Subgroups — #36469650

Order 22784.

462 conjugacy classes of subgroups.

The subgroup lattice has 236 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	C₂×#19151401	C₂	1	2	yes	maximal
3	C₂×#19151374	C₂	1	1	yes	maximal, characteristic
4	#26018343	C₂	1	2	yes	maximal
5	C₄×#107×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₇₈	—	2	1	no	—
8	C₂²×#13083549	C₂²	1	2	yes	—
9	C₄×Dic₃₅₆	—	2	2	no	—
10	C₄×Dic₃₅₆	C₄	1	4	yes	complemented
11	C₂×#15667946	C₂²	1	1	yes	characteristic
12	C₂×#15667937	—	2	1	no	—
13	#19151401	C₄	1	8	yes	complemented
14	#19151374	—	2	2	no	—
15	C₂×#107×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
18	#340×C₈₉	—	2	1	no	—
19	C₂²×Dic₁₇₈	—	2	1	no	—
20	C₂×Dic₃₅₆	—	2	1	no	—
21	C₂×Dic₃₅₆	—	2	2	no	—
22	C₂×Dic₃₅₆	C₂×C₄	1	4	yes	—
23	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₇₈	—	2	4	no	—
25	C₄×Dic₁₇₈	—	4	2	no	—
26	C₂×C₄⋊Dic₈₉	—	2	1	no	—
27	C₂×#13083553	—	4	1	no	—
28	C₂×C₄×Dic₈₉	—	4	1	no	—
29	C₂×#13083549	—	2	2	no	—
30	C₂×#13083549	—	2	2	no	—
31	C₂×#13083549	C₂×C₄	1	4	yes	—
32	#15667946	—	2	1	no	—
33	#15667946	C₂×C₄	1	2	yes	—
34	#15667937	—	2	4	no	—
35	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
38	C₂×C₄²×C₈₉	—	4	1	no	—
39	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
40	#107×C₈₉	—	2	2	no	—
41	#107×C₈₉	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₈₉	D₄	1	2	yes	—
44	#95×C₈₉	—	4	1	no	—
45	C₂×Dic₁₇₈	—	2	2	no	—
46	C₂×Dic₁₇₈	—	2	4	no	—
47	C₂×Dic₁₇₈	—	4	1	no	—
48	C₂×Dic₁₇₈	—	4	2	no	—
49	C₂²×Dic₈₉	—	4	1	no	—
50	C₂²×Dic₈₉	—	4	1	no	—
51	Dic₃₅₆	—	2	2	no	—
52	Dic₃₅₆	—	2	4	no	—
53	Dic₃₅₆	C₄²	1	4	yes	complemented
54	C₄⋊Dic₈₉	—	2	2	no	—
55	C₄⋊Dic₈₉	—	4	1	no	—
56	#13083553	—	4	4	no	—
57	C₄×Dic₈₉	—	4	4	no	—
58	#13083549	—	2	4	no	—
59	C₂²×C₄×C₈₉	—	4	2	no	—
60	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₈₉	—	2	2	no	—
64	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	#41	1	2	yes	—
69	C₄⋊C₄×C₈₉	—	2	4	no	—
70	C₄⋊C₄×C₈₉	—	4	2	no	—
71	C₄²×C₈₉	—	8	2	no	—
72	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₈₉	SD₁₆	1	2	yes	—
74	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₈₉	—	2	1	no	—
76	C₂×C₄×C₈₉	—	2	2	no	—
77	C₂×C₄×C₈₉	—	2	1	no	—
78	C₂×C₄×C₈₉	—	4	4	no	—
79	C₂×C₄×C₈₉	—	8	2	no	—
80	C₂×C₄×C₈₉	—	8	2	no	—
81	C₂×C₄×C₈₉	#102	1	2	yes	—
82	C₂×C₄×C₈₉	#102	1	2	yes	—
83	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₈₉	#135	1	2	yes	—
87	C₂×Dic₈₉	—	4	2	no	—
88	C₂×Dic₈₉	—	4	4	no	—
89	C₂×Dic₈₉	—	4	2	no	—
90	C₂×Dic₈₉	—	8	1	no	—
91	C₂×Dic₈₉	—	8	1	no	—
92	Dic₁₇₈	—	2	4	no	—
93	Dic₁₇₈	—	4	2	no	—
94	Dic₁₇₈	—	4	4	no	—
95	C₈×C₈₉	—	2	2	no	—
96	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
97	C₈×C₈₉	#117	1	1	yes	characteristic
98	C₈×C₈₉	C₄×D₄	1	2	yes	—
99	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
100	C₂²×C₈₉	C₄×D₈	1	2	yes	—
101	C₂²×C₈₉	#509	1	1	yes	characteristic
102	C₂²×C₈₉	#510	1	1	yes	characteristic
103	C₂²×C₈₉	#510	1	2	yes	—
104	C₄×C₈₉	—	2	2	no	—
105	C₄×C₈₉	—	2	2	no	—
106	C₄×C₈₉	—	8	4	no	—
107	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
108	C₄×C₈₉	#416	1	1	yes	characteristic
109	C₄×C₈₉	C₄×SD₁₆	1	2	yes	—
110	Dic₈₉	—	4	4	no	—
111	Dic₈₉	—	8	2	no	—
112	#12638	—	89	1	no	maximal, Sylow
113	C₂×C₈₉	C₄×#102	1	1	yes	characteristic
114	C₂×C₈₉	#2139	1	1	yes	characteristic
115	C₂×C₈₉	#2141	1	1	yes	characteristic
116	C₂×C₈₉	#2175	1	2	yes	—
117	C₂×C₈₉	#2176	1	2	yes	—
118	C₂×C₄×Q₁₆	—	89	1	no	—
119	C₂×#358	—	89	2	no	—
120	C₂×#346	—	89	1	no	—
121	C₄×#107	D₈₉	1	1	yes	characteristic
122	#1380	—	89	2	no	—
123	C₈₉	#12638	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₁₆	—	89	1	no	—
125	C₂×C₄×Q₈	—	178	1	no	—
126	C₂²×C₁₆	—	89	2	no	—
127	C₄×Q₁₆	—	89	4	no	—
128	C₄×Q₁₆	—	178	2	no	—
129	C₂×#107	—	89	1	no	—
130	C₂×#107	C₂×D₈₉	1	2	yes	—
131	C₂×#103	—	178	1	no	—
132	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
133	C₄×C₄⋊C₄	—	2	1	no	—
134	#358	—	89	8	no	—
135	#346	—	178	2	no	—
136	#340	—	2	1	no	—
137	C₂²×Q₈	—	178	1	no	—
138	C₂×Q₁₆	—	89	4	no	—
139	C₂×Q₁₆	—	89	2	no	—
140	C₂×Q₁₆	—	178	1	no	—
141	C₂×Q₁₆	—	178	2	no	—
142	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
144	C₄×Q₈	—	356	2	no	—
145	C₄×Q₈	—	178	4	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	356	1	no	—
148	C₂×C₄⋊C₄	—	178	1	no	—
149	C₂×C₄²	—	4	1	no	—
150	C₂×C₄²	—	356	1	no	—
151	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
152	C₂×C₁₆	—	89	4	no	—
153	C₂×C₁₆	—	178	2	no	—
154	C₂×C₁₆	—	178	2	no	—
155	#107	—	2	2	no	—
156	#107	—	89	2	no	—
157	#107	—	178	1	no	—
158	#107	C₄×D₈₉	1	4	yes	—
159	#103	—	178	4	no	—
160	C₄×C₈	C₄×D₈₉	1	2	yes	—
161	C₄×C₈	D₃₅₆	1	2	yes	—
162	#95	—	4	1	no	—
163	C₂×Q₈	—	356	1	no	—
164	C₂×Q₈	—	356	2	no	—
165	C₂×Q₈	—	178	4	no	—
166	C₂×Q₈	—	178	2	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	356	1	no	—
169	C₂²×C₄	—	356	1	no	—
170	C₂²×C₄	#13083559	1	1	yes	characteristic
171	C₂²×C₄	#13083559	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
173	Q₁₆	—	89	4	no	—
174	Q₁₆	—	178	2	no	—
175	Q₁₆	—	178	4	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	#13083551	1	2	yes	—
178	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
179	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
180	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
181	C₂×C₈	#13083574	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	356	1	no	—
185	C₄⋊C₄	—	356	4	no	—
186	C₄⋊C₄	—	178	2	no	—
187	C₄²	—	8	2	no	—
188	C₄²	—	356	4	no	—
189	C₄²	#13083551	1	2	yes	—
190	C₄²	#13083561	1	2	yes	—
191	C₁₆	—	178	4	no	—
192	C₂³	C₂×#13083559	1	1	yes	characteristic
193	Q₈	—	356	4	no	—
194	Q₈	—	356	2	no	—
195	Q₈	—	178	4	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	8	2	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	—	356	4	no	—
203	C₂×C₄	—	356	2	no	—
204	C₂×C₄	—	356	2	no	—
205	C₂×C₄	—	712	1	no	—
206	C₂×C₄	—	712	1	no	—
207	C₂×C₄	#15667932	1	2	yes	—
208	C₂×C₄	#15667932	1	2	yes	—
209	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
210	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
211	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
212	C₂×C₄	#15668032	1	2	yes	—
213	C₈	—	2	2	no	—
214	C₈	C₄²×D₈₉	1	1	yes	characteristic
215	C₈	#15667969	1	1	yes	characteristic
216	C₈	C₄×D₃₅₆	1	2	yes	—
217	C₂²	C₂×#15667932	1	1	yes	characteristic
218	C₂²	C₄×#13083559	1	2	yes	—
219	C₂²	#19152034	1	1	yes	characteristic
220	C₂²	#19152038	1	2	yes	—
221	C₂²	#19152039	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	—	356	4	no	—
226	C₄	—	712	2	no	—
227	C₄	C₄×#13083551	1	1	yes	characteristic
228	C₄	#19151592	1	1	yes	characteristic
229	C₄	C₄×#13083561	1	2	yes	—
230	C₂	C₄×#15667932	1	1	yes	characteristic
231	C₂	#26021962	1	1	yes	characteristic
232	C₂	#26021971	1	1	yes	characteristic
233	C₂	#26022116	1	2	yes	—
234	C₂	#26022121	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#13083559
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₄×#107×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄×#107	D₈₉
hypocenter	C₈₉	#12638
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₃₅₆	C₄	semidirect
13. #19151401	C₄	semidirect
53. Dic₃₅₆	C₄²	semidirect
123. C₈₉	#12638	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
C₂×#19151401	2
C₂×#19151374	2
#26018343	2
C₄×#107×C₈₉	2
#12638	89