Subgroups — #36707815

Order 22784.

674 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152030	C₂	1	1	yes	maximal, characteristic
2	#26029129	C₂	1	2	yes	maximal, complemented
3	#26022118	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₇₁₂	C₂	1	2	yes	maximal, complemented
5	C₄×D₁₄₂₄	C₂	1	2	yes	maximal, complemented
6	C₂×#19151424	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021965	C₂	1	2	yes	maximal, complemented
8	C₈₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668034	C₂²	1	2	yes	—
10	#19152078	—	2	2	no	—
11	#19152030	C₄	1	8	yes	complemented
12	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
13	C₂×#15667957	C₂²	1	2	yes	—
14	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
15	#19151700	—	2	4	no	—
16	C₄×Dic₃₅₆	C₂²	1	4	yes	—
17	C₄×D₇₁₂	C₂²	1	4	yes	—
18	C₂×#15667946	C₂²	1	2	yes	—
19	#19151600	—	2	2	no	—
20	#19151424	C₂²	1	2	yes	—
21	#19151424	C₂²	1	2	yes	—
22	#19151403	C₂²	1	4	yes	—
23	#19151400	C₂²	1	4	yes	—
24	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	#405×C₈₉	C₂²	1	1	yes	characteristic
27	#346×C₈₉	C₂²	1	2	yes	—
28	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
29	C₂×#13083574	—	2	2	no	—
30	#15668034	—	2	4	no	—
31	#15668034	C₂×C₄	1	8	yes	complemented
32	C₂×Dic₃₅₆	C₂³	1	2	yes	—
33	C₂×#13083563	—	2	2	no	—
34	C₂×D₇₁₂	C₂³	1	2	yes	—
35	C₄×Dic₁₇₈	—	2	4	no	—
36	C₄×D₃₅₆	—	2	4	no	—
37	#15667972	—	4	4	no	—
38	#15667970	—	4	2	no	—
39	C₂×C₄⋊Dic₈₉	—	2	2	no	—
40	C₄⋊C₄×D₈₉	—	4	2	no	—
41	Dic₇₁₂	C₂×C₄	1	4	yes	complemented
42	#15667957	C₂×C₄	1	8	yes	complemented
43	D₁₄₂₄	C₂×C₄	1	4	yes	complemented
44	#15667946	—	2	2	no	—
45	#15667946	C₂³	1	4	yes	—
46	#15667937	—	2	4	no	—
47	#15667934	—	2	4	no	—
48	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
50	#117×C₈₉	D₄	1	1	yes	characteristic
51	M₃₂×C₈₉	—	2	2	no	—
52	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
53	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
54	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
55	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
56	#97×C₈₉	D₄	1	2	yes	—
57	C₄×C₈×C₈₉	C₂³	1	2	yes	—
58	#13083574	—	2	8	no	—
59	#13083574	—	4	4	no	—
60	C₂×Dic₁₇₈	—	2	2	no	—
61	C₂×D₃₅₆	—	2	2	no	—
62	C₂×C₂²⋊D₈₉	—	4	2	no	—
63	C₂²×Dic₈₉	—	4	2	no	—
64	C₂×C₄×D₈₉	—	4	2	no	—
65	C₂×C₄×D₈₉	—	4	4	no	—
66	Dic₃₅₆	—	2	2	no	—
67	Dic₃₅₆	C₂²×C₄	1	4	yes	—
68	#13083563	—	2	8	no	—
69	D₇₁₂	—	2	2	no	—
70	D₇₁₂	C₂²×C₄	1	4	yes	—
71	C₄⋊Dic₈₉	—	2	4	no	—
72	C₄⋊Dic₈₉	—	4	2	no	—
73	#13083553	—	4	4	no	—
74	#13083551	—	4	4	no	—
75	C₄×Dic₈₉	—	4	4	no	—
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	M₁₆×C₈₉	—	2	2	no	—
78	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₈₉	#41	1	2	yes	—
82	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
84	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
85	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
86	C₄²×C₈₉	C₂×D₄	1	2	yes	—
87	C₁₆×C₈₉	—	2	2	no	—
88	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
89	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
90	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
91	C₂²×D₈₉	—	4	2	no	—
92	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₈₉	#135	1	2	yes	—
97	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
98	C₂²⋊D₈₉	—	4	8	no	—
99	C₂×Dic₈₉	—	4	2	no	—
100	C₂×Dic₈₉	—	4	4	no	—
101	C₂×Dic₈₉	—	16	2	no	—
102	D₃₅₆	—	2	4	no	—
103	D₃₅₆	—	4	2	no	—
104	C₄×D₈₉	—	4	8	no	—
105	C₄×D₈₉	—	16	4	no	—
106	Dic₁₇₈	—	2	4	no	—
107	Dic₁₇₈	—	4	2	no	—
108	C₈×C₈₉	—	2	2	no	—
109	C₈×C₈₉	C₄×D₄	1	2	yes	—
110	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
111	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
112	C₂²×C₈₉	—	2	1	no	—
113	C₂²×C₈₉	—	2	1	no	—
114	C₂²×C₈₉	C₄×D₈	1	2	yes	—
115	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
116	C₂×D₈₉	—	4	4	no	—
117	C₂×D₈₉	—	8	2	no	—
118	C₂×D₈₉	—	8	2	no	—
119	C₄×C₈₉	—	2	2	no	—
120	C₄×C₈₉	—	2	2	no	—
121	C₄×C₈₉	C₄×D₈	1	2	yes	—
122	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
123	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
124	Dic₈₉	—	4	4	no	—
125	Dic₈₉	—	8	4	no	—
126	#33459	—	89	1	no	maximal, Sylow
127	C₂×C₈₉	—	2	2	no	—
128	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
129	C₂×C₈₉	#3415	1	1	yes	characteristic
130	C₂×C₈₉	#3417	1	1	yes	characteristic
131	D₈₉	—	8	4	no	—
132	C₂×#508	—	89	1	no	—
133	#2951	—	89	2	no	—
134	#2175	—	89	4	no	—
135	C₄×Q₃₂	—	89	2	no	—
136	C₄×D₁₆	—	89	2	no	—
137	C₂×#366	—	89	1	no	—
138	#2139	—	89	2	no	—
139	#2108	D₈₉	1	1	yes	characteristic, complemented
140	C₈₉	#33459	1	1	yes	characteristic, complemented, Sylow
141	C₂×#135	—	89	2	no	—
142	#520	—	178	2	no	—
143	#508	—	89	8	no	—
144	C₂×Q₃₂	—	89	1	no	—
145	C₂×SD₃₂	—	89	2	no	—
146	C₂×D₁₆	—	89	1	no	—
147	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
148	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
149	#440	—	178	4	no	—
150	C₄×Q₁₆	—	89	4	no	—
151	C₄×D₈	—	89	4	no	—
152	C₂×#107	—	89	2	no	—
153	#417	—	178	2	no	—
154	#405	C₂×D₈₉	1	1	yes	characteristic
155	#366	—	89	2	no	—
156	#366	—	89	2	no	—
157	#358	—	89	4	no	—
158	#357	—	89	4	no	—
159	#346	C₂×D₈₉	1	2	yes	—
160	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
161	C₂×#41	—	178	2	no	—
162	#135	—	89	8	no	—
163	#135	—	178	4	no	—
164	C₂×Q₁₆	—	89	2	no	—
165	C₂×SD₁₆	—	178	2	no	—
166	C₂×D₈	—	89	2	no	—
167	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
169	C₄×Q₈	—	178	4	no	—
170	C₄×D₄	—	356	4	no	—
171	C₄×D₄	—	178	4	no	—
172	#117	—	356	2	no	—
173	#117	D₃₅₆	1	1	yes	characteristic
174	C₂×C₄⋊C₄	—	356	2	no	—
175	C₂×C₄⋊C₄	—	178	2	no	—
176	Q₃₂	—	89	4	no	—
177	SD₃₂	—	89	8	no	—
178	D₁₆	—	89	4	no	—
179	M₃₂	—	2	2	no	—
180	C₂×C₁₆	C₄×D₈₉	1	4	yes	complemented
181	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
182	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
183	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
184	#107	—	89	4	no	—
185	#107	—	178	2	no	—
186	#103	—	178	4	no	—
187	#102	—	178	4	no	—
188	#97	D₃₅₆	1	2	yes	—
189	C₄×C₈	C₂²×D₈₉	1	2	yes	—
190	#41	—	356	4	no	—
191	#41	—	178	8	no	—
192	C₂×Q₈	—	178	2	no	—
193	C₂×D₄	—	356	2	no	—
194	C₂×D₄	—	178	2	no	—
195	C₂²×C₄	—	356	2	no	—
196	C₂²×C₄	—	356	2	no	—
197	C₂²×C₄	—	356	4	no	—
198	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
199	Q₁₆	—	89	4	no	—
200	Q₁₆	—	178	2	no	—
201	SD₁₆	—	178	8	no	—
202	D₈	—	89	4	no	—
203	D₈	—	178	2	no	—
204	M₁₆	—	2	2	no	—
205	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
206	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
207	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
208	C₂×C₈	#13083574	1	2	yes	—
209	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
210	C₄⋊C₄	—	356	2	no	—
211	C₄⋊C₄	—	356	4	no	—
212	C₄⋊C₄	—	178	4	no	—
213	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
214	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
215	C₂²⋊C₄	—	356	4	no	—
216	C₂²⋊C₄	D₇₁₂	1	2	yes	—
217	C₄²	—	356	4	no	—
218	C₄²	C₂×D₃₅₆	1	2	yes	—
219	C₁₆	—	2	2	no	—
220	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
221	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
222	C₂³	—	356	2	no	—
223	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
224	Q₈	—	356	2	no	—
225	Q₈	—	178	4	no	—
226	D₄	—	356	8	no	—
227	D₄	—	356	2	no	—
228	D₄	—	178	4	no	—
229	C₂×C₄	—	356	2	no	—
230	C₂×C₄	—	356	8	no	—
231	C₂×C₄	—	356	4	no	—
232	C₂×C₄	—	1424	4	no	—
233	C₂×C₄	—	1424	2	no	—
234	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
235	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
236	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
237	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
238	C₂×C₄	#15668034	1	2	yes	—
239	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	C₄×D₃₅₆	1	2	yes	—
242	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
243	C₈	C₂×#13083574	1	1	yes	characteristic
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	356	4	no	—
247	C₂²	—	712	2	no	—
248	C₂²	—	712	2	no	—
249	C₂²	C₄×D₇₁₂	1	2	yes	—
250	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	356	4	no	—
254	C₄	—	712	4	no	—
255	C₄	C₄×D₇₁₂	1	2	yes	—
256	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
257	C₄	C₂×#15668034	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	712	4	no	—
260	C₂	C₂×C₄×D₇₁₂	1	1	yes	characteristic
261	C₂	#26033393	1	1	yes	characteristic
262	C₂	#26033405	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2108	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#19152030	C₂
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2108	D₈₉
hypocenter	C₈₉	#33459
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #26029129	C₂	semidirect
3. #26022118	C₂	semidirect
4. C₄×Dic₇₁₂	C₂	semidirect
5. C₄×D₁₄₂₄	C₂	semidirect
6. C₂×#19151424	C₂	semidirect
7. #26021965	C₂	semidirect
8. C₈₉×#2108	C₂	semidirect
11. #19152030	C₄	semidirect
31. #15668034	C₂×C₄	semidirect
41. Dic₇₁₂	C₂×C₄	semidirect
42. #15667957	C₂×C₄	semidirect
43. D₁₄₂₄	C₂×C₄	semidirect
52. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
139. #2108	D₈₉	semidirect
140. C₈₉	#33459	semidirect
180. C₂×C₁₆	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152030	2
#26029129	2
#26022118	2
C₄×Dic₇₁₂	2
C₄×D₁₄₂₄	2
C₂×#19151424	2
#26021965	2
C₈₉×#2108	2
#33459	89