Subgroups — #26032237

Order 11392.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152257	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151960	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151933	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151911	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151890	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151842	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151832	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151813	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151741	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151736	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151698	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	C₂×#15667935	C₂	1	1	yes	maximal, characteristic, complemented
15	#547×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
18	C₂×#13083562	C₂²	1	1	yes	characteristic
19	C₂×#13083562	C₂²	1	1	yes	characteristic
20	C₂×#13083559	C₂²	1	1	yes	characteristic
21	C₂×#13083558	C₂²	1	1	yes	characteristic
22	C₂²×#10623494	C₂²	1	1	yes	characteristic
23	#15668000	C₂²	1	1	yes	characteristic
24	#15667998	—	2	1	no	—
25	#15667991	C₂²	1	1	yes	characteristic
26	#15667990	—	2	1	no	—
27	#15667986	C₂²	1	1	yes	characteristic
28	#15667984	—	2	1	no	—
29	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
30	#15667972	—	2	1	no	—
31	#15667945	C₂²	1	1	yes	characteristic
32	#15667945	C₂²	1	1	yes	characteristic
33	#15667945	C₂²	1	1	yes	characteristic
34	C₄⋊#10623494	C₂²	1	1	yes	characteristic
35	#15667935	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	#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	C₂×#41×C₈₉	—	2	1	no	—
49	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
50	#123×C₈₉	—	2	1	no	—
51	#123×C₈₉	—	2	1	no	—
52	C₂²⋊Q₈×C₈₉	—	2	1	no	—
53	#121×C₈₉	C₂²	1	1	yes	characteristic
54	#121×C₈₉	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
56	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
59	#13083574	—	2	2	no	—
60	#13083574	—	4	1	no	—
61	C₂×Dic₁₇₈	—	2	2	no	—
62	C₂×Dic₁₇₈	—	2	1	no	—
63	C₂×Dic₁₇₈	—	4	1	no	—
64	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
65	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
66	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₉	—	2	1	no	—
68	C₂×C₂²⋊D₈₉	—	2	1	no	—
69	C₂²×Dic₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	#13083562	—	2	2	no	—
72	#13083562	—	4	1	no	—
73	#13083559	—	4	1	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	C₂³	1	1	yes	characteristic
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	C₂²⋊Dic₈₉	—	2	1	no	—
84	#13083551	—	2	1	no	—
85	#13083551	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	C₄×Dic₈₉	—	2	1	no	—
88	#41×C₈₉	—	4	4	no	—
89	C₂×Q₈×C₈₉	—	2	1	no	—
90	C₂×D₄×C₈₉	—	2	1	no	—
91	C₂×D₄×C₈₉	—	2	1	no	—
92	C₂×D₄×C₈₉	D₄	1	1	yes	characteristic
93	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₈₉	—	2	1	no	—
95	C₂²×C₄×C₈₉	—	2	1	no	—
96	C₂²×C₄×C₈₉	—	2	1	no	—
97	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₉	—	2	1	no	—
100	C₄⋊C₄×C₈₉	—	2	1	no	—
101	C₄⋊C₄×C₈₉	—	2	1	no	—
102	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
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	C₂²⋊C₄×C₈₉	—	2	1	no	—
108	C₂²⋊C₄×C₈₉	—	2	1	no	—
109	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
110	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
111	C₄²×C₈₉	C₂³	1	1	yes	characteristic
112	C₂³×C₈₉	—	2	1	no	—
113	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₈₉	—	2	1	no	—
116	Q₈×C₈₉	—	4	2	no	—
117	D₄×C₈₉	—	2	2	no	—
118	D₄×C₈₉	—	2	1	no	—
119	D₄×C₈₉	—	2	1	no	—
120	D₄×C₈₉	—	4	2	no	—
121	D₄×C₈₉	—	4	2	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	—	2	1	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	4	2	no	—
131	C₂×C₄×C₈₉	—	4	2	no	—
132	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
137	C₂²⋊D₈₉	—	4	2	no	—
138	C₂²⋊D₈₉	—	4	2	no	—
139	C₂×Dic₈₉	—	2	1	no	—
140	C₂×Dic₈₉	—	2	1	no	—
141	C₂×Dic₈₉	—	2	1	no	—
142	C₂×Dic₈₉	—	4	2	no	—
143	D₃₅₆	—	2	1	no	—
144	D₃₅₆	—	2	1	no	—
145	C₄×D₈₉	—	4	2	no	—
146	Dic₁₇₈	—	2	1	no	—
147	Dic₁₇₈	—	2	1	no	—
148	Dic₁₇₈	—	2	1	no	—
149	Dic₁₇₈	—	4	2	no	—
150	Dic₁₇₈	#41	1	2	yes	complemented
151	#10623494	—	2	1	no	—
152	#10623494	—	2	1	no	—
153	#10623494	—	2	1	no	—
154	#10623494	—	2	1	no	—
155	C₂²×C₈₉	—	2	1	no	—
156	C₂²×C₈₉	—	2	1	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	—	2	1	no	—
159	C₂²×C₈₉	—	2	1	no	—
160	C₂²×C₈₉	—	4	1	no	—
161	C₂²×C₈₉	—	4	1	no	—
162	C₂²×C₈₉	—	4	1	no	—
163	C₂²×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
164	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
165	C₂×D₈₉	—	4	1	no	—
166	C₂×D₈₉	—	4	1	no	—
167	C₂×D₈₉	—	4	1	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	—	2	1	no	—
171	C₄×C₈₉	—	2	1	no	—
172	C₄×C₈₉	—	4	1	no	—
173	C₄×C₈₉	—	4	1	no	—
174	C₄×C₈₉	—	4	1	no	—
175	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
176	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
177	Dic₈₉	—	2	2	no	—
178	Dic₈₉	—	4	1	no	—
179	Dic₈₉	—	4	1	no	—
180	C₂×C₈₉	—	2	2	no	—
181	C₂×C₈₉	—	4	1	no	—
182	C₂×C₈₉	—	8	1	no	—
183	C₂×C₈₉	#546	1	1	yes	characteristic
184	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₂×C₈₉	#578	1	1	yes	characteristic
186	D₈₉	—	8	1	no	—
187	#3308	—	89	1	no	maximal, Sylow
188	C₈₉	#3308	1	1	yes	characteristic, complemented, Sylow
189	#548	—	89	1	no	—
190	#547	D₈₉	1	1	yes	characteristic, complemented
191	#489	—	89	1	no	—
192	#484	—	89	1	no	—
193	#480	—	89	1	no	—
194	#477	—	89	1	no	—
195	#468	—	89	1	no	—
196	#466	—	89	1	no	—
197	#463	—	89	1	no	—
198	#449	—	89	1	no	—
199	#448	—	89	1	no	—
200	#440	—	89	1	no	—
201	#435	—	89	1	no	—
202	#417	—	89	1	no	—
203	C₂×#103	—	89	1	no	—
204	C₂×#41	—	2	1	no	—
205	C₂×#41	—	89	1	no	—
206	C₂²×Q₈	—	89	1	no	—
207	C₂×SD₁₆	—	89	1	no	—
208	C₂×SD₁₆	—	89	1	no	—
209	C₂×D₈	—	89	1	no	—
210	C₂×M₁₆	—	89	1	no	—
211	C₂²×C₈	—	89	1	no	—
212	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
213	#124	—	89	1	no	—
214	#124	—	178	1	no	—
215	#123	—	2	1	no	—
216	#123	—	2	1	no	—
217	C₂²⋊Q₈	—	2	1	no	—
218	C₂²⋊Q₈	—	89	1	no	—
219	C₂²⋊Q₈	—	178	1	no	—
220	#121	—	89	1	no	—
221	#121	—	178	1	no	—
222	#121	C₂×D₈₉	1	1	yes	characteristic
223	#121	C₂×D₈₉	1	1	yes	characteristic
224	C₄×Q₈	—	89	1	no	—
225	C₄×D₄	—	178	1	no	—
226	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
227	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
228	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
229	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
230	#107	—	89	1	no	—
231	#107	—	89	1	no	—
232	#107	—	89	1	no	—
233	C₄⋊C₈	—	89	1	no	—
234	#103	—	89	1	no	—
235	#103	—	89	1	no	—
236	#103	—	89	1	no	—
237	#103	—	89	1	no	—
238	#103	—	89	1	no	—
239	#103	—	89	1	no	—
240	#102	—	89	1	no	—
241	#102	—	89	1	no	—
242	#102	—	89	1	no	—
243	#102	—	89	1	no	—
244	C₂²⋊C₈	—	89	1	no	—
245	C₂²⋊C₈	—	89	1	no	—
246	#97	—	89	1	no	—
247	#41	—	4	4	no	—
248	#41	—	356	1	no	—
249	#41	—	178	2	no	—
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	—	89	1	no	—
252	C₂×Q₈	—	89	1	no	—
253	C₂×Q₈	—	356	1	no	—
254	C₂×Q₈	—	178	2	no	—
255	C₂×Q₈	—	178	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	89	1	no	—
259	C₂×D₄	—	178	1	no	—
260	C₂×D₄	—	178	1	no	—
261	C₂×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
262	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	2	1	no	—
266	C₂²×C₄	—	178	1	no	—
267	C₂²×C₄	—	178	1	no	—
268	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
269	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
270	SD₁₆	—	356	1	no	—
271	SD₁₆	—	178	2	no	—
272	D₈	—	356	1	no	—
273	M₁₆	—	356	1	no	—
274	C₂×C₈	—	89	1	no	—
275	C₂×C₈	—	89	1	no	—
276	C₂×C₈	—	89	1	no	—
277	C₂×C₈	—	89	1	no	—
278	C₂×C₈	—	356	1	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	89	1	no	—
283	C₄⋊C₄	—	178	1	no	—
284	C₄⋊C₄	—	178	1	no	—
285	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
286	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
287	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
288	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
289	C₂²⋊C₄	—	2	1	no	—
290	C₂²⋊C₄	—	2	1	no	—
291	C₂²⋊C₄	—	2	1	no	—
292	C₂²⋊C₄	—	178	1	no	—
293	C₂²⋊C₄	—	178	1	no	—
294	C₂²⋊C₄	—	178	1	no	—
295	C₂²⋊C₄	—	178	1	no	—
296	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
297	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
298	C₄²	—	178	1	no	—
299	C₄²	C₂²×D₈₉	1	1	yes	characteristic
300	C₂³	—	2	1	no	—
301	C₂³	—	178	1	no	—
302	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
303	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
304	Q₈	—	4	2	no	—
305	Q₈	—	89	2	no	—
306	Q₈	—	356	2	no	—
307	Q₈	—	178	1	no	—
308	Q₈	—	178	1	no	—
309	Q₈	—	178	1	no	—
310	D₄	—	2	2	no	—
311	D₄	—	2	1	no	—
312	D₄	—	2	1	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	356	2	no	—
316	D₄	—	356	2	no	—
317	D₄	—	178	1	no	—
318	D₄	—	178	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	356	2	no	—
330	C₂×C₄	—	356	2	no	—
331	C₂×C₄	—	178	1	no	—
332	C₂×C₄	—	178	1	no	—
333	C₂×C₄	—	178	1	no	—
334	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
335	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
336	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
337	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
338	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
339	C₈	—	178	1	no	—
340	C₈	—	178	1	no	—
341	C₈	—	178	1	no	—
342	C₈	—	178	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	356	1	no	—
352	C₂²	—	356	1	no	—
353	C₂²	—	356	1	no	—
354	C₂²	#15668038	1	2	yes	—
355	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	4	1	no	—
361	C₄	—	4	1	no	—
362	C₄	—	4	1	no	—
363	C₄	—	356	1	no	—
364	C₄	—	356	1	no	—
365	C₄	—	178	2	no	—
366	C₄	#41×D₈₉	1	1	yes	characteristic
367	C₄	#15668058	1	1	yes	characteristic
368	C₂	—	2	2	no	—
369	C₂	—	4	1	no	—
370	C₂	—	8	1	no	—
371	C₂	—	712	1	no	—
372	C₂	#19152236	1	1	yes	characteristic
373	C₂	C₂×#15668038	1	1	yes	characteristic
374	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	#547×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#547	D₈₉
hypocenter	C₈₉	#3308
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #547

Complements

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

N	H ≅ G/N	type
1. #19152257	C₂	semidirect
2. #19151960	C₂	semidirect
3. #19151933	C₂	semidirect
4. #19151911	C₂	semidirect
5. #19151890	C₂	semidirect
6. #19151842	C₂	semidirect
7. #19151832	C₂	semidirect
8. #19151813	C₂	semidirect
9. #19151741	C₂	semidirect
10. #19151736	C₂	semidirect
11. #19151698	C₂	semidirect
12. #19151670	C₂	semidirect
13. #19151597	C₂	semidirect
14. C₂×#15667935	C₂	semidirect
15. #547×C₈₉	C₂	semidirect
150. Dic₁₇₈	#41	semidirect
188. C₈₉	#3308	semidirect
190. #547	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152257	2
#19151960	2
#19151933	2
#19151911	2
#19151890	2
#19151842	2
#19151832	2
#19151813	2
#19151741	2
#19151736	2
#19151698	2
#19151670	2
#19151597	2
C₂×#15667935	2
#547×C₈₉	2
#3308	89