Subgroups — #36701096

Order 22784.

688 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26025239	C₂	1	2	yes	maximal
2	#26021743	C₂	1	2	yes	maximal
3	#26021579	C₂	1	2	yes	maximal
4	#26021578	C₂	1	2	yes	maximal
5	#26021188	C₂	1	1	yes	maximal, characteristic
6	#26020985	C₂	1	1	yes	maximal, characteristic
7	#26020825	C₂	1	2	yes	maximal
8	#26020822	C₂	1	2	yes	maximal
9	C₈₉×#1942	C₂	1	1	yes	maximal, characteristic
10	C₂×#15668011	—	2	2	no	—
11	C₂×#15668011	C₂²	1	2	yes	—
12	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
13	C₂×#15667946	C₂²	1	1	yes	characteristic
14	C₂×#15667946	C₂²	1	1	yes	characteristic
15	C₂×#15667945	C₂²	1	2	yes	—
16	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
17	#19151532	—	2	2	no	—
18	#19151531	—	2	2	no	—
19	#19151502	—	2	2	no	—
20	#19151497	C₂²	1	2	yes	—
21	#19151496	C₂²	1	2	yes	—
22	#19151494	C₂²	1	2	yes	—
23	C₄⋊C₄×Dic₈₉	C₂²	1	2	yes	—
24	#19151361	C₂²	1	2	yes	—
25	#19151360	C₂²	1	2	yes	—
26	#19151360	C₂²	1	2	yes	—
27	#19151360	C₂²	1	2	yes	—
28	#19151352	C₂²	1	2	yes	—
29	#19151352	C₂²	1	2	yes	—
30	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
33	#389×C₈₉	C₂²	1	2	yes	—
34	#340×C₈₉	C₂²	1	2	yes	—
35	C₂²×Dic₁₇₈	—	2	2	no	—
36	C₂²×Dic₁₇₈	D₄	1	1	yes	characteristic
37	C₂²×#10623494	C₂³	1	2	yes	—
38	#15668011	—	4	2	no	—
39	#15668011	—	4	2	no	—
40	#15668011	—	4	8	no	—
41	C₄×Dic₁₇₈	—	4	1	no	—
42	C₄×Dic₁₇₈	—	4	1	no	—
43	C₂×C₄⋊Dic₈₉	—	2	2	no	—
44	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
45	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
47	C₂×#13083553	—	2	2	no	—
48	C₂×#13083553	—	2	2	no	—
49	C₂×#13083553	—	2	2	no	—
50	C₂×#13083553	D₄	1	2	yes	—
51	C₂×C₄×Dic₈₉	—	2	2	no	—
52	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
53	#15667946	—	4	1	no	—
54	#15667946	Q₈	1	4	yes	complemented
55	#15667945	—	4	2	no	—
56	C₄⋊#10623494	—	4	1	no	—
57	#15667919	—	2	2	no	—
58	#15667919	—	2	2	no	—
59	#15667919	—	2	2	no	—
60	#15667919	—	2	2	no	—
61	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
64	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
66	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
67	#107×C₈₉	—	4	1	no	—
68	#107×C₈₉	Q₈	1	4	yes	complemented
69	C₄×C₈×C₈₉	—	4	1	no	—
70	#95×C₈₉	—	2	2	no	—
71	C₂×Dic₁₇₈	—	4	1	no	—
72	C₂×Dic₁₇₈	—	4	1	no	—
73	C₂×Dic₁₇₈	—	4	4	no	—
74	C₂×Dic₁₇₈	—	4	4	no	—
75	C₂×Dic₁₇₈	—	4	4	no	—
76	C₂×Dic₁₇₈	D₈	1	4	yes	—
77	C₂²×Dic₈₉	—	2	2	no	—
78	C₂²×Dic₈₉	—	2	2	no	—
79	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
80	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
81	C₂×#10623494	—	2	2	no	—
82	C₂×#10623494	—	2	2	no	—
83	C₂×#10623494	—	2	2	no	—
84	C₄⋊Dic₈₉	—	2	2	no	—
85	C₄⋊Dic₈₉	—	2	2	no	—
86	C₄⋊Dic₈₉	—	2	2	no	—
87	C₄⋊Dic₈₉	—	4	1	no	—
88	C₄⋊Dic₈₉	—	4	4	no	—
89	C₄⋊Dic₈₉	C₂×Q₈	1	4	yes	—
90	#13083553	—	4	2	no	—
91	#13083553	—	4	4	no	—
92	#13083553	—	4	4	no	—
93	#13083553	—	4	4	no	—
94	C₄×Dic₈₉	—	2	2	no	—
95	C₄×Dic₈₉	—	2	2	no	—
96	C₄×Dic₈₉	—	4	4	no	—
97	C₂²×C₄×C₈₉	—	2	2	no	—
98	C₂²×C₄×C₈₉	—	2	2	no	—
99	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
102	C₂×C₈×C₈₉	—	2	1	no	—
103	C₂×C₈×C₈₉	—	2	1	no	—
104	C₂×C₈×C₈₉	—	2	1	no	—
105	C₂×C₈×C₈₉	—	2	1	no	—
106	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
107	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
108	C₄⋊C₄×C₈₉	—	2	2	no	—
109	C₄⋊C₄×C₈₉	—	2	2	no	—
110	C₄⋊C₄×C₈₉	—	2	2	no	—
111	C₄⋊C₄×C₈₉	—	4	4	no	—
112	C₄⋊C₄×C₈₉	C₂×Q₈	1	4	yes	—
113	C₄²×C₈₉	—	2	1	no	—
114	C₄²×C₈₉	—	2	1	no	—
115	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	4	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	4	2	no	—
125	C₂×C₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
134	C₂×Dic₈₉	—	2	2	no	—
135	C₂×Dic₈₉	—	2	2	no	—
136	C₂×Dic₈₉	—	2	2	no	—
137	C₂×Dic₈₉	—	2	4	no	—
138	C₂×Dic₈₉	—	2	2	no	—
139	C₂×Dic₈₉	—	2	2	no	—
140	C₂×Dic₈₉	—	4	2	no	—
141	C₂×Dic₈₉	—	4	2	no	—
142	C₂×Dic₈₉	—	4	2	no	—
143	C₂×Dic₈₉	—	4	2	no	—
144	C₂×Dic₈₉	—	4	2	no	—
145	C₂×Dic₈₉	C₂×D₈	1	4	yes	—
146	Dic₁₇₈	—	4	4	no	—
147	Dic₁₇₈	—	8	8	no	—
148	C₈×C₈₉	—	2	2	no	—
149	C₈×C₈₉	—	2	2	no	—
150	#10623494	—	4	2	no	—
151	C₂²×C₈₉	#534	1	1	yes	characteristic
152	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
153	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
154	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
155	C₂²×C₈₉	#575	1	1	yes	characteristic
156	C₂²×C₈₉	#578	1	1	yes	characteristic
157	C₂²×C₈₉	#578	1	1	yes	characteristic
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	—	4	4	no	—
160	C₄×C₈₉	—	4	1	no	—
161	C₄×C₈₉	—	8	2	no	—
162	C₄×C₈₉	#544	1	1	yes	characteristic
163	C₄×C₈₉	#544	1	1	yes	characteristic
164	C₄×C₈₉	Q₈²	1	1	yes	characteristic
165	C₄×C₈₉	#563	1	1	yes	characteristic
166	Dic₈₉	—	2	4	no	—
167	Dic₈₉	—	4	4	no	—
168	Dic₈₉	—	4	2	no	—
169	Dic₈₉	—	8	2	no	—
170	#32682	—	89	1	no	maximal, Sylow
171	C₂×C₈₉	#2734	1	1	yes	characteristic
172	C₂×C₈₉	#3186	1	1	yes	characteristic
173	C₂×C₈₉	#3202	1	1	yes	characteristic
174	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
175	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
176	C₂×C₈₉	#3390	1	1	yes	characteristic
177	C₂×C₈₉	#3390	1	1	yes	characteristic
178	#2546	—	89	2	no	—
179	#2086	—	89	2	no	—
180	#2057	—	89	2	no	—
181	#2057	—	89	2	no	—
182	#1984	—	89	1	no	—
183	#1942	—	89	1	no	—
184	#1942	D₈₉	1	1	yes	characteristic
185	#1918	—	89	2	no	—
186	#1918	—	89	2	no	—
187	C₈₉	#32682	1	1	yes	characteristic, complemented, Sylow
188	C₂×C₄⋊Q₈	—	89	2	no	—
189	C₂×C₄⋊Q₈	—	178	2	no	—
190	C₂×C₄×Q₈	—	89	1	no	—
191	C₂×#107	—	89	1	no	—
192	C₂×#107	—	89	1	no	—
193	C₂×#107	—	89	2	no	—
194	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
195	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
196	C₂×C₄⋊C₈	—	89	1	no	—
197	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
198	#398	—	178	2	no	—
199	#398	—	178	2	no	—
200	#391	—	178	2	no	—
201	#389	—	89	2	no	—
202	#389	—	89	2	no	—
203	#389	—	89	2	no	—
204	#389	C₂×D₈₉	1	2	yes	—
205	C₄×C₄⋊C₄	—	89	2	no	—
206	#340	—	89	2	no	—
207	#340	—	89	2	no	—
208	#340	—	89	2	no	—
209	#340	—	89	2	no	—
210	#340	C₂×D₈₉	1	2	yes	—
211	#336	—	89	2	no	—
212	#336	—	89	2	no	—
213	C₂²×Q₈	—	89	1	no	—
214	C₂²×Q₈	—	178	2	no	—
215	C₂²×C₈	—	89	2	no	—
216	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
217	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
218	C₄⋊Q₈	—	356	8	no	—
219	C₄⋊Q₈	—	356	2	no	—
220	C₄⋊Q₈	—	356	2	no	—
221	C₄×Q₈	—	356	1	no	—
222	C₄×Q₈	—	356	1	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	89	2	no	—
225	C₂×C₄⋊C₄	—	89	2	no	—
226	C₂×C₄⋊C₄	—	89	1	no	—
227	C₂×C₄⋊C₄	—	89	2	no	—
228	C₂×C₄⋊C₄	—	178	2	no	—
229	C₂×C₄⋊C₄	—	178	2	no	—
230	C₂×C₄⋊C₄	—	178	2	no	—
231	C₂×C₄⋊C₄	—	178	2	no	—
232	C₂×C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
233	C₂×C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
234	C₂×C₄²	—	89	2	no	—
235	C₂×C₄²	—	178	2	no	—
236	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
237	#107	—	4	1	no	—
238	#107	—	89	4	no	—
239	#107	—	356	2	no	—
240	#107	—	356	1	no	—
241	#107	Dic₁₇₈	1	4	yes	complemented
242	C₄⋊C₈	—	356	1	no	—
243	C₄×C₈	—	4	1	no	—
244	#95	—	2	2	no	—
245	#95	—	178	2	no	—
246	#95	—	178	2	no	—
247	#95	—	178	2	no	—
248	#95	—	178	2	no	—
249	C₂×Q₈	—	89	4	no	—
250	C₂×Q₈	—	356	1	no	—
251	C₂×Q₈	—	356	1	no	—
252	C₂×Q₈	—	356	4	no	—
253	C₂×Q₈	—	356	4	no	—
254	C₂×Q₈	—	356	4	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	89	2	no	—
258	C₂²×C₄	—	89	2	no	—
259	C₂²×C₄	—	178	2	no	—
260	C₂²×C₄	—	178	2	no	—
261	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
262	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
263	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	178	2	no	—
269	C₂×C₈	—	178	2	no	—
270	C₂×C₈	—	178	2	no	—
271	C₂×C₈	Q₈×D₈₉	1	2	yes	—
272	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	—	89	4	no	—
278	C₄⋊C₄	—	356	2	no	—
279	C₄⋊C₄	—	356	1	no	—
280	C₄⋊C₄	—	356	4	no	—
281	C₄⋊C₄	—	356	4	no	—
282	C₄⋊C₄	—	356	4	no	—
283	C₄⋊C₄	—	356	4	no	—
284	C₄⋊C₄	—	178	2	no	—
285	C₄⋊C₄	—	178	2	no	—
286	C₄⋊C₄	—	178	2	no	—
287	C₄⋊C₄	C₂×Dic₁₇₈	1	4	yes	—
288	C₄²	—	2	1	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	356	4	no	—
291	C₄²	—	178	2	no	—
292	C₄²	—	178	2	no	—
293	C₂³	C₂×D₄×D₈₉	1	1	yes	characteristic
294	Q₈	—	356	4	no	—
295	Q₈	—	712	8	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	89	4	no	—
308	C₂×C₄	—	356	2	no	—
309	C₂×C₄	—	356	2	no	—
310	C₂×C₄	—	356	2	no	—
311	C₂×C₄	—	356	2	no	—
312	C₂×C₄	—	356	2	no	—
313	C₂×C₄	—	178	2	no	—
314	C₂×C₄	—	178	2	no	—
315	C₂×C₄	—	178	2	no	—
316	C₂×C₄	—	178	4	no	—
317	C₂×C₄	—	178	2	no	—
318	C₂×C₄	—	178	2	no	—
319	C₂×C₄	D₈×D₈₉	1	2	yes	—
320	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
321	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
322	C₂×C₄	#15668057	1	1	yes	characteristic
323	C₂×C₄	#15668058	1	1	yes	characteristic
324	C₂×C₄	#15668059	1	1	yes	characteristic
325	C₂×C₄	#15668060	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	2	2	no	—
328	C₈	—	356	2	no	—
329	C₂²	#19152156	1	1	yes	characteristic
330	C₂²	#19152263	1	1	yes	characteristic
331	C₂²	D₄×Dic₁₇₈	1	1	yes	characteristic
332	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
333	C₂²	#19152395	1	1	yes	characteristic
334	C₂²	#19152411	1	1	yes	characteristic
335	C₂²	#19152412	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	4	no	—
339	C₄	—	8	2	no	—
340	C₄	—	356	2	no	—
341	C₄	—	356	4	no	—
342	C₄	—	178	4	no	—
343	C₄	—	712	2	no	—
344	C₄	#19152227	1	1	yes	characteristic
345	C₄	#19152229	1	1	yes	characteristic
346	C₄	Q₈×Dic₁₇₈	1	1	yes	characteristic
347	C₄	#19152352	1	1	yes	characteristic
348	C₂	#26027283	1	1	yes	characteristic
349	C₂	#26030999	1	1	yes	characteristic
350	C₂	#26031142	1	1	yes	characteristic
351	C₂	#26033104	1	1	yes	characteristic
352	C₂	D₈×Dic₁₇₈	1	1	yes	characteristic
353	C₂	#26033217	1	1	yes	characteristic
354	C₂	#26033222	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₈₉
commutator, cosocle	C₂²×C₄×C₈₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1942	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
hypercenter	#1942	D₈₉
hypocenter	C₈₉	#32682
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1942

Complements

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

N	H ≅ G/N	type
54. #15667946	Q₈	semidirect
68. #107×C₈₉	Q₈	semidirect
187. C₈₉	#32682	semidirect
241. #107	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26025239	2
#26021743	2
#26021579	2
#26021578	2
#26021188	2
#26020985	2
#26020825	2
#26020822	2
C₈₉×#1942	2
#32682	89