Subgroups — #26031815

Order 11392.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668041	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13083564	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₂²⋊D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#19152254	C₂	1	1	yes	maximal, characteristic
5	#19151989	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151855	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151825	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151818	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151807	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151757	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151756	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151755	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151702	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151666	C₂	1	1	yes	maximal, characteristic, complemented
15	#448×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083573	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
19	#15668041	—	2	4	no	—
20	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
22	C₂×#13083564	—	2	4	no	—
23	C₂×#13083564	C₂²	1	1	yes	characteristic
24	C₂×#13083564	C₂²	1	1	yes	characteristic
25	C₂×#13083564	C₂²	1	1	yes	characteristic
26	C₂×#13083564	C₂²	1	1	yes	characteristic
27	C₂×#13083562	C₂²	1	1	yes	characteristic
28	C₂×#13083557	C₂²	1	1	yes	characteristic
29	C₂²×#10623494	C₂²	1	1	yes	characteristic
30	#15668010	—	2	1	no	—
31	#15668010	C₂²	1	1	yes	characteristic
32	#15667999	C₂²	1	1	yes	characteristic
33	#15667998	—	2	1	no	—
34	#15667992	C₂²	1	1	yes	characteristic
35	#15667991	—	2	1	no	—
36	#15667991	C₂²	1	1	yes	characteristic
37	#15667989	—	2	1	no	—
38	#15667988	—	2	1	no	—
39	#15667988	C₂²	1	1	yes	characteristic
40	#15667984	—	2	1	no	—
41	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
42	#15667977	C₂²	1	1	yes	characteristic
43	D₄×Dic₈₉	—	2	1	no	—
44	C₄×C₂²⋊D₈₉	—	2	1	no	—
45	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
46	#15667943	C₂²	1	1	yes	characteristic
47	#15667941	C₂²	1	1	yes	characteristic
48	#15667937	C₂²	1	1	yes	characteristic
49	#15667937	C₂²	1	1	yes	characteristic
50	#15667935	C₂²	1	1	yes	characteristic
51	#15667933	C₂²	1	1	yes	characteristic
52	#15667924	C₂²	1	1	yes	characteristic
53	#15667922	C₂²	1	1	yes	characteristic
54	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
55	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
56	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
57	#121×C₈₉	C₂²	1	1	yes	characteristic
58	#103×C₈₉	C₂²	1	1	yes	characteristic
59	#103×C₈₉	C₂²	1	1	yes	characteristic
60	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
61	#13083573	—	2	2	no	—
62	#13083573	—	4	1	no	—
63	C₂×Dic₁₇₈	—	2	2	no	—
64	C₂×Dic₁₇₈	—	2	1	no	—
65	C₂×Dic₁₇₈	—	2	1	no	—
66	C₂×Dic₁₇₈	—	4	1	no	—
67	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
68	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
69	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
70	Q₈×D₈₉	—	2	2	no	—
71	Q₈×D₈₉	—	4	1	no	—
72	C₂×C₂²⋊D₈₉	—	2	1	no	—
73	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
74	C₂²×Dic₈₉	—	2	1	no	—
75	C₂²×Dic₈₉	—	2	1	no	—
76	C₂×C₄×D₈₉	—	2	1	no	—
77	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
78	Dic₃₅₆	—	2	2	no	—
79	Dic₃₅₆	—	4	1	no	—
80	#13083564	—	2	2	no	—
81	#13083564	—	2	2	no	—
82	#13083564	—	2	2	no	—
83	#13083564	D₄	1	4	yes	complemented
84	#13083562	—	2	2	no	—
85	#13083557	—	2	2	no	—
86	C₂×#10623494	—	2	2	no	—
87	C₂×#10623494	C₂³	1	1	yes	characteristic
88	C₂×#10623494	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₈₉	—	2	1	no	—
90	C₄⋊Dic₈₉	—	2	1	no	—
91	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
92	#13083553	—	2	1	no	—
93	#13083553	—	2	1	no	—
94	#13083553	—	2	1	no	—
95	#13083553	—	2	1	no	—
96	#13083553	—	2	1	no	—
97	#13083553	D₄	1	1	yes	characteristic
98	C₂²⋊Dic₈₉	—	2	1	no	—
99	C₂²⋊Dic₈₉	—	2	1	no	—
100	C₂²⋊Dic₈₉	—	2	1	no	—
101	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
102	#13083551	—	2	1	no	—
103	#13083551	—	2	1	no	—
104	#13083551	D₄	1	1	yes	characteristic
105	C₄×Dic₈₉	—	2	1	no	—
106	C₄×Dic₈₉	—	2	1	no	—
107	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
108	C₂×Q₈×C₈₉	—	2	1	no	—
109	C₂×Q₈×C₈₉	—	2	2	no	—
110	C₂×Q₈×C₈₉	—	4	1	no	—
111	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
112	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
113	C₂×D₄×C₈₉	—	2	1	no	—
114	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
115	C₂²×C₄×C₈₉	—	2	1	no	—
116	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
117	SD₁₆×C₈₉	—	2	2	no	—
118	SD₁₆×C₈₉	—	4	1	no	—
119	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
121	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
122	C₂²⋊C₄×C₈₉	—	2	1	no	—
123	C₂³×C₈₉	—	2	1	no	—
124	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
126	Q₈×C₈₉	—	2	1	no	—
127	Q₈×C₈₉	—	2	1	no	—
128	Q₈×C₈₉	—	2	1	no	—
129	Q₈×C₈₉	—	4	2	no	—
130	Q₈×C₈₉	C₂×D₄	1	2	yes	—
131	D₄×C₈₉	—	2	1	no	—
132	D₄×C₈₉	—	2	1	no	—
133	D₄×C₈₉	—	4	2	no	—
134	C₂×C₄×C₈₉	—	2	1	no	—
135	C₂×C₄×C₈₉	—	2	1	no	—
136	C₂×C₄×C₈₉	—	2	1	no	—
137	C₂×C₄×C₈₉	—	2	1	no	—
138	C₂×C₄×C₈₉	—	2	1	no	—
139	C₂×C₄×C₈₉	—	4	2	no	—
140	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
142	C₂²⋊D₈₉	—	2	2	no	—
143	C₂²⋊D₈₉	—	4	2	no	—
144	C₂×Dic₈₉	—	2	1	no	—
145	C₂×Dic₈₉	—	2	1	no	—
146	C₂×Dic₈₉	—	2	1	no	—
147	C₂×Dic₈₉	—	2	1	no	—
148	C₂×Dic₈₉	—	4	2	no	—
149	C₂×Dic₈₉	—	4	2	no	—
150	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
151	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
152	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
153	C₄×D₈₉	—	2	1	no	—
154	C₄×D₈₉	—	2	1	no	—
155	C₄×D₈₉	—	4	2	no	—
156	Dic₁₇₈	—	2	1	no	—
157	Dic₁₇₈	—	2	1	no	—
158	Dic₁₇₈	—	2	1	no	—
159	Dic₁₇₈	—	2	1	no	—
160	Dic₁₇₈	—	2	1	no	—
161	Dic₁₇₈	—	4	2	no	—
162	Dic₁₇₈	—	4	2	no	—
163	Dic₁₇₈	C₂×D₄	1	2	yes	—
164	C₈×C₈₉	—	2	1	no	—
165	C₈×C₈₉	—	2	1	no	—
166	#10623494	—	2	1	no	—
167	#10623494	C₂×D₄	1	2	yes	—
168	C₂²×C₈₉	—	2	1	no	—
169	C₂²×C₈₉	—	2	1	no	—
170	C₂²×C₈₉	—	4	1	no	—
171	C₂²×C₈₉	—	4	1	no	—
172	C₂²×C₈₉	—	4	1	no	—
173	C₂²×C₈₉	#137	1	2	yes	complemented
174	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
175	C₂×D₈₉	—	2	1	no	—
176	C₂×D₈₉	—	2	1	no	—
177	C₂×D₈₉	—	2	1	no	—
178	C₄×C₈₉	—	2	1	no	—
179	C₄×C₈₉	—	2	2	no	—
180	C₄×C₈₉	—	4	1	no	—
181	C₄×C₈₉	—	4	1	no	—
182	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
183	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
184	Dic₈₉	—	2	2	no	—
185	Dic₈₉	—	2	1	no	—
186	Dic₈₉	—	2	1	no	—
187	Dic₈₉	—	2	1	no	—
188	Dic₈₉	—	4	1	no	—
189	Dic₈₉	—	4	1	no	—
190	Dic₈₉	—	4	1	no	—
191	C₂×C₈₉	—	2	2	no	—
192	C₂×C₈₉	—	8	1	no	—
193	C₂×C₈₉	D₄²	1	1	yes	characteristic
194	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
195	C₂×C₈₉	#578	1	1	yes	characteristic
196	D₈₉	—	4	1	no	—
197	#3270	—	89	1	no	maximal, Sylow
198	C₈₉	#3270	1	1	yes	characteristic, complemented, Sylow
199	C₂×#137	—	89	1	no	—
200	C₂²×Q₁₆	—	89	1	no	—
201	D₄×Q₈	—	89	1	no	—
202	#548	—	89	1	no	—
203	#497	—	89	1	no	—
204	#470	—	89	1	no	—
205	#465	—	89	1	no	—
206	#464	—	89	1	no	—
207	#462	—	89	1	no	—
208	#452	—	89	1	no	—
209	#451	—	89	1	no	—
210	#451	—	89	1	no	—
211	#448	D₈₉	1	1	yes	characteristic, complemented
212	#441	—	89	1	no	—
213	#435	—	89	1	no	—
214	C₂×#41	—	89	1	no	—
215	C₂²×Q₈	—	89	1	no	—
216	C₂²×Q₈	—	89	1	no	—
217	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
218	#137	—	178	4	no	—
219	C₂×Q₁₆	—	89	1	no	—
220	C₂×Q₁₆	—	89	1	no	—
221	C₂×Q₁₆	—	89	1	no	—
222	C₂×Q₁₆	—	89	1	no	—
223	C₂×Q₁₆	—	89	1	no	—
224	C₂×Q₁₆	—	89	1	no	—
225	C₂×Q₁₆	—	178	4	no	—
226	C₂×SD₁₆	—	89	1	no	—
227	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
228	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
229	C₂×M₁₆	—	89	1	no	—
230	C₂²×C₈	—	89	1	no	—
231	C₄⋊Q₈	—	89	1	no	—
232	C₄⋊Q₈	—	178	1	no	—
233	#124	—	89	1	no	—
234	#124	—	178	1	no	—
235	C₂²⋊Q₈	—	89	1	no	—
236	C₂²⋊Q₈	—	89	1	no	—
237	C₂²⋊Q₈	—	89	1	no	—
238	C₂²⋊Q₈	—	178	1	no	—
239	C₂²⋊Q₈	—	178	1	no	—
240	C₂²⋊Q₈	—	178	1	no	—
241	#121	—	178	1	no	—
242	#121	C₂×D₈₉	1	1	yes	characteristic
243	C₄×Q₈	—	89	1	no	—
244	C₄×Q₈	—	89	1	no	—
245	C₄×D₄	—	89	1	no	—
246	C₄×D₄	—	178	1	no	—
247	C₄×D₄	—	178	1	no	—
248	#106	—	89	1	no	—
249	C₄⋊C₈	—	89	1	no	—
250	#103	—	89	1	no	—
251	#103	—	89	1	no	—
252	#103	—	89	1	no	—
253	#103	C₂×D₈₉	1	1	yes	characteristic
254	#103	C₂×D₈₉	1	1	yes	characteristic
255	#102	—	89	1	no	—
256	C₂²⋊C₈	—	89	1	no	—
257	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
258	#97	—	89	1	no	—
259	#41	—	356	1	no	—
260	#41	—	178	2	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×Q₈	—	89	1	no	—
265	C₂×Q₈	—	89	1	no	—
266	C₂×Q₈	—	89	1	no	—
267	C₂×Q₈	—	356	1	no	—
268	C₂×Q₈	—	356	1	no	—
269	C₂×Q₈	—	178	1	no	—
270	C₂×Q₈	—	178	2	no	—
271	C₂×Q₈	—	178	2	no	—
272	C₂×Q₈	—	178	1	no	—
273	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
274	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
275	C₂×D₄	—	2	1	no	—
276	C₂×D₄	—	89	1	no	—
277	C₂×D₄	—	178	1	no	—
278	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	89	1	no	—
281	C₂²×C₄	—	178	1	no	—
282	C₂²×C₄	—	178	1	no	—
283	C₂²×C₄	—	178	1	no	—
284	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
285	Q₁₆	—	89	4	no	—
286	Q₁₆	—	356	1	no	—
287	Q₁₆	—	178	2	no	—
288	Q₁₆	—	178	2	no	—
289	Q₁₆	—	178	2	no	—
290	Q₁₆	—	178	2	no	—
291	SD₁₆	—	2	2	no	—
292	SD₁₆	—	4	1	no	—
293	SD₁₆	—	178	2	no	—
294	M₁₆	—	178	2	no	—
295	C₂×C₈	—	89	1	no	—
296	C₂×C₈	—	89	1	no	—
297	C₂×C₈	—	178	2	no	—
298	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
299	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
300	C₄⋊C₄	—	89	1	no	—
301	C₄⋊C₄	—	89	1	no	—
302	C₄⋊C₄	—	178	1	no	—
303	C₄⋊C₄	—	178	1	no	—
304	C₄⋊C₄	—	178	1	no	—
305	C₄⋊C₄	—	178	1	no	—
306	C₄⋊C₄	—	178	1	no	—
307	C₄⋊C₄	—	178	1	no	—
308	C₄⋊C₄	—	178	1	no	—
309	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
310	C₂²⋊C₄	—	2	1	no	—
311	C₂²⋊C₄	—	89	1	no	—
312	C₂²⋊C₄	—	89	1	no	—
313	C₂²⋊C₄	—	178	1	no	—
314	C₂²⋊C₄	—	178	1	no	—
315	C₂²⋊C₄	—	178	1	no	—
316	C₂²⋊C₄	—	178	1	no	—
317	C₂²⋊C₄	—	178	1	no	—
318	C₄²	—	89	1	no	—
319	C₄²	—	178	1	no	—
320	C₄²	—	178	1	no	—
321	C₂³	—	2	1	no	—
322	C₂³	—	89	1	no	—
323	C₂³	D₄×D₈₉	1	1	yes	characteristic
324	Q₈	—	2	1	no	—
325	Q₈	—	2	1	no	—
326	Q₈	—	2	1	no	—
327	Q₈	—	4	2	no	—
328	Q₈	—	89	2	no	—
329	Q₈	—	356	2	no	—
330	Q₈	—	356	2	no	—
331	Q₈	—	178	1	no	—
332	Q₈	—	178	1	no	—
333	Q₈	—	178	1	no	—
334	Q₈	—	178	1	no	—
335	Q₈	—	178	1	no	—
336	Q₈	D₄×D₈₉	1	2	yes	—
337	D₄	—	2	1	no	—
338	D₄	—	2	1	no	—
339	D₄	—	4	2	no	—
340	D₄	—	356	2	no	—
341	D₄	—	178	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	89	1	no	—
349	C₂×C₄	—	89	1	no	—
350	C₂×C₄	—	89	1	no	—
351	C₂×C₄	—	356	2	no	—
352	C₂×C₄	—	356	2	no	—
353	C₂×C₄	—	356	2	no	—
354	C₂×C₄	—	178	1	no	—
355	C₂×C₄	—	178	1	no	—
356	C₂×C₄	—	178	1	no	—
357	C₂×C₄	—	178	1	no	—
358	C₂×C₄	—	178	1	no	—
359	C₂×C₄	—	178	1	no	—
360	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
361	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
362	C₈	—	2	1	no	—
363	C₈	—	2	1	no	—
364	C₈	—	89	2	no	—
365	C₈	—	178	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	4	1	no	—
370	C₂²	—	4	1	no	—
371	C₂²	—	178	1	no	—
372	C₂²	—	178	1	no	—
373	C₂²	—	178	1	no	—
374	C₂²	#15668041	1	2	yes	complemented
375	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	2	2	no	—
378	C₄	—	4	1	no	—
379	C₄	—	4	1	no	—
380	C₄	—	356	1	no	—
381	C₄	—	356	1	no	—
382	C₄	—	356	1	no	—
383	C₄	—	178	1	no	—
384	C₄	—	178	1	no	—
385	C₄	—	178	1	no	—
386	C₄	—	178	2	no	—
387	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
388	C₄	#15668057	1	1	yes	characteristic
389	C₂	—	2	2	no	—
390	C₂	—	8	1	no	—
391	C₂	—	356	1	no	—
392	C₂	#19152230	1	1	yes	characteristic
393	C₂	C₂×#15668041	1	1	yes	characteristic
394	C₂	#19152414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#448×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152254	C₂
hypercenter	#448	D₈₉
hypocenter	C₈₉	#3270
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #448

Complements

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

N	H ≅ G/N	type
1. C₂×#15668041	C₂	semidirect
2. C₂²×#13083564	C₂	semidirect
3. Q₈×C₂²⋊D₈₉	C₂	semidirect
5. #19151989	C₂	semidirect
6. #19151855	C₂	semidirect
7. #19151825	C₂	semidirect
8. #19151818	C₂	semidirect
9. #19151807	C₂	semidirect
10. #19151757	C₂	semidirect
11. #19151756	C₂	semidirect
12. #19151755	C₂	semidirect
13. #19151702	C₂	semidirect
14. #19151666	C₂	semidirect
15. #448×C₈₉	C₂	semidirect
83. #13083564	D₄	semidirect
173. C₂²×C₈₉	#137	semidirect
198. C₈₉	#3270	semidirect
211. #448	D₈₉	semidirect
374. C₂²	#15668041	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668041	2
C₂²×#13083564	2
Q₈×C₂²⋊D₈₉	2
#19152254	2
#19151989	2
#19151855	2
#19151825	2
#19151818	2
#19151807	2
#19151757	2
#19151756	2
#19151755	2
#19151702	2
#19151666	2
#448×C₈₉	2
#3270	89