186	#484	D₈₉	1	1	yes	characteristic, complemented
187	#479	—	89	1	no	—
188	#468	—	89	1	no	—
189	#465	—	89	1	no	—
190	#461	—	89	1	no	—
191	#448	—	89	1	no	—
192	#448	—	89	1	no	—
193	#440	—	89	1	no	—
194	#435	—	89	1	no	—
195	#419	—	89	1	no	—
196	C₂×#103	—	89	1	no	—
197	C₂²×Q₈	—	89	1	no	—
198	C₂²×Q₈	—	89	1	no	—
199	C₂×SD₁₆	—	89	1	no	—
200	C₂×SD₁₆	—	89	1	no	—
201	C₂×SD₁₆	—	89	1	no	—
202	C₂×M₁₆	—	89	1	no	—
203	C₂²×C₈	—	89	1	no	—
204	C₄⋊Q₈	—	89	1	no	—
205	C₄⋊Q₈	—	178	1	no	—
206	#126	—	178	1	no	—
207	#125	—	89	1	no	—
208	#123	—	178	1	no	—
209	C₂²⋊Q₈	—	89	1	no	—
210	C₂²⋊Q₈	—	178	1	no	—
211	C₂²⋊Q₈	—	178	1	no	—
212	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
213	#121	—	89	1	no	—
214	#121	—	89	1	no	—
215	#121	—	178	1	no	—
216	C₄×Q₈	—	89	1	no	—
217	C₄×D₄	—	89	1	no	—
218	C₄×D₄	—	89	1	no	—
219	C₄×D₄	—	178	1	no	—
220	C₄×D₄	—	178	1	no	—
221	#117	—	89	1	no	—
222	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
223	#107	C₂×D₈₉	1	1	yes	characteristic
224	#107	C₂×D₈₉	1	1	yes	characteristic
225	#106	—	89	1	no	—
226	C₄⋊C₈	—	89	1	no	—
227	#103	—	89	1	no	—
228	#103	—	89	1	no	—
229	#103	—	89	1	no	—
230	#103	—	89	1	no	—
231	#103	—	89	1	no	—
232	#103	—	89	1	no	—
233	#103	—	89	1	no	—
234	#103	C₂×D₈₉	1	1	yes	characteristic
235	#103	C₂×D₈₉	1	1	yes	characteristic
236	#102	—	89	1	no	—
237	C₂²⋊C₈	—	89	1	no	—
238	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
239	#97	—	89	1	no	—
240	C₂×Q₈	—	89	1	no	—
241	C₂×Q₈	—	89	1	no	—
242	C₂×Q₈	—	356	1	no	—
243	C₂×Q₈	—	356	1	no	—
244	C₂×Q₈	—	178	1	no	—
245	C₂×Q₈	—	178	2	no	—
246	C₂×Q₈	—	178	2	no	—
247	C₂×Q₈	—	178	1	no	—
248	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
249	C₂×D₄	—	89	1	no	—
250	C₂×D₄	—	89	1	no	—
251	C₂×D₄	—	178	1	no	—
252	C₂×D₄	—	178	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	89	1	no	—
255	C₂²×C₄	—	178	1	no	—
256	C₂²×C₄	—	178	1	no	—
257	C₂²×C₄	—	178	1	no	—
258	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
259	SD₁₆	—	356	1	no	—
260	SD₁₆	—	356	1	no	—
261	SD₁₆	—	178	2	no	—
262	M₁₆	—	356	1	no	—
263	C₂×C₈	—	89	1	no	—
264	C₂×C₈	—	89	1	no	—
265	C₂×C₈	—	356	1	no	—
266	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	89	1	no	—
271	C₄⋊C₄	—	89	1	no	—
272	C₄⋊C₄	—	178	1	no	—
273	C₄⋊C₄	—	178	1	no	—
274	C₄⋊C₄	—	178	1	no	—
275	C₄⋊C₄	—	178	1	no	—
276	C₄⋊C₄	—	178	1	no	—
277	C₄⋊C₄	—	178	1	no	—
278	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
279	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
280	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
281	C₂²⋊C₄	—	2	1	no	—
282	C₂²⋊C₄	—	89	1	no	—
283	C₂²⋊C₄	—	89	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	C₄²	—	89	1	no	—
290	C₄²	—	178	1	no	—
291	C₄²	—	178	1	no	—
292	C₂³	—	89	1	no	—
293	C₂³	—	178	1	no	—
294	C₂³	D₄×D₈₉	1	1	yes	characteristic
295	Q₈	—	2	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	89	2	no	—
298	Q₈	—	356	2	no	—
299	Q₈	—	356	2	no	—
300	Q₈	—	178	1	no	—
301	Q₈	—	178	1	no	—
302	Q₈	—	178	1	no	—
303	D₄	—	356	2	no	—
304	D₄	—	178	2	no	—
305	D₄	—	178	1	no	—
306	D₄	—	178	1	no	—
307	D₄	—	712	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	8	1	no	—
315	C₂×C₄	—	89	1	no	—
316	C₂×C₄	—	89	1	no	—
317	C₂×C₄	—	89	1	no	—
318	C₂×C₄	—	356	2	no	—
319	C₂×C₄	—	356	2	no	—
320	C₂×C₄	—	178	1	no	—
321	C₂×C₄	—	178	1	no	—
322	C₂×C₄	—	178	1	no	—
323	C₂×C₄	—	178	1	no	—
324	C₂×C₄	—	178	1	no	—
325	C₂×C₄	—	712	1	no	—
326	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
327	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	178	1	no	—
331	C₈	—	178	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	356	1	no	—
335	C₂²	—	356	1	no	—
336	C₂²	—	356	1	no	—
337	C₂²	—	178	1	no	—
338	C₂²	—	178	1	no	—
339	C₂²	—	178	1	no	—
340	C₂²	#15668044	1	2	yes	—
341	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	356	1	no	—
348	C₄	—	356	1	no	—
349	C₄	—	178	1	no	—
350	C₄	—	178	1	no	—
351	C₄	—	178	1	no	—
352	C₄	—	178	2	no	—
353	C₄	#41×D₈₉	1	1	yes	characteristic
354	C₄	#15668058	1	1	yes	characteristic
355	C₂	—	2	2	no	—
356	C₂	—	356	1	no	—
357	C₂	—	712	1	no	—
358	C₂	#19152240	1	1	yes	characteristic
359	C₂	C₂×#15668044	1	1	yes	characteristic
360	C₂	#19152408	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#484×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152296	C₂
hypercenter	#484	D₈₉
hypocenter	C₈₉	#3320
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #484

Complements

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

N	H ≅ G/N	type
2. #19152261	C₂	semidirect
3. #19151965	C₂	semidirect
4. #19151941	C₂	semidirect
5. #19151904	C₂	semidirect
6. #19151845	C₂	semidirect
7. #19151829	C₂	semidirect
8. #19151802	C₂	semidirect
9. #19151739	C₂	semidirect
10. #19151738	C₂	semidirect
11. #19151697	C₂	semidirect
12. #19151666	C₂	semidirect
13. #19151605	C₂	semidirect
14. C₂×#15667935	C₂	semidirect
15. #484×C₈₉	C₂	semidirect
181. C₈₉	#3320	semidirect
186. #484	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152296	2
#19152261	2
#19151965	2
#19151941	2
#19151904	2
#19151845	2
#19151829	2
#19151802	2
#19151739	2
#19151738	2
#19151697	2
#19151666	2
#19151605	2
C₂×#15667935	2
#484×C₈₉	2
#3320	89