Subgroups — #26031662

Order 11392.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152222	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152199	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151908	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151897	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151875	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151852	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151843	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151833	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151828	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151814	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151813	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151797	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151792	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151623	C₂	1	1	yes	maximal, characteristic, complemented
15	#476×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083563	C₂²	1	1	yes	characteristic
17	C₂×#13083562	C₂²	1	1	yes	characteristic
18	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
19	C₂×#13083559	C₂²	1	1	yes	characteristic
20	C₂×#13083557	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
22	#15668008	C₂²	1	1	yes	characteristic
23	#15668007	—	2	1	no	—
24	#15668003	C₂²	1	1	yes	characteristic
25	#15668001	—	2	1	no	—
26	#15668000	C₂²	1	1	yes	characteristic
27	#15667997	—	2	1	no	—
28	#15667995	—	2	1	no	—
29	#15667994	—	2	1	no	—
30	#15667992	C₂²	1	1	yes	characteristic
31	#15667988	—	2	1	no	—
32	#15667987	C₂²	1	1	yes	characteristic
33	#15667987	C₂²	1	1	yes	characteristic
34	#15667986	—	2	1	no	—
35	#15667985	—	2	1	no	—
36	#15667983	C₂²	1	1	yes	characteristic
37	#15667977	C₂²	1	1	yes	characteristic
38	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
39	#15667974	C₂²	1	1	yes	characteristic
40	#15667969	C₂²	1	1	yes	characteristic
41	#15667945	C₂²	1	1	yes	characteristic
42	#15667945	C₂²	1	1	yes	characteristic
43	C₄⋊#10623494	C₂²	1	1	yes	characteristic
44	#15667941	C₂²	1	1	yes	characteristic
45	#15667941	C₂²	1	1	yes	characteristic
46	#15667937	C₂²	1	1	yes	characteristic
47	#15667935	C₂²	1	1	yes	characteristic
48	#15667934	C₂²	1	1	yes	characteristic
49	#15667932	C₂²	1	1	yes	characteristic
50	#15667932	C₂²	1	1	yes	characteristic
51	#15667932	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
54	#106×C₈₉	C₂²	1	1	yes	characteristic
55	#106×C₈₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	#102×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
60	C₂×D₃₅₆	—	2	1	no	—
61	C₂×D₃₅₆	—	2	1	no	—
62	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
63	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₈₉	—	2	1	no	—
65	C₂×C₂²⋊D₈₉	—	2	1	no	—
66	C₂×C₂²⋊D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
70	#13083563	—	4	1	no	—
71	#13083562	—	4	1	no	—
72	D₇₁₂	—	4	1	no	—
73	#13083559	—	4	1	no	—
74	#13083557	—	4	1	no	—
75	C₂×#10623494	C₂³	1	1	yes	characteristic
76	C₂×#10623494	C₂³	1	1	yes	characteristic
77	C₈×D₈₉	—	4	1	no	—
78	C₄⋊Dic₈₉	—	2	1	no	—
79	C₄⋊Dic₈₉	—	2	1	no	—
80	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	#13083553	D₄	1	1	yes	characteristic
84	#13083553	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₈₉	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	#13083551	—	2	1	no	—
88	#13083551	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	#13083551	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	#13083551	D₄	1	1	yes	characteristic
94	#13083551	D₄	1	1	yes	characteristic
95	C₄×Dic₈₉	—	2	1	no	—
96	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₉	—	2	1	no	—
98	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	—	2	1	no	—
100	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	—	2	1	no	—
103	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₈₉	—	2	1	no	—
107	C₄²×C₈₉	C₂³	1	1	yes	characteristic
108	C₂³×C₈₉	—	2	1	no	—
109	C₂²×D₈₉	—	2	1	no	—
110	C₂²×D₈₉	—	2	1	no	—
111	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
112	Q₈×C₈₉	—	8	1	no	—
113	D₄×C₈₉	—	2	1	no	—
114	D₄×C₈₉	—	2	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	—	8	1	no	—
119	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₈₉	—	8	1	no	—
123	C₂²⋊D₈₉	—	8	1	no	—
124	C₂²⋊D₈₉	—	8	1	no	—
125	C₂×Dic₈₉	—	2	1	no	—
126	C₂×Dic₈₉	—	2	1	no	—
127	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
130	D₃₅₆	—	2	1	no	—
131	D₃₅₆	—	2	1	no	—
132	D₃₅₆	—	2	1	no	—
133	D₃₅₆	—	2	1	no	—
134	D₃₅₆	—	8	1	no	—
135	D₃₅₆	—	8	1	no	—
136	C₄×D₈₉	—	2	1	no	—
137	C₄×D₈₉	—	2	1	no	—
138	C₄×D₈₉	—	8	1	no	—
139	C₄×D₈₉	—	8	1	no	—
140	Dic₁₇₈	—	2	1	no	—
141	Dic₁₇₈	—	2	1	no	—
142	C₈×C₈₉	—	2	1	no	—
143	C₈×C₈₉	—	2	1	no	—
144	#10623494	—	2	1	no	—
145	#10623494	—	2	1	no	—
146	C₂²×C₈₉	—	4	1	no	—
147	C₂²×C₈₉	—	4	1	no	—
148	C₂²×C₈₉	—	4	1	no	—
149	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₈₉	—	2	1	no	—
151	C₂×D₈₉	—	2	1	no	—
152	C₂×D₈₉	—	2	1	no	—
153	C₂×D₈₉	—	4	1	no	—
154	C₂×D₈₉	—	4	1	no	—
155	C₂×D₈₉	—	4	1	no	—
156	C₂×D₈₉	—	4	1	no	—
157	C₂×D₈₉	—	4	1	no	—
158	C₂×D₈₉	—	4	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	H(5,𝔽₂)	1	1	yes	characteristic
165	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
166	Dic₈₉	—	2	1	no	—
167	Dic₈₉	—	2	1	no	—
168	Dic₈₉	—	2	1	no	—
169	Dic₈₉	—	4	1	no	—
170	Dic₈₉	—	4	1	no	—
171	C₂×C₈₉	—	8	1	no	—
172	C₂×C₈₉	#537	1	1	yes	characteristic
173	C₂×C₈₉	#576	1	1	yes	characteristic
174	C₂×C₈₉	#577	1	1	yes	characteristic
175	D₈₉	—	4	1	no	—
176	D₈₉	—	8	1	no	—
177	D₈₉	—	8	1	no	—
178	#3257	—	89	1	no	maximal, Sylow
179	C₈₉	#3257	1	1	yes	characteristic, complemented, Sylow
180	#542	—	89	1	no	—
181	#540	—	89	1	no	—
182	#479	—	89	1	no	—
183	#478	—	89	1	no	—
184	#476	D₈₉	1	1	yes	characteristic, complemented
185	#474	—	89	1	no	—
186	#469	—	89	1	no	—
187	#468	—	89	1	no	—
188	#466	—	89	1	no	—
189	#465	—	89	1	no	—
190	#463	—	89	1	no	—
191	#463	—	89	1	no	—
192	#460	—	89	1	no	—
193	#459	—	89	1	no	—
194	#424	—	89	1	no	—
195	C₂×SD₁₆	—	89	1	no	—
196	C₂×SD₁₆	—	89	1	no	—
197	C₂×D₈	—	89	1	no	—
198	C₂×D₈	—	89	1	no	—
199	C₂×M₁₆	—	89	1	no	—
200	C₂²×C₈	—	89	1	no	—
201	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
202	#127	—	89	1	no	—
203	#126	—	178	1	no	—
204	#125	—	89	1	no	—
205	#124	—	89	1	no	—
206	#124	—	178	1	no	—
207	#123	—	178	1	no	—
208	#123	—	178	1	no	—
209	#123	—	178	1	no	—
210	C₂²⋊Q₈	—	89	1	no	—
211	C₂²⋊Q₈	—	178	1	no	—
212	#121	—	89	1	no	—
213	#121	—	89	1	no	—
214	#121	—	89	1	no	—
215	#121	—	178	1	no	—
216	#121	—	178	1	no	—
217	C₄×Q₈	—	89	1	no	—
218	C₄×D₄	—	89	1	no	—
219	C₄×D₄	—	89	1	no	—
220	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
221	#117	—	89	1	no	—
222	#107	—	89	1	no	—
223	#107	—	89	1	no	—
224	#106	C₂×D₈₉	1	1	yes	characteristic
225	#106	C₂×D₈₉	1	1	yes	characteristic
226	C₄⋊C₈	—	89	1	no	—
227	C₄⋊C₈	—	89	1	no	—
228	C₄⋊C₈	—	89	1	no	—
229	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
230	#103	—	89	1	no	—
231	#103	—	89	1	no	—
232	#102	—	89	1	no	—
233	#102	—	89	1	no	—
234	#102	—	89	1	no	—
235	#102	—	89	1	no	—
236	#102	C₂×D₈₉	1	1	yes	characteristic
237	#102	C₂×D₈₉	1	1	yes	characteristic
238	C₂×Q₈	—	2	1	no	—
239	C₂×Q₈	—	89	1	no	—
240	C₂×D₄	—	89	1	no	—
241	C₂×D₄	—	89	1	no	—
242	C₂×D₄	—	178	1	no	—
243	C₂×D₄	—	178	1	no	—
244	C₂×D₄	—	178	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	—	89	1	no	—
250	C₂²×C₄	—	178	1	no	—
251	C₂²×C₄	—	178	1	no	—
252	SD₁₆	—	356	1	no	—
253	SD₁₆	—	356	1	no	—
254	D₈	—	356	1	no	—
255	D₈	—	356	1	no	—
256	M₁₆	—	356	1	no	—
257	C₂×C₈	—	89	1	no	—
258	C₂×C₈	—	89	1	no	—
259	C₂×C₈	—	356	1	no	—
260	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	89	1	no	—
264	C₄⋊C₄	—	89	1	no	—
265	C₄⋊C₄	—	89	1	no	—
266	C₄⋊C₄	—	178	1	no	—
267	C₄⋊C₄	—	178	1	no	—
268	C₄⋊C₄	—	178	1	no	—
269	C₄⋊C₄	—	178	1	no	—
270	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
272	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	89	1	no	—
275	C₂²⋊C₄	—	89	1	no	—
276	C₂²⋊C₄	—	178	1	no	—
277	C₂²⋊C₄	—	178	1	no	—
278	C₂²⋊C₄	—	178	1	no	—
279	C₂²⋊C₄	—	178	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	C₄²	—	89	1	no	—
285	C₄²	—	178	1	no	—
286	C₄²	C₂²×D₈₉	1	1	yes	characteristic
287	C₂³	—	2	1	no	—
288	C₂³	—	89	1	no	—
289	C₂³	—	178	1	no	—
290	C₂³	—	178	1	no	—
291	Q₈	—	8	1	no	—
292	Q₈	—	178	1	no	—
293	Q₈	—	178	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	178	1	no	—
297	D₄	—	178	1	no	—
298	D₄	—	178	1	no	—
299	D₄	—	178	1	no	—
300	D₄	—	712	1	no	—
301	D₄	—	712	1	no	—
302	D₄	—	712	1	no	—
303	D₄	—	712	1	no	—
304	D₄	—	712	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	8	1	no	—
309	C₂×C₄	—	89	1	no	—
310	C₂×C₄	—	89	1	no	—
311	C₂×C₄	—	89	1	no	—
312	C₂×C₄	—	178	1	no	—
313	C₂×C₄	—	178	1	no	—
314	C₂×C₄	—	178	1	no	—
315	C₂×C₄	—	178	1	no	—
316	C₂×C₄	—	712	1	no	—
317	C₂×C₄	—	712	1	no	—
318	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
319	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
320	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	178	1	no	—
324	C₈	—	178	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	356	1	no	—
329	C₂²	—	356	1	no	—
330	C₂²	—	356	1	no	—
331	C₂²	—	356	1	no	—
332	C₂²	—	356	1	no	—
333	C₂²	—	356	1	no	—
334	C₂²	—	178	1	no	—
335	C₂²	—	178	1	no	—
336	C₂²	—	178	1	no	—
337	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	356	1	no	—
344	C₄	—	356	1	no	—
345	C₄	—	178	1	no	—
346	C₄	—	178	1	no	—
347	C₄	—	178	1	no	—
348	C₄	#15668058	1	1	yes	characteristic
349	C₄	#15668060	1	1	yes	characteristic
350	C₂	—	8	1	no	—
351	C₂	—	356	1	no	—
352	C₂	—	712	1	no	—
353	C₂	—	712	1	no	—
354	C₂	#19152178	1	1	yes	characteristic
355	C₂	#19152402	1	1	yes	characteristic
356	C₂	#19152404	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#476×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#476	D₈₉
hypocenter	C₈₉	#3257
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

Complements

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

N	H ≅ G/N	type
1. #19152222	C₂	semidirect
2. #19152199	C₂	semidirect
3. #19151908	C₂	semidirect
4. #19151897	C₂	semidirect
5. #19151875	C₂	semidirect
6. #19151852	C₂	semidirect
7. #19151843	C₂	semidirect
8. #19151833	C₂	semidirect
9. #19151828	C₂	semidirect
10. #19151814	C₂	semidirect
11. #19151813	C₂	semidirect
12. #19151797	C₂	semidirect
13. #19151792	C₂	semidirect
14. #19151623	C₂	semidirect
15. #476×C₈₉	C₂	semidirect
179. C₈₉	#3257	semidirect
184. #476	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152222	2
#19152199	2
#19151908	2
#19151897	2
#19151875	2
#19151852	2
#19151843	2
#19151833	2
#19151828	2
#19151814	2
#19151813	2
#19151797	2
#19151792	2
#19151623	2
#476×C₈₉	2
#3257	89