Subgroups — #36464370

Order 22784.

342 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	#26022051	C₂	1	1	yes	maximal, characteristic
3	#26018230	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2162	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₇₈	C₄	1	1	yes	characteristic
6	C₈×Dic₁₇₈	—	2	1	no	—
7	C₈×Dic₁₇₈	—	2	1	no	—
8	C₈×Dic₁₇₈	C₄	1	2	yes	—
9	C₈×Dic₁₇₈	C₂²	1	2	yes	—
10	C₂×C₄⋊#10623494	C₄	1	1	yes	characteristic
11	C₂×#15667941	—	2	1	no	—
12	C₂×C₈×Dic₈₉	—	2	1	no	—
13	C₄⋊#13083549	—	2	1	no	—
14	C₂²⋊#13083549	—	2	1	no	—
15	C₄×#13083549	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
18	C₂²⋊C₁₆×C₈₉	—	2	1	no	—
19	C₄×C₁₆×C₈₉	—	2	1	no	—
20	C₂²×Dic₁₇₈	C₈	1	1	yes	characteristic
21	C₂²×#10623494	—	2	1	no	—
22	C₄×Dic₁₇₈	—	2	1	no	—
23	C₄×Dic₁₇₈	—	2	1	no	—
24	C₄×Dic₁₇₈	C₈	1	2	yes	—
25	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊Dic₈₉	C₈	1	1	yes	characteristic
27	C₂×#13083553	—	2	1	no	—
28	C₂×C₄×Dic₈₉	—	2	1	no	—
29	C₂×#13083549	—	2	2	no	—
30	C₄⋊#10623494	—	2	1	no	—
31	C₄⋊#10623494	C₂×C₄	1	2	yes	—
32	#15667941	—	2	2	no	—
33	#15667941	—	2	2	no	—
34	C₈×Dic₈₉	—	2	2	no	—
35	C₈×Dic₈₉	—	2	2	no	—
36	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₆×C₈₉	—	2	2	no	—
40	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
42	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
43	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₂×Dic₁₇₈	—	2	1	no	—
45	C₂×Dic₁₇₈	—	2	2	no	—
46	C₂×Dic₁₇₈	C₁₆	1	4	yes	complemented
47	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
48	C₂²×Dic₈₉	—	2	1	no	—
49	C₂²×Dic₈₉	—	2	1	no	—
50	C₂×#10623494	—	2	2	no	—
51	C₂×#10623494	—	4	2	no	—
52	C₄⋊Dic₈₉	—	2	1	no	—
53	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
54	#13083553	—	2	2	no	—
55	#13083553	—	2	2	no	—
56	C₄×Dic₈₉	—	2	2	no	—
57	C₄×Dic₈₉	—	2	2	no	—
58	#13083549	—	4	2	no	—
59	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
62	C₂×C₈×C₈₉	—	2	2	no	—
63	C₂×C₈×C₈₉	—	4	1	no	—
64	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
70	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
71	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
72	C₁₆×C₈₉	—	4	2	no	—
73	C₂³×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₈₉	—	2	1	no	—
75	C₂×C₄×C₈₉	—	2	1	no	—
76	C₂×C₄×C₈₉	—	4	1	no	—
77	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₂×C₁₆	1	2	yes	—
81	C₂×C₄×C₈₉	M₃₂	1	2	yes	—
82	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
85	C₂×Dic₈₉	—	2	4	no	—
86	C₂×Dic₈₉	—	2	2	no	—
87	C₂×Dic₈₉	—	2	2	no	—
88	C₂×Dic₈₉	—	4	2	no	—
89	Dic₁₇₈	—	2	4	no	—
90	Dic₁₇₈	—	2	2	no	—
91	Dic₁₇₈	C₂×C₁₆	1	4	yes	—
92	C₈×C₈₉	—	2	1	no	—
93	C₈×C₈₉	—	2	1	no	—
94	C₈×C₈₉	—	2	1	no	—
95	C₈×C₈₉	#137	1	2	yes	—
96	#10623494	—	4	2	no	—
97	C₂²×C₈₉	—	2	1	no	—
98	C₂²×C₈₉	—	2	1	no	—
99	C₂²×C₈₉	C₂²⋊C₁₆	1	2	yes	—
100	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
101	C₄×C₈₉	—	2	1	no	—
102	C₄×C₈₉	—	2	1	no	—
103	C₄×C₈₉	—	2	1	no	—
104	C₄×C₈₉	C₂²⋊C₁₆	1	2	yes	—
105	C₄×C₈₉	#419	1	1	yes	characteristic
106	C₄×C₈₉	#421	1	1	yes	characteristic
107	C₄×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
108	C₄×C₈₉	C₂×M₃₂	1	1	yes	characteristic
109	Dic₈₉	—	2	4	no	—
110	Dic₈₉	—	4	2	no	—
111	#11637	—	89	1	no	maximal, Sylow
112	C₂×C₈₉	—	2	2	no	—
113	C₂×C₈₉	#1479	1	1	yes	characteristic
114	C₂×C₈₉	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
115	C₂×C₈₉	#2117	1	1	yes	characteristic
116	C₂×C₈×Q₈	—	89	1	no	—
117	#2162	—	89	1	no	—
118	#2162	D₈₉	1	1	yes	characteristic
119	Q₈⋊C₁₆	—	89	4	no	—
120	C₈₉	#11637	1	1	yes	characteristic, complemented, Sylow
121	C₂×C₄×Q₈	—	89	1	no	—
122	C₈×Q₈	—	89	2	no	—
123	C₈×Q₈	—	89	2	no	—
124	C₈×Q₈	—	178	1	no	—
125	C₈×Q₈	—	178	1	no	—
126	C₂×C₄⋊C₈	—	89	1	no	—
127	C₂×C₄⋊C₈	—	178	1	no	—
128	C₂×C₄×C₈	—	178	1	no	—
129	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
130	C₄⋊C₁₆	—	178	1	no	—
131	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
132	C₂²⋊C₁₆	—	2	1	no	—
133	C₂²⋊C₁₆	—	178	1	no	—
134	C₄×C₁₆	—	2	1	no	—
135	C₄×C₁₆	—	89	2	no	—
136	C₂²×Q₈	—	89	1	no	—
137	C₂²×C₈	—	178	1	no	—
138	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
140	C₄×Q₈	—	89	2	no	—
141	C₄×Q₈	—	89	2	no	—
142	C₄×Q₈	—	178	1	no	—
143	C₄×Q₈	—	178	1	no	—
144	C₂×C₄⋊C₄	—	89	1	no	—
145	C₂×C₄⋊C₄	—	178	1	no	—
146	C₂×C₄²	—	178	1	no	—
147	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	178	2	no	—
150	C₄⋊C₈	—	89	2	no	—
151	C₄⋊C₈	—	178	2	no	—
152	C₄⋊C₈	—	178	2	no	—
153	C₄⋊C₈	—	178	1	no	—
154	C₄×C₈	—	178	2	no	—
155	C₄×C₈	—	178	2	no	—
156	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
157	C₄×C₈	D₃₅₆	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
159	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
160	C₂×Q₈	—	89	2	no	—
161	C₂×Q₈	—	89	4	no	—
162	C₂×Q₈	—	178	1	no	—
163	C₂×Q₈	—	178	2	no	—
164	C₂²×C₄	—	178	1	no	—
165	C₂²×C₄	—	178	1	no	—
166	C₂²×C₄	#13083551	1	1	yes	characteristic
167	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
168	C₂²×C₄	#13083557	1	1	yes	characteristic
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	356	2	no	—
172	C₂×C₈	—	178	2	no	—
173	C₂×C₈	#13083551	1	1	yes	characteristic
174	C₂×C₈	#13083551	1	1	yes	characteristic
175	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
176	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
177	C₄⋊C₄	—	89	2	no	—
178	C₄⋊C₄	—	178	2	no	—
179	C₄⋊C₄	—	178	1	no	—
180	C₄⋊C₄	—	178	2	no	—
181	C₄²	—	178	2	no	—
182	C₄²	—	178	2	no	—
183	C₄²	#13083551	1	1	yes	characteristic
184	C₄²	C₈×D₈₉	1	1	yes	characteristic
185	C₄²	#13083557	1	1	yes	characteristic
186	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
187	C₁₆	—	4	2	no	—
188	C₁₆	—	356	2	no	—
189	C₂³	#15667924	1	1	yes	characteristic
190	Q₈	—	89	4	no	—
191	Q₈	—	178	4	no	—
192	Q₈	—	178	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	356	2	no	—
197	C₂×C₄	—	178	2	no	—
198	C₂×C₄	—	178	4	no	—
199	C₂×C₄	—	178	2	no	—
200	C₂×C₄	#15667924	1	1	yes	characteristic
201	C₂×C₄	#15667924	1	1	yes	characteristic
202	C₂×C₄	#15667924	1	1	yes	characteristic
203	C₂×C₄	C₁₆×D₈₉	1	2	yes	—
204	C₂×C₄	#15667951	1	2	yes	—
205	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
206	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
207	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	356	2	no	—
212	C₈	#15668042	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	#19151377	1	2	yes	—
216	C₂²	C₂×#15667924	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	356	2	no	—
221	C₄	—	178	4	no	—
222	C₄	#19151377	1	2	yes	—
223	C₄	#19151606	1	1	yes	characteristic
224	C₄	#19151614	1	1	yes	characteristic
225	C₄	C₂×C₁₆×D₈₉	1	1	yes	characteristic
226	C₄	C₂×#15667951	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	#26018646	1	1	yes	characteristic
229	C₂	C₂×#19151377	1	1	yes	characteristic
230	C₂	#26021881	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₈₉
commutator	C₄×C₈₉	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#2162	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂³×C₈₉	C₂²⋊C₈
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#2162	D₈₉
hypocenter	C₈₉	#11637
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
3. #26018230	C₂	semidirect
46. C₂×Dic₁₇₈	C₁₆	semidirect
120. C₈₉	#11637	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₇₈	2
#26022051	2
#26018230	2
C₈₉×#2162	2
#11637	89