Subgroups — #36708002

Order 22784.

552 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022161	C₂	1	2	yes	maximal, complemented
2	#26022078	C₂	1	4	yes	maximal, complemented
3	D₄×#13083549	C₂	1	2	yes	maximal, complemented
4	#26022055	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022055	C₂	1	2	yes	maximal, complemented
6	C₄×#15667952	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151374	C₂	1	2	yes	maximal, complemented
8	C₈×#41×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#15667952	C₂²	1	2	yes	—
10	C₂²×#13083549	C₂²	1	4	yes	—
11	C₄⋊#13083549	C₂²	1	2	yes	—
12	C₄⋊#13083549	C₂²	1	4	yes	—
13	C₂²⋊#13083549	C₂²	1	4	yes	—
14	C₂²⋊#13083549	C₂²	1	2	yes	—
15	#19151374	C₂²	1	4	yes	—
16	#19151374	C₂²	1	2	yes	—
17	#19151374	C₂²	1	1	yes	characteristic
18	#19151374	C₂²	1	1	yes	characteristic
19	C₄×#13083549	C₂²	1	2	yes	—
20	C₄×#41×C₈₉	C₄	1	1	yes	characteristic
21	C₈×Q₈×C₈₉	C₄	1	1	yes	characteristic
22	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₈×D₄×C₈₉	C₄	1	2	yes	—
24	C₈×D₄×C₈₉	C₄	1	1	yes	characteristic
25	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₈₉	C₂²	1	2	yes	—
27	#431×C₈₉	C₄	1	1	yes	characteristic
28	#431×C₈₉	C₄	1	2	yes	—
29	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	#15667952	—	2	4	no	—
32	C₂×#13083549	—	2	8	no	—
33	C₂×#13083549	C₂³	1	2	yes	—
34	C₂×#13083549	C₂³	1	4	yes	—
35	C₂×#13083549	C₂³	1	2	yes	—
36	C₂×#41×C₈₉	C₈	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
41	C₄×Q₈×C₈₉	C₈	1	1	yes	characteristic
42	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₈₉	C₈	1	2	yes	—
44	C₄×D₄×C₈₉	C₈	1	1	yes	characteristic
45	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
47	#117×C₈₉	C₈	1	1	yes	characteristic
48	#117×C₈₉	C₈	1	2	yes	—
49	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
52	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
55	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
56	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
57	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₈₉	C₂³	1	2	yes	—
65	#13083549	—	2	4	no	—
66	#13083549	—	2	2	no	—
67	#13083549	#41	1	4	yes	complemented
68	#41×C₈₉	—	2	4	no	—
69	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
70	C₂×D₄×C₈₉	C₂×C₈	1	2	yes	—
71	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
76	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₈₉	—	2	4	no	—
79	C₂×C₈×C₈₉	—	2	2	no	—
80	C₂×C₈×C₈₉	—	2	4	no	—
81	C₂×C₈×C₈₉	—	2	2	no	—
82	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
89	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
90	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
91	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
92	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
93	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
94	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
95	C₄²×C₈₉	C₂×C₈	1	2	yes	—
96	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
97	C₄²×C₈₉	C₂×C₈	1	2	yes	—
98	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
99	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
101	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
103	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
104	Q₈×C₈₉	—	2	2	no	—
105	D₄×C₈₉	—	2	4	no	—
106	D₄×C₈₉	M₃₂	1	4	yes	complemented
107	C₂×C₄×C₈₉	—	2	4	no	—
108	C₂×C₄×C₈₉	—	2	2	no	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	2	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	2	no	—
113	C₂×C₄×C₈₉	—	4	2	no	—
114	C₂×C₄×C₈₉	—	4	1	no	—
115	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
124	C₈×C₈₉	—	2	2	no	—
125	C₈×C₈₉	—	2	1	no	—
126	C₈×C₈₉	—	2	1	no	—
127	C₈×C₈₉	—	2	2	no	—
128	C₈×C₈₉	#131	1	2	yes	—
129	C₈×C₈₉	C₂×#41	1	2	yes	—
130	C₂²×C₈₉	—	2	2	no	—
131	C₂²×C₈₉	—	2	2	no	—
132	C₂²×C₈₉	—	2	1	no	—
133	C₂²×C₈₉	—	2	1	no	—
134	C₂²×C₈₉	—	2	1	no	—
135	C₂²×C₈₉	C₂×M₃₂	1	4	yes	—
136	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
137	C₄×C₈₉	—	2	1	no	—
138	C₄×C₈₉	—	2	2	no	—
139	C₄×C₈₉	—	2	1	no	—
140	C₄×C₈₉	—	2	2	no	—
141	C₄×C₈₉	—	2	1	no	—
142	C₄×C₈₉	—	2	2	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	C₂×M₃₂	1	2	yes	—
145	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
146	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
147	#33484	—	89	1	no	maximal, Sylow
148	C₂×C₈₉	—	2	4	no	—
149	C₂×C₈₉	—	4	1	no	—
150	C₂×C₈₉	C₈×#41	1	1	yes	characteristic
151	C₂×C₈₉	C₂²×M₃₂	1	1	yes	characteristic
152	C₂×C₈₉	#3407	1	1	yes	characteristic
153	C₈×#41	D₈₉	1	1	yes	characteristic
154	#2183	—	89	2	no	—
155	#2168	—	89	4	no	—
156	D₄×C₁₆	—	89	2	no	—
157	#2163	—	89	1	no	—
158	#2163	—	89	2	no	—
159	C₄×M₃₂	—	89	1	no	—
160	C₂×#346	—	89	2	no	—
161	C₈₉	#33484	1	1	yes	characteristic, complemented, Sylow
162	C₄×#41	Dic₈₉	1	1	yes	characteristic
163	C₂×M₃₂	—	89	2	no	—
164	C₂²×C₁₆	—	89	4	no	—
165	C₈×Q₈	Dic₈₉	1	1	yes	characteristic
166	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
167	C₈×D₄	Dic₈₉	1	2	yes	—
168	C₈×D₄	Dic₈₉	1	1	yes	characteristic
169	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
170	C₈×D₄	C₂×D₈₉	1	2	yes	—
171	#431	Dic₈₉	1	1	yes	characteristic
172	#431	Dic₈₉	1	2	yes	—
173	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
174	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
175	C₄⋊C₁₆	—	89	2	no	—
176	C₄⋊C₁₆	—	89	4	no	—
177	C₂²⋊C₁₆	—	89	4	no	—
178	C₂²⋊C₁₆	—	89	2	no	—
179	#346	—	89	1	no	—
180	#346	—	89	2	no	—
181	#346	—	89	4	no	—
182	#346	—	89	1	no	—
183	C₄×C₁₆	—	89	2	no	—
184	C₂×#41	#10623494	1	1	yes	characteristic
185	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
186	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
188	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
189	C₄×Q₈	#10623494	1	1	yes	characteristic
190	C₄×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
191	C₄×D₄	#10623494	1	2	yes	—
192	C₄×D₄	#10623494	1	1	yes	characteristic
193	C₄×D₄	C₂×Dic₈₉	1	2	yes	—
194	C₄×D₄	C₂×Dic₈₉	1	1	yes	characteristic
195	#117	#10623494	1	1	yes	characteristic
196	#117	#10623494	1	2	yes	—
197	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
198	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
199	M₃₂	—	178	4	no	—
200	C₂×C₁₆	—	89	2	no	—
201	C₂×C₁₆	—	89	2	no	—
202	C₂×C₁₆	—	89	4	no	—
203	C₂×C₁₆	—	178	8	no	—
204	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
205	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
206	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
207	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
208	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
209	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
210	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
211	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
212	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
213	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
214	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₈₉	1	2	yes	—
218	#41	—	2	4	no	—
219	C₂×Q₈	C₂×#10623494	1	1	yes	characteristic
220	C₂×D₄	C₂×#10623494	1	2	yes	—
221	C₂×D₄	C₂×#10623494	1	1	yes	characteristic
222	C₂²×C₄	C₂×#10623494	1	2	yes	—
223	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
224	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
225	C₂²×C₄	C₂×#10623494	1	2	yes	—
226	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
227	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
233	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
234	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
235	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
236	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
237	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
238	C₄⋊C₄	C₂×#10623494	1	2	yes	—
239	C₄⋊C₄	C₂×#10623494	1	1	yes	characteristic
240	C₄⋊C₄	C₂×#10623494	1	2	yes	—
241	C₄⋊C₄	C₂×#10623494	1	1	yes	characteristic
242	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
243	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
244	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
245	C₄²	C₂×#10623494	1	2	yes	—
246	C₄²	C₂×#10623494	1	1	yes	characteristic
247	C₄²	C₂×#10623494	1	2	yes	—
248	C₄²	C₂×#10623494	1	1	yes	characteristic
249	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
250	C₄²	C₂²×Dic₈₉	1	2	yes	—
251	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
252	C₁₆	—	89	4	no	—
253	C₁₆	—	178	4	no	—
254	C₁₆	—	178	2	no	—
255	C₂³	C₂²×#10623494	1	1	yes	characteristic
256	C₂³	C₂²×#10623494	1	2	yes	—
257	Q₈	—	2	2	no	—
258	D₄	—	2	4	no	—
259	D₄	#15667952	1	4	yes	complemented
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
269	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
270	C₂×C₄	C₂²×#10623494	1	2	yes	—
271	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
272	C₂×C₄	C₂²×#10623494	1	2	yes	—
273	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
274	C₂×C₄	C₂²×#10623494	1	2	yes	—
275	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
276	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	2	no	—
281	C₈	#15668019	1	2	yes	—
282	C₈	#41×D₈₉	1	2	yes	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	C₂×#15667952	1	4	yes	—
289	C₂²	C₂³×#10623494	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	C₂×#15667952	1	2	yes	—
298	C₄	#41×Dic₈₉	1	1	yes	characteristic
299	C₄	C₂×#15668019	1	1	yes	characteristic
300	C₂	—	2	4	no	—
301	C₂	—	4	1	no	—
302	C₂	#41×#10623494	1	1	yes	characteristic
303	C₂	C₂²×#15667952	1	1	yes	characteristic
304	C₂	#26033355	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈×#41×C₈₉	C₂
prime radical	C₂×#41×C₈₉	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₈×#41	D₈₉
hypocenter	C₈₉	#33484
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. #26022161	C₂	semidirect
2. #26022078	C₂	semidirect
3. D₄×#13083549	C₂	semidirect
4. #26022055	C₂	semidirect
5. #26022055	C₂	semidirect
6. C₄×#15667952	C₂	semidirect
7. C₂×#19151374	C₂	semidirect
67. #13083549	#41	semidirect
106. D₄×C₈₉	M₃₂	semidirect
161. C₈₉	#33484	semidirect
259. D₄	#15667952	semidirect

Maximal subgroups

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

subgroup	index
#26022161	2
#26022078	2
D₄×#13083549	2
#26022055	2
C₄×#15667952	2
C₂×#19151374	2
C₈×#41×C₈₉	2
#33484	89