Subgroups — #36453286

Order 22784.

502 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×C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
2	#26021188	C₂	1	2	yes	maximal
3	C₂×#19151351	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151330	C₂	1	2	yes	maximal
5	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₅₆	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₇₈	C₂²	1	2	yes	—
8	C₄×Dic₃₅₆	—	2	2	no	—
9	C₄×Dic₃₅₆	—	2	2	no	—
10	C₂×#15667946	C₄	1	1	yes	characteristic
11	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
12	C₂×#15667937	C₄	1	2	yes	—
13	#19151352	—	2	4	no	—
14	#19151351	—	2	2	no	—
15	#19151330	—	2	4	no	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₈²×C₈₉	—	2	2	no	—
20	C₂²×Dic₁₇₈	C₂×C₄	1	2	yes	—
21	C₂×Dic₃₅₆	—	2	2	no	—
22	C₂×Dic₃₅₆	—	2	2	no	—
23	C₂×Dic₃₅₆	C₈	1	4	yes	complemented
24	C₂²×#10623494	—	2	2	no	—
25	C₄×Dic₁₇₈	—	2	2	no	—
26	C₄×Dic₁₇₈	—	2	2	no	—
27	C₄×Dic₁₇₈	—	2	2	no	—
28	C₄×Dic₁₇₈	—	2	2	no	—
29	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
30	C₂×#13083553	—	2	2	no	—
31	C₂×C₄×Dic₈₉	—	2	2	no	—
32	#15667946	—	2	2	no	—
33	C₄⋊#10623494	—	2	2	no	—
34	C₄⋊#10623494	—	2	2	no	—
35	#15667937	—	2	4	no	—
36	C₂²×C₈×C₈₉	—	2	1	no	—
37	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	D₄	1	2	yes	—
40	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
41	C₄×C₈×C₈₉	—	2	1	no	—
42	C₄×C₈×C₈₉	—	2	1	no	—
43	C₄×C₈×C₈₉	—	2	1	no	—
44	C₄×C₈×C₈₉	D₄	1	2	yes	—
45	C₄×C₈×C₈₉	D₄	1	2	yes	—
46	C₄×C₈×C₈₉	D₄	1	2	yes	—
47	C₂×Dic₁₇₈	—	2	2	no	—
48	C₂×Dic₁₇₈	—	2	2	no	—
49	C₂×Dic₁₇₈	—	2	2	no	—
50	C₂×Dic₁₇₈	—	2	2	no	—
51	C₂×Dic₁₇₈	—	2	2	no	—
52	C₂×Dic₁₇₈	C₂×C₈	1	4	yes	—
53	C₂²×Dic₈₉	—	2	2	no	—
54	C₂²×Dic₈₉	—	2	2	no	—
55	Dic₃₅₆	—	2	8	no	—
56	C₂×#10623494	—	4	2	no	—
57	C₂×#10623494	—	4	4	no	—
58	C₄⋊Dic₈₉	—	2	2	no	—
59	C₄⋊Dic₈₉	—	2	2	no	—
60	#13083553	—	4	2	no	—
61	#13083553	—	4	2	no	—
62	C₄×Dic₈₉	—	4	2	no	—
63	C₄×Dic₈₉	—	4	2	no	—
64	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
67	C₂×C₈×C₈₉	—	2	4	no	—
68	C₂×C₈×C₈₉	—	2	2	no	—
69	C₂×C₈×C₈₉	—	2	2	no	—
70	C₂×C₈×C₈₉	—	2	1	no	—
71	C₂×C₈×C₈₉	—	2	1	no	—
72	C₂×C₈×C₈₉	—	2	2	no	—
73	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
77	C₂×C₈×C₈₉	Q₁₆	1	4	yes	complemented
78	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₈₉	#41	1	1	yes	characteristic
82	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₈₉	—	2	2	no	—
84	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
87	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
88	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
89	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
90	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
93	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
95	C₂×C₄×C₈₉	#135	1	2	yes	—
96	C₂×Dic₈₉	—	2	4	no	—
97	C₂×Dic₈₉	—	4	2	no	—
98	C₂×Dic₈₉	—	4	2	no	—
99	C₂×Dic₈₉	—	4	2	no	—
100	C₂×Dic₈₉	—	4	4	no	—
101	Dic₁₇₈	—	2	4	no	—
102	Dic₁₇₈	—	2	4	no	—
103	Dic₁₇₈	—	4	2	no	—
104	Dic₁₇₈	—	4	2	no	—
105	C₈×C₈₉	—	2	4	no	—
106	C₈×C₈₉	—	2	2	no	—
107	C₈×C₈₉	—	2	4	no	—
108	C₈×C₈₉	C₂²⋊C₈	1	4	yes	—
109	#10623494	—	8	2	no	—
110	C₂²×C₈₉	#390	1	1	yes	characteristic
111	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
112	C₂²×C₈₉	#436	1	1	yes	characteristic
113	C₂²×C₈₉	C₄×Q₁₆	1	2	yes	—
114	C₂²×C₈₉	#443	1	2	yes	—
115	C₄×C₈₉	—	2	1	no	—
116	C₄×C₈₉	—	2	1	no	—
117	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₄×C₈₉	#408	1	1	yes	characteristic
119	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
120	C₄×C₈₉	#436	1	1	yes	characteristic
121	C₄×C₈₉	#467	1	2	yes	—
122	C₄×C₈₉	#494	1	1	yes	characteristic
123	C₄×C₈₉	#495	1	1	yes	characteristic
124	Dic₈₉	—	4	4	no	—
125	Dic₈₉	—	8	2	no	—
126	#9656	—	89	1	no	maximal, Sylow
127	C₂×C₈₉	C₈×Q₁₆	1	2	yes	—
128	C₂×C₈₉	#1591	1	2	yes	—
129	C₂×C₈₉	#1954	1	1	yes	characteristic
130	C₂×C₈₉	#1969	1	1	yes	characteristic
131	C₂×C₈₉	#1974	1	1	yes	characteristic
132	C₂×C₄×Q₁₆	—	89	1	no	—
133	#1984	—	89	2	no	—
134	C₂×#335	—	89	1	no	—
135	C₂×Q₈⋊C₈	—	89	2	no	—
136	C₂×C₈²	D₈₉	1	1	yes	characteristic
137	C₈₉	#9656	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₁₆	—	89	1	no	—
139	C₂×C₄×Q₈	—	89	2	no	—
140	C₄×Q₁₆	—	178	2	no	—
141	C₄×Q₁₆	—	178	2	no	—
142	C₂×#107	—	89	1	no	—
143	C₂×C₄⋊C₈	—	89	2	no	—
144	C₂×#103	—	89	2	no	—
145	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
148	#336	—	178	4	no	—
149	#335	—	178	2	no	—
150	Q₈⋊C₈	—	178	4	no	—
151	C₈²	—	2	2	no	—
152	C₂²×Q₈	—	89	2	no	—
153	C₂×Q₁₆	—	89	4	no	—
154	C₂×Q₁₆	—	178	2	no	—
155	C₂×Q₁₆	—	178	2	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	178	2	no	—
158	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
159	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
160	C₂²×C₈	D₃₅₆	1	2	yes	—
161	C₄×Q₈	—	178	2	no	—
162	C₄×Q₈	—	178	2	no	—
163	C₄×Q₈	—	178	2	no	—
164	C₄×Q₈	—	178	2	no	—
165	C₂×C₄⋊C₄	—	89	2	no	—
166	C₂×C₄⋊C₄	—	178	2	no	—
167	C₂×C₄²	—	178	2	no	—
168	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
169	#107	—	178	2	no	—
170	C₄⋊C₈	—	178	2	no	—
171	C₄⋊C₈	—	178	2	no	—
172	#103	—	178	4	no	—
173	C₄×C₈	—	2	1	no	—
174	C₄×C₈	—	2	1	no	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	D₃₅₆	1	2	yes	—
177	C₄×C₈	D₃₅₆	1	2	yes	—
178	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
179	C₂×Q₈	—	89	4	no	—
180	C₂×Q₈	—	178	2	no	—
181	C₂×Q₈	—	178	2	no	—
182	C₂×Q₈	—	178	2	no	—
183	C₂×Q₈	—	178	2	no	—
184	C₂×Q₈	—	178	2	no	—
185	C₂²×C₄	—	178	2	no	—
186	C₂²×C₄	—	178	2	no	—
187	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
189	C₂²×C₄	#13083574	1	1	yes	characteristic
190	Q₁₆	—	178	8	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	356	2	no	—
198	C₂×C₈	—	356	4	no	—
199	C₂×C₈	#13083551	1	2	yes	—
200	C₂×C₈	#13083551	1	2	yes	—
201	C₂×C₈	C₈×D₈₉	1	2	yes	—
202	C₂×C₈	#13083557	1	2	yes	—
203	C₂×C₈	Dic₃₅₆	1	4	yes	complemented
204	C₄⋊C₄	—	356	2	no	—
205	C₄⋊C₄	—	356	2	no	—
206	C₄⋊C₄	—	178	2	no	—
207	C₄⋊C₄	—	178	2	no	—
208	C₄²	—	356	2	no	—
209	C₄²	—	356	2	no	—
210	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
211	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
212	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
213	C₄²	#13083574	1	1	yes	characteristic
214	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
215	Q₈	—	356	2	no	—
216	Q₈	—	356	2	no	—
217	Q₈	—	178	4	no	—
218	Q₈	—	178	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	356	2	no	—
221	C₂×C₄	—	356	2	no	—
222	C₂×C₄	—	356	4	no	—
223	C₂×C₄	—	356	2	no	—
224	C₂×C₄	—	178	4	no	—
225	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
226	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
227	C₂×C₄	#15667986	1	1	yes	characteristic
228	C₂×C₄	#15667986	1	1	yes	characteristic
229	C₂×C₄	#15668000	1	1	yes	characteristic
230	C₂×C₄	#15668009	1	1	yes	characteristic
231	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
232	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
233	C₂×C₄	#15668018	1	1	yes	characteristic
234	C₂×C₄	#15668018	1	1	yes	characteristic
235	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
236	C₂×C₄	#15668034	1	2	yes	—
237	C₈	—	2	4	no	—
238	C₈	—	2	2	no	—
239	C₈	—	2	4	no	—
240	C₈	—	712	2	no	—
241	C₈	#15667924	1	4	yes	—
242	C₂²	#19151500	1	1	yes	characteristic
243	C₂²	C₈×D₃₅₆	1	1	yes	characteristic
244	C₂²	#19151677	1	1	yes	characteristic
245	C₂²	C₄×Dic₃₅₆	1	2	yes	—
246	C₂²	#19151716	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	356	4	no	—
250	C₄	—	712	2	no	—
251	C₄	C₂×#15667924	1	1	yes	characteristic
252	C₄	#19151564	1	1	yes	characteristic
253	C₄	C₈×D₃₅₆	1	1	yes	characteristic
254	C₄	#19151677	1	1	yes	characteristic
255	C₄	#19151839	1	2	yes	—
256	C₄	#19151978	1	1	yes	characteristic
257	C₄	#19151982	1	1	yes	characteristic
258	C₂	C₈×Dic₃₅₆	1	2	yes	—
259	C₂	#26019150	1	2	yes	—
260	C₂	#26021046	1	1	yes	characteristic
261	C₂	#26021113	1	1	yes	characteristic
262	C₂	#26021140	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×C₈²×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×C₈²	D₈₉
hypocenter	C₈₉	#9656
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
23. C₂×Dic₃₅₆	C₈	semidirect
77. C₂×C₈×C₈₉	Q₁₆	semidirect
137. C₈₉	#9656	semidirect
203. C₂×C₈	Dic₃₅₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₃₅₆	2
#26021188	2
C₂×#19151351	2
C₂×#19151330	2
C₂×C₈²×C₈₉	2
#9656	89