Subgroups — #36563123

Order 22784.

802 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#10623494	C₂	1	2	yes	maximal
2	#26021193	C₂	1	2	yes	maximal
3	#26020976	C₂	1	4	yes	maximal
4	#26020406	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×#10623494	C₂	1	4	yes	maximal
6	#26019939	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#2399	C₂	1	1	yes	maximal, characteristic
8	Q₈×#10623494	—	2	8	no	—
9	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
10	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
11	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
12	C₂×#15667923	C₂²	1	1	yes	characteristic
13	C₂×#15667923	C₂²	1	2	yes	—
14	C₂×C₄×#10623494	C₂²	1	4	yes	—
15	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
16	C₂×C₄×#10623494	C₂²	1	2	yes	—
17	#19151353	C₂²	1	8	yes	—
18	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
19	#398×C₈₉	C₄	1	4	yes	—
20	#395×C₈₉	C₄	1	4	yes	—
21	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
22	#376×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×#10623494	C₂³	1	4	yes	—
24	C₂²×#10623494	C₂³	1	1	yes	characteristic
25	C₂²×#10623494	C₂³	1	2	yes	—
26	C₂²×#10623494	C₂³	1	1	yes	characteristic
27	C₄⋊#10623494	—	2	8	no	—
28	C₄⋊#10623494	—	2	4	no	—
29	C₄⋊#10623494	—	2	8	no	—
30	#15667923	—	2	2	no	—
31	#15667923	—	2	4	no	—
32	C₄×#10623494	—	2	8	no	—
33	C₄×#10623494	D₄	1	4	yes	—
34	C₄×#10623494	Q₈	1	8	yes	complemented
35	C₂²×Q₈×C₈₉	C₂×C₄	1	2	yes	—
36	C₄×Q₈×C₈₉	—	2	4	no	—
37	C₄×Q₈×C₈₉	—	2	4	no	—
38	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	8	yes	—
39	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
42	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
44	#95×C₈₉	C₂×C₄	1	4	yes	—
45	#95×C₈₉	C₂×C₄	1	4	yes	—
46	#95×C₈₉	C₂×C₄	1	4	yes	—
47	C₂×#10623494	—	2	2	no	—
48	C₂×#10623494	—	2	2	no	—
49	C₂×#10623494	—	2	2	no	—
50	C₂×#10623494	—	2	4	no	—
51	C₂×#10623494	—	2	8	no	—
52	C₂×#10623494	C₂×D₄	1	2	yes	—
53	C₂×#10623494	C₂×D₄	1	2	yes	—
54	C₂×#10623494	C₂×Q₈	1	4	yes	—
55	C₂×#10623494	C₂×Q₈	1	4	yes	—
56	C₂×Q₈×C₈₉	—	2	4	no	—
57	C₂×Q₈×C₈₉	—	2	8	no	—
58	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₈₉	—	2	8	no	—
64	C₄⋊C₄×C₈₉	—	2	4	no	—
65	C₄⋊C₄×C₈₉	—	2	16	no	—
66	C₄²×C₈₉	—	2	4	no	—
67	C₄²×C₈₉	—	2	4	no	—
68	C₄²×C₈₉	C₂×D₄	1	2	yes	—
69	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
70	C₄²×C₈₉	#41	1	2	yes	—
71	C₄²×C₈₉	#41	1	4	yes	—
72	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
73	Q₈×C₈₉	—	2	16	no	—
74	C₂×C₄×C₈₉	—	2	4	no	—
75	C₂×C₄×C₈₉	—	2	8	no	—
76	C₂×C₄×C₈₉	—	2	4	no	—
77	C₂×C₄×C₈₉	—	2	4	no	—
78	C₂×C₄×C₈₉	—	2	4	no	—
79	C₂×C₄×C₈₉	—	2	8	no	—
80	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
81	C₂×C₄×C₈₉	C₄×Q₈	1	8	yes	—
82	C₂×C₄×C₈₉	#131	1	8	yes	—
83	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
85	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
87	#10623494	—	2	2	no	—
88	#10623494	—	2	4	no	—
89	#10623494	—	4	4	no	—
90	#10623494	C₄⋊Q₈	1	4	yes	—
91	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
92	C₂²×C₈₉	C₂×C₄×Q₈	1	2	yes	—
93	C₂²×C₈₉	C₂×#131	1	4	yes	—
94	C₄×C₈₉	—	2	4	no	—
95	C₄×C₈₉	—	2	8	no	—
96	C₄×C₈₉	—	2	8	no	—
97	C₄×C₈₉	—	4	4	no	—
98	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
99	C₄×C₈₉	#532	1	1	yes	characteristic
100	C₄×C₈₉	#533	1	2	yes	—
101	#21715	—	89	1	no	maximal, Sylow
102	C₂×C₈₉	C₄×C₄⋊Q₈	1	1	yes	characteristic
103	C₂×C₈₉	#2929	1	2	yes	—
104	C₂×C₈₉	#2960	1	4	yes	—
105	C₂×C₈×Q₈	—	89	2	no	—
106	#2399	D₈₉	1	1	yes	characteristic
107	#1985	—	89	2	no	—
108	#1940	—	89	4	no	—
109	#1843	—	89	1	no	—
110	C₈×C₄⋊C₄	—	89	4	no	—
111	#1726	—	89	1	no	—
112	C₈₉	#21715	1	1	yes	characteristic, complemented, Sylow
113	C₂×C₄×Q₈	C₂×D₈₉	1	2	yes	—
114	C₈×Q₈	—	178	8	no	—
115	C₂×C₄⋊C₈	—	89	2	no	—
116	C₂×C₄⋊C₈	—	89	4	no	—
117	C₂×C₄⋊C₈	—	89	4	no	—
118	C₂×#97	—	89	2	no	—
119	C₂×#97	—	89	1	no	—
120	C₂×C₄×C₈	—	89	1	no	—
121	C₂×C₄×C₈	—	89	2	no	—
122	C₂×C₄×C₈	—	89	4	no	—
123	#398	Dic₈₉	1	4	yes	—
124	#395	Dic₈₉	1	4	yes	—
125	C₄×C₄⋊C₄	C₂×D₈₉	1	4	yes	—
126	#376	C₂×D₈₉	1	1	yes	characteristic
127	#336	—	89	8	no	—
128	C₂²×Q₈	C₂×Dic₈₉	1	2	yes	—
129	C₂²×C₈	—	89	1	no	—
130	C₂²×C₈	—	89	2	no	—
131	C₂²×C₈	—	89	1	no	—
132	C₂²×C₈	—	89	4	no	—
133	C₄×Q₈	—	2	4	no	—
134	C₄×Q₈	—	2	4	no	—
135	C₂×C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
136	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
137	C₂×C₄⋊C₄	C₂×Dic₈₉	1	8	yes	—
138	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
139	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
140	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
141	C₄⋊C₈	—	178	8	no	—
142	C₄⋊C₈	—	178	4	no	—
143	C₄⋊C₈	—	178	8	no	—
144	#97	—	178	2	no	—
145	#97	—	178	4	no	—
146	C₄×C₈	—	89	4	no	—
147	C₄×C₈	—	89	8	no	—
148	C₄×C₈	—	178	8	no	—
149	#95	C₂×Dic₈₉	1	4	yes	—
150	#95	C₂×Dic₈₉	1	4	yes	—
151	#95	C₂×Dic₈₉	1	4	yes	—
152	C₂×Q₈	—	2	4	no	—
153	C₂×Q₈	—	2	8	no	—
154	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
155	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
156	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
157	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
158	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
159	C₂×C₈	—	89	2	no	—
160	C₂×C₈	—	89	4	no	—
161	C₂×C₈	—	89	2	no	—
162	C₂×C₈	—	89	4	no	—
163	C₂×C₈	—	178	4	no	—
164	C₂×C₈	—	178	8	no	—
165	C₂×C₈	—	178	2	no	—
166	C₂×C₈	—	178	2	no	—
167	C₂×C₈	—	178	2	no	—
168	C₄⋊C₄	—	2	8	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	16	no	—
171	C₄²	—	2	4	no	—
172	C₄²	—	2	4	no	—
173	C₄²	D₄×D₈₉	1	2	yes	—
174	C₄²	Q₈×D₈₉	1	4	yes	—
175	C₄²	#13083573	1	2	yes	—
176	C₄²	#13083575	1	4	yes	—
177	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
178	Q₈	—	2	16	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	D₄×Dic₈₉	1	4	yes	—
186	C₂×C₄	Q₈×Dic₈₉	1	8	yes	—
187	C₂×C₄	#15668019	1	8	yes	—
188	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
189	C₂×C₄	C₂×Q₈×D₈₉	1	2	yes	—
190	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
191	C₂×C₄	C₂×#13083575	1	2	yes	—
192	C₈	—	89	4	no	—
193	C₈	—	356	4	no	—
194	C₈	—	178	2	no	—
195	C₈	—	178	4	no	—
196	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
197	C₂²	C₂×Q₈×Dic₈₉	1	2	yes	—
198	C₂²	C₂×#15668019	1	4	yes	—
199	C₄	—	2	8	no	—
200	C₄	—	2	4	no	—
201	C₄	—	2	8	no	—
202	C₄	—	4	4	no	—
203	C₄	C₄⋊Q₈×D₈₉	1	1	yes	characteristic
204	C₄	#19152149	1	1	yes	characteristic
205	C₄	#19152153	1	2	yes	—
206	C₂	C₄⋊Q₈×Dic₈₉	1	1	yes	characteristic
207	C₂	#26028947	1	2	yes	—
208	C₂	#26029187	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator, socle, prime radical	C₂³×C₈₉	C₂³×C₄
Fitting	C₈₉×#2399	C₂
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#2399	D₈₉
hypocenter	C₈₉	#21715
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #2399

Complements

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

N	H ≅ G/N	type
34. C₄×#10623494	Q₈	semidirect
112. C₈₉	#21715	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#10623494	2
#26021193	2
#26020976	2
#26020406	2
C₄⋊C₄×#10623494	2
#26019939	2
C₈₉×#2399	2
#21715	89