Subgroups — #26032113

Order 11392.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₂²⋊D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152249	C₂	1	1	yes	maximal, characteristic
3	#19151947	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151928	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151927	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151881	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151830	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151827	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151802	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151738	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151737	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151683	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈×C₂²⋊D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#103×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	—	2	1	no	—
17	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	C₂×#13083561	C₂²	1	1	yes	characteristic
20	C₂×#13083561	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
22	#15668011	C₂²	1	1	yes	characteristic
23	#15668010	—	2	1	no	—
24	#15668010	C₂²	1	1	yes	characteristic
25	#15667997	—	2	1	no	—
26	#15667995	—	2	1	no	—
27	#15667991	—	2	1	no	—
28	#15667991	C₂²	1	1	yes	characteristic
29	#15667990	—	2	1	no	—
30	#15667988	—	2	1	no	—
31	#15667988	C₂²	1	1	yes	characteristic
32	#15667987	C₂²	1	1	yes	characteristic
33	#15667986	C₂²	1	1	yes	characteristic
34	Q₈×Dic₈₉	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	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
39	#15667944	C₂²	1	1	yes	characteristic
40	#15667943	C₂²	1	1	yes	characteristic
41	#15667943	C₂²	1	1	yes	characteristic
42	#15667941	C₂²	1	1	yes	characteristic
43	#15667937	C₂²	1	1	yes	characteristic
44	#15667936	C₂²	1	1	yes	characteristic
45	#15667936	C₂²	1	1	yes	characteristic
46	#15667935	C₂²	1	1	yes	characteristic
47	#15667935	C₂²	1	1	yes	characteristic
48	#15667934	C₂²	1	1	yes	characteristic
49	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
55	#103×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	#103×C₈₉	C₂²	1	1	yes	characteristic
59	#13083574	—	4	4	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	—	2	1	no	—
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	Q₈×D₈₉	—	2	2	no	—
64	Q₈×D₈₉	—	4	1	no	—
65	C₂×D₃₅₆	—	2	1	no	—
66	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₉	—	2	1	no	—
68	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
69	C₂×C₄×D₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	2	1	no	—
72	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
73	#13083563	—	4	1	no	—
74	#13083561	—	2	2	no	—
75	#13083561	—	4	1	no	—
76	C₂×#10623494	C₂³	1	1	yes	characteristic
77	C₂×#10623494	C₂³	1	1	yes	characteristic
78	C₈×D₈₉	—	4	1	no	—
79	C₄⋊Dic₈₉	—	2	1	no	—
80	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
82	#13083553	—	2	1	no	—
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	#13083553	—	2	1	no	—
86	#13083553	—	2	1	no	—
87	#13083553	—	2	1	no	—
88	#13083553	D₄	1	1	yes	characteristic
89	C₂²⋊Dic₈₉	—	2	1	no	—
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	D₄	1	1	yes	characteristic
96	C₄×Dic₈₉	—	2	1	no	—
97	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₈₉	—	2	1	no	—
99	C₂×Q₈×C₈₉	—	2	2	no	—
100	C₂×Q₈×C₈₉	—	4	1	no	—
101	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	—	2	1	no	—
104	C₂²×C₄×C₈₉	—	2	1	no	—
105	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₈₉	—	4	1	no	—
107	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₈₉	—	2	1	no	—
110	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
112	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₈₉	—	2	1	no	—
114	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₈₉	—	2	1	no	—
116	Q₈×C₈₉	—	2	1	no	—
117	Q₈×C₈₉	—	2	1	no	—
118	Q₈×C₈₉	—	4	2	no	—
119	Q₈×C₈₉	#41	1	2	yes	complemented
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	—	2	1	no	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₈₉	—	4	2	no	—
131	C₂²⋊D₈₉	SD₁₆	1	4	yes	complemented
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	—	2	1	no	—
134	C₂×Dic₈₉	—	2	1	no	—
135	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
138	D₃₅₆	—	2	1	no	—
139	D₃₅₆	—	2	1	no	—
140	D₃₅₆	—	4	2	no	—
141	C₄×D₈₉	—	2	1	no	—
142	C₄×D₈₉	—	2	1	no	—
143	C₄×D₈₉	—	4	2	no	—
144	C₄×D₈₉	—	4	2	no	—
145	C₄×D₈₉	—	4	2	no	—
146	Dic₁₇₈	—	2	1	no	—
147	Dic₁₇₈	—	2	1	no	—
148	Dic₁₇₈	—	4	2	no	—
149	Dic₁₇₈	—	4	2	no	—
150	C₈×C₈₉	—	2	1	no	—
151	C₈×C₈₉	—	2	1	no	—
152	#10623494	—	2	1	no	—
153	#10623494	—	2	1	no	—
154	C₂²×C₈₉	—	2	1	no	—
155	C₂²×C₈₉	—	2	1	no	—
156	C₂²×C₈₉	C₂×SD₁₆	1	2	yes	—
157	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×D₈₉	—	2	1	no	—
159	C₂×D₈₉	—	2	1	no	—
160	C₂×D₈₉	—	4	1	no	—
161	C₂×D₈₉	—	4	1	no	—
162	C₂×D₈₉	—	4	1	no	—
163	C₂×D₈₉	C₂×SD₁₆	1	2	yes	—
164	C₄×C₈₉	—	2	2	no	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	4	1	no	—
167	C₄×C₈₉	—	4	1	no	—
168	C₄×C₈₉	—	4	1	no	—
169	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
170	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
171	Dic₈₉	—	2	1	no	—
172	Dic₈₉	—	2	1	no	—
173	Dic₈₉	—	4	1	no	—
174	Dic₈₉	—	4	1	no	—
175	Dic₈₉	—	4	1	no	—
176	Dic₈₉	C₂×SD₁₆	1	2	yes	—
177	C₂×C₈₉	—	2	2	no	—
178	C₂×C₈₉	#546	1	1	yes	characteristic
179	C₂×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
180	C₂×C₈₉	#578	1	1	yes	characteristic
181	D₈₉	—	2	2	no	—
182	D₈₉	—	8	1	no	—
183	#3298	—	89	1	no	maximal, Sylow
184	C₈₉	#3298	1	1	yes	characteristic, complemented, Sylow
185	D₄×Q₈	—	89	1	no	—
186	#547	—	89	1	no	—
187	#487	—	89	1	no	—
188	#483	—	89	1	no	—
189	#483	—	89	1	no	—
190	#475	—	89	1	no	—
191	#465	—	89	1	no	—
192	#465	—	89	1	no	—
193	#461	—	89	1	no	—
194	#448	—	89	1	no	—
195	#448	—	89	1	no	—
196	C₄×SD₁₆	—	89	1	no	—
197	C₈×D₄	—	89	1	no	—
198	C₂×#103	—	89	1	no	—
199	C₂×#103	D₈₉	1	1	yes	characteristic, complemented
200	C₂×#41	—	178	1	no	—
201	C₂²×Q₈	—	89	1	no	—
202	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
203	C₂×SD₁₆	—	89	1	no	—
204	C₂×SD₁₆	—	89	1	no	—
205	C₂×SD₁₆	—	89	1	no	—
206	C₂²×C₈	—	89	1	no	—
207	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
208	C₄⋊Q₈	—	89	1	no	—
209	C₄⋊Q₈	—	89	1	no	—
210	C₄⋊Q₈	—	178	1	no	—
211	#123	—	178	1	no	—
212	#123	—	178	1	no	—
213	C₂²⋊Q₈	—	89	1	no	—
214	C₂²⋊Q₈	—	89	1	no	—
215	C₂²⋊Q₈	—	178	1	no	—
216	C₂²⋊Q₈	—	178	1	no	—
217	C₂²⋊Q₈	—	178	1	no	—
218	#121	—	89	1	no	—
219	#121	—	89	1	no	—
220	C₄×Q₈	—	89	1	no	—
221	C₄×D₄	—	89	1	no	—
222	C₄×D₄	—	89	1	no	—
223	C₄×D₄	—	178	1	no	—
224	C₂×C₄⋊C₄	—	89	1	no	—
225	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
226	#106	—	89	1	no	—
227	#106	—	89	1	no	—
228	#106	—	89	1	no	—
229	C₄⋊C₈	—	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	—	89	1	no	—
235	#103	C₂×D₈₉	1	1	yes	characteristic
236	#103	C₂×D₈₉	1	1	yes	characteristic
237	#103	C₂×D₈₉	1	1	yes	characteristic
238	#103	C₂×D₈₉	1	1	yes	characteristic
239	#102	—	89	1	no	—
240	C₂²⋊C₈	—	89	1	no	—
241	C₂²⋊C₈	—	89	1	no	—
242	C₄×C₈	—	89	1	no	—
243	#41	—	356	4	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	89	1	no	—
248	C₂×Q₈	—	356	1	no	—
249	C₂×Q₈	—	178	1	no	—
250	C₂×Q₈	—	178	1	no	—
251	C₂×Q₈	—	178	2	no	—
252	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
253	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
254	C₂×D₄	—	89	1	no	—
255	C₂×D₄	—	89	1	no	—
256	C₂×D₄	—	178	1	no	—
257	C₂×D₄	—	178	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	89	1	no	—
261	C₂²×C₄	—	178	1	no	—
262	C₂²×C₄	—	178	1	no	—
263	C₂²×C₄	—	178	1	no	—
264	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
265	SD₁₆	—	356	1	no	—
266	SD₁₆	—	356	1	no	—
267	SD₁₆	—	178	2	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	89	1	no	—
270	C₂×C₈	—	89	1	no	—
271	C₂×C₈	—	356	1	no	—
272	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
273	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	89	1	no	—
276	C₄⋊C₄	—	89	1	no	—
277	C₄⋊C₄	—	89	1	no	—
278	C₄⋊C₄	—	178	1	no	—
279	C₄⋊C₄	—	178	1	no	—
280	C₄⋊C₄	—	178	1	no	—
281	C₄⋊C₄	—	178	1	no	—
282	C₄⋊C₄	—	178	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	C₂²⋊C₄	—	89	1	no	—
288	C₂²⋊C₄	—	89	1	no	—
289	C₂²⋊C₄	—	178	1	no	—
290	C₂²⋊C₄	—	178	1	no	—
291	C₂²⋊C₄	—	178	1	no	—
292	C₂²⋊C₄	—	178	1	no	—
293	C₂²⋊C₄	—	178	1	no	—
294	C₄²	—	89	1	no	—
295	C₄²	—	178	1	no	—
296	C₂³	—	89	1	no	—
297	C₂³	—	178	1	no	—
298	C₂³	D₄×D₈₉	1	1	yes	characteristic
299	Q₈	—	2	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	356	2	no	—
304	Q₈	—	356	2	no	—
305	Q₈	—	178	1	no	—
306	Q₈	—	178	1	no	—
307	Q₈	#13083574	1	2	yes	complemented
308	D₄	—	89	4	no	—
309	D₄	—	356	2	no	—
310	D₄	—	356	2	no	—
311	D₄	—	178	1	no	—
312	D₄	—	178	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	89	1	no	—
322	C₂×C₄	—	89	1	no	—
323	C₂×C₄	—	89	1	no	—
324	C₂×C₄	—	356	2	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	356	2	no	—
327	C₂×C₄	—	178	1	no	—
328	C₂×C₄	—	178	1	no	—
329	C₂×C₄	—	178	1	no	—
330	C₂×C₄	—	178	1	no	—
331	C₂×C₄	—	178	1	no	—
332	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	178	1	no	—
337	C₈	—	178	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	89	2	no	—
341	C₂²	—	356	1	no	—
342	C₂²	—	356	1	no	—
343	C₂²	—	356	1	no	—
344	C₂²	—	178	1	no	—
345	C₂²	—	178	1	no	—
346	C₂²	SD₁₆×D₈₉	1	2	yes	—
347	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	89	2	no	—
354	C₄	—	356	1	no	—
355	C₄	—	356	1	no	—
356	C₄	—	356	1	no	—
357	C₄	—	178	1	no	—
358	C₄	—	178	1	no	—
359	C₄	C₂×#13083574	1	1	yes	characteristic
360	C₄	#15668057	1	1	yes	characteristic
361	C₂	—	2	2	no	—
362	C₂	—	178	2	no	—
363	C₂	—	712	1	no	—
364	C₂	#19152238	1	1	yes	characteristic
365	C₂	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
366	C₂	#19152411	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×#103×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152249	C₂
hypercenter	C₂×#103	D₈₉
hypocenter	C₈₉	#3298
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#103

Complements

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

N	H ≅ G/N	type
1. Q₈×C₂²⋊D₈₉	C₂	semidirect
3. #19151947	C₂	semidirect
4. #19151928	C₂	semidirect
5. #19151927	C₂	semidirect
6. #19151881	C₂	semidirect
7. #19151830	C₂	semidirect
8. #19151827	C₂	semidirect
9. #19151802	C₂	semidirect
10. #19151738	C₂	semidirect
11. #19151737	C₂	semidirect
12. #19151683	C₂	semidirect
13. C₈×C₂²⋊D₈₉	C₂	semidirect
14. #103×D₈₉	C₂	semidirect
15. C₂×#103×C₈₉	C₂	semidirect
119. Q₈×C₈₉	#41	semidirect
131. C₂²⋊D₈₉	SD₁₆	semidirect
184. C₈₉	#3298	semidirect
199. C₂×#103	D₈₉	semidirect
307. Q₈	#13083574	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₂²⋊D₈₉	2
#19152249	2
#19151947	2
#19151928	2
#19151927	2
#19151881	2
#19151830	2
#19151827	2
#19151802	2
#19151738	2
#19151737	2
#19151683	2
C₈×C₂²⋊D₈₉	2
#103×D₈₉	2
C₂×#103×C₈₉	2
#3298	89