Subgroups — #26032342

Order 11392.

382 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	#19152253	C₂	1	1	yes	maximal, characteristic
2	#19151968	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151939	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151933	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151895	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151854	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151837	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151808	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151758	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151758	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151696	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151670	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151597	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151592	C₂	1	1	yes	maximal, characteristic, complemented
15	#552×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	C₂²	1	1	yes	characteristic
17	C₂×#13083574	C₂²	1	1	yes	characteristic
18	C₂×#13083564	C₂²	1	1	yes	characteristic
19	C₂×#13083564	C₂²	1	1	yes	characteristic
20	C₂×#13083561	C₂²	1	1	yes	characteristic
21	C₂×#13083558	C₂²	1	1	yes	characteristic
22	C₂²×#10623494	C₂²	1	1	yes	characteristic
23	#15668012	C₂²	1	1	yes	characteristic
24	#15667994	—	2	1	no	—
25	#15667994	—	2	1	no	—
26	#15667991	C₂²	1	1	yes	characteristic
27	#15667991	C₂²	1	1	yes	characteristic
28	#15667983	—	2	1	no	—
29	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×D₈₉	—	2	1	no	—
31	#15667945	C₂²	1	1	yes	characteristic
32	#15667945	C₂²	1	1	yes	characteristic
33	#15667943	C₂²	1	1	yes	characteristic
34	C₄⋊#10623494	C₂²	1	1	yes	characteristic
35	#15667936	C₂²	1	1	yes	characteristic
36	#15667935	C₂²	1	1	yes	characteristic
37	#15667935	C₂²	1	1	yes	characteristic
38	#15667935	C₂²	1	1	yes	characteristic
39	#15667935	C₂²	1	1	yes	characteristic
40	#15667935	C₂²	1	1	yes	characteristic
41	#15667935	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	—	2	1	no	—
50	#125×C₈₉	C₂²	1	1	yes	characteristic
51	#123×C₈₉	—	2	1	no	—
52	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
56	#117×C₈₉	—	2	1	no	—
57	#117×C₈₉	C₂²	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
59	#13083574	—	2	2	no	—
60	#13083574	—	2	2	no	—
61	#13083574	—	4	1	no	—
62	#13083574	—	4	1	no	—
63	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
64	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
65	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₈₉	—	2	1	no	—
67	C₂×C₂²⋊D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	#13083564	—	4	1	no	—
72	#13083564	—	4	1	no	—
73	#13083561	—	2	2	no	—
74	#13083558	—	4	1	no	—
75	C₂×#10623494	—	4	1	no	—
76	C₂×#10623494	C₂³	1	1	yes	characteristic
77	C₂×#10623494	C₂³	1	1	yes	characteristic
78	C₂×#10623494	C₂³	1	1	yes	characteristic
79	C₂×#10623494	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₉	—	2	1	no	—
81	C₄⋊Dic₈₉	—	2	1	no	—
82	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	C₂²⋊Dic₈₉	—	2	1	no	—
86	C₂²⋊Dic₈₉	—	2	1	no	—
87	#13083551	—	2	1	no	—
88	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
89	C₂×D₄×C₈₉	D₄	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	—	2	1	no	—
91	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₈₉	—	2	1	no	—
94	C₄⋊C₄×C₈₉	—	2	1	no	—
95	C₄⋊C₄×C₈₉	—	2	1	no	—
96	C₄⋊C₄×C₈₉	—	2	1	no	—
97	C₄⋊C₄×C₈₉	—	2	1	no	—
98	C₄⋊C₄×C₈₉	—	2	1	no	—
99	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
100	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
103	C₂²⋊C₄×C₈₉	—	2	1	no	—
104	C₂²⋊C₄×C₈₉	—	2	1	no	—
105	C₂²⋊C₄×C₈₉	—	2	1	no	—
106	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
107	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
108	C₄²×C₈₉	—	2	1	no	—
109	C₄²×C₈₉	—	2	1	no	—
110	C₄²×C₈₉	C₂³	1	1	yes	characteristic
111	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₈₉	—	2	1	no	—
114	Q₈×C₈₉	—	2	1	no	—
115	Q₈×C₈₉	—	2	1	no	—
116	D₄×C₈₉	—	4	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	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	4	2	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	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
131	C₂²⋊D₈₉	—	4	2	no	—
132	C₂²⋊D₈₉	—	4	2	no	—
133	C₂×Dic₈₉	—	2	1	no	—
134	C₂×Dic₈₉	—	2	1	no	—
135	C₂×Dic₈₉	—	2	1	no	—
136	D₃₅₆	—	2	1	no	—
137	D₃₅₆	#41	1	2	yes	—
138	C₄×D₈₉	—	4	2	no	—
139	C₄×D₈₉	—	4	2	no	—
140	C₄×D₈₉	—	4	2	no	—
141	Dic₁₇₈	—	2	1	no	—
142	Dic₁₇₈	—	2	1	no	—
143	Dic₁₇₈	—	2	1	no	—
144	Dic₁₇₈	—	2	1	no	—
145	#10623494	—	2	1	no	—
146	#10623494	—	2	1	no	—
147	#10623494	—	2	1	no	—
148	#10623494	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	—	2	1	no	—
152	C₂²×C₈₉	—	2	1	no	—
153	C₂²×C₈₉	—	2	1	no	—
154	C₂²×C₈₉	—	2	1	no	—
155	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₈₉	—	2	2	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	—	2	1	no	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	—	4	1	no	—
164	C₄×C₈₉	—	4	1	no	—
165	C₄×C₈₉	—	4	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	—	4	1	no	—
170	Dic₈₉	—	4	1	no	—
171	Dic₈₉	—	4	1	no	—
172	C₂×C₈₉	—	4	1	no	—
173	C₂×C₈₉	—	4	1	no	—
174	C₂×C₈₉	#546	1	1	yes	characteristic
175	C₂×C₈₉	#575	1	1	yes	characteristic
176	C₂×C₈₉	#578	1	1	yes	characteristic
177	D₈₉	—	4	2	no	—
178	#3315	—	89	1	no	maximal, Sylow
179	C₈₉	#3315	1	1	yes	characteristic, complemented, Sylow
180	#552	D₈₉	1	1	yes	characteristic, complemented
181	#547	—	89	1	no	—
182	#491	—	89	1	no	—
183	#485	—	89	1	no	—
184	#484	—	89	1	no	—
185	#478	—	89	1	no	—
186	#470	—	89	1	no	—
187	#467	—	89	1	no	—
188	#462	—	89	1	no	—
189	#452	—	89	1	no	—
190	#452	—	89	1	no	—
191	#440	—	89	1	no	—
192	#435	—	89	1	no	—
193	#417	—	89	1	no	—
194	#416	—	89	1	no	—
195	C₂×#41	—	89	1	no	—
196	C₂×#41	—	89	1	no	—
197	C₂×Q₁₆	—	89	1	no	—
198	C₂×Q₁₆	—	89	1	no	—
199	C₂×SD₁₆	—	89	1	no	—
200	C₂×M₁₆	—	89	1	no	—
201	C₂²×C₈	—	89	1	no	—
202	C₄⋊Q₈	—	89	1	no	—
203	#126	—	2	1	no	—
204	#125	—	2	1	no	—
205	#125	C₂×D₈₉	1	1	yes	characteristic
206	#123	—	2	1	no	—
207	#123	—	178	1	no	—
208	#123	—	178	1	no	—
209	C₂²⋊Q₈	—	89	1	no	—
210	C₂²⋊Q₈	—	89	1	no	—
211	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
212	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
213	#121	—	178	1	no	—
214	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
215	C₄×D₄	—	89	1	no	—
216	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
217	#117	—	2	1	no	—
218	#117	C₂×D₈₉	1	1	yes	characteristic
219	C₂×C₄⋊C₄	—	178	1	no	—
220	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
221	#107	—	89	1	no	—
222	#107	—	89	1	no	—
223	#106	—	89	1	no	—
224	C₄⋊C₈	—	89	1	no	—
225	#103	—	89	1	no	—
226	#103	—	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	#102	—	89	1	no	—
233	#102	—	89	1	no	—
234	#102	—	89	1	no	—
235	C₂²⋊C₈	—	89	1	no	—
236	C₂²⋊C₈	—	89	1	no	—
237	#97	—	89	1	no	—
238	#41	—	356	1	no	—
239	#41	—	356	1	no	—
240	#41	—	178	2	no	—
241	#41	—	178	2	no	—
242	C₂×Q₈	—	89	1	no	—
243	C₂×Q₈	—	89	1	no	—
244	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
245	C₂×D₄	—	89	1	no	—
246	C₂×D₄	—	178	1	no	—
247	C₂×D₄	—	178	1	no	—
248	C₂×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	178	1	no	—
251	C₂²×C₄	—	178	1	no	—
252	C₂²×C₄	—	178	1	no	—
253	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
254	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
255	Q₁₆	—	356	1	no	—
256	Q₁₆	—	356	1	no	—
257	SD₁₆	—	178	2	no	—
258	M₁₆	—	356	1	no	—
259	C₂×C₈	—	89	1	no	—
260	C₂×C₈	—	89	1	no	—
261	C₂×C₈	—	89	1	no	—
262	C₂×C₈	—	89	1	no	—
263	C₂×C₈	—	356	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	89	1	no	—
271	C₄⋊C₄	—	178	1	no	—
272	C₄⋊C₄	—	178	1	no	—
273	C₄⋊C₄	—	178	1	no	—
274	C₄⋊C₄	—	178	1	no	—
275	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
277	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
278	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	2	1	no	—
282	C₂²⋊C₄	—	178	1	no	—
283	C₂²⋊C₄	—	178	1	no	—
284	C₂²⋊C₄	—	178	1	no	—
285	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
286	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
287	C₄²	—	2	1	no	—
288	C₄²	—	2	1	no	—
289	C₄²	C₂²×D₈₉	1	1	yes	characteristic
290	C₂³	—	178	1	no	—
291	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
292	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	178	1	no	—
296	Q₈	—	178	1	no	—
297	Q₈	—	178	1	no	—
298	Q₈	—	178	1	no	—
299	D₄	—	4	1	no	—
300	D₄	—	89	2	no	—
301	D₄	—	356	2	no	—
302	D₄	—	356	2	no	—
303	D₄	—	178	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	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	—	4	2	no	—
313	C₂×C₄	—	356	2	no	—
314	C₂×C₄	—	356	2	no	—
315	C₂×C₄	—	356	2	no	—
316	C₂×C₄	—	178	1	no	—
317	C₂×C₄	—	178	1	no	—
318	C₂×C₄	—	178	1	no	—
319	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
320	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
321	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
322	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
324	C₈	—	178	1	no	—
325	C₈	—	178	1	no	—
326	C₈	—	178	1	no	—
327	C₈	—	178	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	356	1	no	—
335	C₂²	—	356	1	no	—
336	C₂²	—	178	2	no	—
337	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	356	1	no	—
347	C₄	—	356	1	no	—
348	C₄	—	356	1	no	—
349	C₄	#41×D₈₉	1	1	yes	characteristic
350	C₄	#15668058	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	4	1	no	—
353	C₂	—	356	2	no	—
354	C₂	#19152236	1	1	yes	characteristic
355	C₂	#19152396	1	1	yes	characteristic
356	C₂	#19152413	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#552×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152253	C₂
hypercenter	#552	D₈₉
hypocenter	C₈₉	#3315
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #552

Complements

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

N	H ≅ G/N	type
2. #19151968	C₂	semidirect
3. #19151939	C₂	semidirect
4. #19151933	C₂	semidirect
5. #19151895	C₂	semidirect
6. #19151854	C₂	semidirect
7. #19151837	C₂	semidirect
8. #19151808	C₂	semidirect
9. #19151758	C₂	semidirect
10. #19151758	C₂	semidirect
11. #19151696	C₂	semidirect
12. #19151670	C₂	semidirect
13. #19151597	C₂	semidirect
14. #19151592	C₂	semidirect
15. #552×C₈₉	C₂	semidirect
179. C₈₉	#3315	semidirect
180. #552	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152253	2
#19151968	2
#19151939	2
#19151933	2
#19151895	2
#19151854	2
#19151837	2
#19151808	2
#19151758	2
#19151696	2
#19151670	2
#19151597	2
#19151592	2
#552×C₈₉	2
#3315	89