Subgroups — #36523665

Order 22784.

702 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	#26028820	C₂	1	2	yes	maximal
2	#26019027	C₂	1	6	yes	maximal, complemented
3	#26019011	C₂	1	3	yes	maximal, complemented
4	C₂×#19151351	C₂	1	3	yes	maximal, complemented
5	C₈×#41×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668019	—	2	2	no	—
7	#19151625	—	2	6	no	—
8	#19151622	C₂²	1	6	yes	—
9	C₂×C₄⋊#10623494	C₂²	1	6	yes	—
10	#19151351	C₂²	1	1	yes	characteristic
11	#19151351	C₂²	1	3	yes	—
12	#19151351	C₂²	1	6	yes	—
13	#19151351	C₂²	1	6	yes	—
14	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₈₉	C₄	1	1	yes	characteristic
16	C₈×Q₈×C₈₉	C₄	1	1	yes	characteristic
17	C₈×D₄×C₈₉	C₄	1	3	yes	—
18	C₈×D₄×C₈₉	C₄	1	3	yes	—
19	#431×C₈₉	C₂²	1	3	yes	—
20	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
21	#15668019	—	4	8	no	—
22	C₂×#13083558	—	2	6	no	—
23	C₂²×#10623494	—	2	6	no	—
24	C₄⋊#10623494	—	2	6	no	—
25	C₄⋊#10623494	—	2	6	no	—
26	C₄⋊#10623494	C₂³	1	2	yes	—
27	C₄⋊#10623494	C₂³	1	6	yes	—
28	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	C₂×C₄	1	3	yes	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	3	yes	—
31	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₄×Q₈×C₈₉	Q₈	1	1	yes	characteristic
33	C₄×D₄×C₈₉	C₂×C₄	1	3	yes	—
34	C₄×D₄×C₈₉	Q₈	1	3	yes	—
35	#117×C₈₉	D₄	1	3	yes	—
36	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
37	C₄⋊C₈×C₈₉	C₂×C₄	1	3	yes	—
38	C₄⋊C₈×C₈₉	C₂×C₄	1	3	yes	—
39	C₂²⋊C₈×C₈₉	C₂×C₄	1	6	yes	—
40	C₄×C₈×C₈₉	C₂×C₄	1	3	yes	—
41	C₄×C₈×C₈₉	C₂×C₄	1	3	yes	—
42	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₈₉	C₂³	1	3	yes	—
44	#13083558	—	4	12	no	—
45	C₂×#10623494	—	2	2	no	—
46	C₂×#10623494	—	2	6	no	—
47	C₂×#10623494	—	4	12	no	—
48	#41×C₈₉	D₈	1	4	yes	—
49	#41×C₈₉	Q₁₆	1	4	yes	—
50	C₂×Q₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂×D₄×C₈₉	C₄⋊C₄	1	3	yes	—
52	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
54	C₂²×C₄×C₈₉	C₂×Q₈	1	3	yes	—
55	C₂×C₈×C₈₉	—	2	6	no	—
56	C₂×C₈×C₈₉	—	4	3	no	—
57	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₈₉	C₂²×C₄	1	3	yes	—
59	C₂×C₈×C₈₉	C₂²×C₄	1	3	yes	—
60	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
61	C₄⋊C₄×C₈₉	C₄⋊C₄	1	3	yes	—
62	C₄⋊C₄×C₈₉	C₄⋊C₄	1	3	yes	—
63	C₂²⋊C₄×C₈₉	C₄⋊C₄	1	6	yes	—
64	C₄²×C₈₉	C₂²×C₄	1	3	yes	—
65	C₄²×C₈₉	C₂×D₄	1	3	yes	—
66	C₄²×C₈₉	C₂×Q₈	1	3	yes	—
67	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
68	C₂³×C₈₉	C₂×C₄⋊C₄	1	3	yes	—
69	Q₈×C₈₉	#107	1	4	yes	—
70	D₄×C₈₉	#107	1	12	yes	—
71	C₂×C₄×C₈₉	—	2	3	no	—
72	C₂×C₄×C₈₉	—	2	3	no	—
73	C₂×C₄×C₈₉	—	4	3	no	—
74	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	3	yes	—
77	C₂×C₄×C₈₉	C₂×D₈	1	6	yes	—
78	C₂×C₄×C₈₉	C₂×Q₁₆	1	6	yes	—
79	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
82	C₈×C₈₉	—	2	3	no	—
83	C₈×C₈₉	—	2	1	no	—
84	C₈×C₈₉	—	2	3	no	—
85	C₈×C₈₉	#131	1	2	yes	complemented
86	#10623494	—	4	2	no	—
87	#10623494	—	4	6	no	—
88	C₂²×C₈₉	—	2	3	no	—
89	C₂²×C₈₉	—	2	3	no	—
90	C₂²×C₈₉	C₂×#107	1	6	yes	—
91	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₈₉	—	2	3	no	—
93	C₄×C₈₉	—	2	1	no	—
94	C₄×C₈₉	—	2	3	no	—
95	C₄×C₈₉	C₂×#107	1	6	yes	—
96	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
97	C₄×C₈₉	#568	1	1	yes	characteristic
98	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
99	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
100	#17828	—	89	1	no	maximal, Sylow
101	C₂×C₈₉	—	2	6	no	—
102	C₂×C₈₉	#2906	1	1	yes	characteristic
103	C₂×C₈₉	C₂²×#107	1	1	yes	characteristic
104	C₂×C₈₉	#2916	1	1	yes	characteristic
105	C₈×#41	D₈₉	1	1	yes	characteristic
106	#2906	—	89	2	no	—
107	#1571	—	89	6	no	—
108	#1567	—	89	3	no	—
109	C₂×#335	—	89	3	no	—
110	C₈₉	#17828	1	1	yes	characteristic, complemented, Sylow
111	C₂×#131	—	178	2	no	—
112	C₄×#41	C₂×D₈₉	1	1	yes	characteristic
113	C₈×Q₈	Dic₈₉	1	1	yes	characteristic
114	C₈×Q₈	Dic₈₉	1	1	yes	characteristic
115	C₈×D₄	Dic₈₉	1	3	yes	—
116	C₈×D₄	Dic₈₉	1	3	yes	—
117	#431	C₂×D₈₉	1	3	yes	—
118	#424	—	178	6	no	—
119	#423	—	89	6	no	—
120	C₂×C₄⋊C₈	—	89	6	no	—
121	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
122	#335	—	89	1	no	—
123	#335	—	89	3	no	—
124	#335	—	89	6	no	—
125	#335	—	89	6	no	—
126	C₂×#41	D₃₅₆	1	1	yes	characteristic
127	#131	—	356	8	no	—
128	C₂×M₁₆	—	178	6	no	—
129	C₂²×C₈	—	178	6	no	—
130	C₂²×C₈	C₂×Dic₈₉	1	3	yes	—
131	C₂²×C₈	C₂×Dic₈₉	1	3	yes	—
132	C₄×Q₈	Dic₁₇₈	1	1	yes	characteristic
133	C₄×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
134	C₄×D₄	Dic₁₇₈	1	3	yes	—
135	C₄×D₄	C₂×Dic₈₉	1	3	yes	—
136	#117	D₃₅₆	1	3	yes	—
137	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
138	C₄⋊C₈	—	89	2	no	—
139	C₄⋊C₈	—	89	6	no	—
140	C₄⋊C₈	—	178	6	no	—
141	C₄⋊C₈	—	178	6	no	—
142	C₄⋊C₈	C₂×Dic₈₉	1	3	yes	—
143	C₄⋊C₈	C₂×Dic₈₉	1	3	yes	—
144	C₂²⋊C₈	C₂×Dic₈₉	1	6	yes	—
145	C₄×C₈	C₂×Dic₈₉	1	3	yes	—
146	C₄×C₈	C₂×Dic₈₉	1	3	yes	—
147	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
148	C₄×C₈	C₂²×D₈₉	1	3	yes	—
149	#41	D₇₁₂	1	4	yes	—
150	#41	Dic₃₅₆	1	4	yes	—
151	C₂×Q₈	C₄⋊Dic₈₉	1	1	yes	characteristic
152	C₂×D₄	C₄⋊Dic₈₉	1	3	yes	—
153	C₂²×C₄	C₂²×Dic₈₉	1	3	yes	—
154	C₂²×C₄	C₂×D₃₅₆	1	3	yes	—
155	C₂²×C₄	C₂×Dic₁₇₈	1	3	yes	—
156	M₁₆	—	356	12	no	—
157	C₂×C₈	—	2	6	no	—
158	C₂×C₈	—	4	3	no	—
159	C₂×C₈	—	356	12	no	—
160	C₂×C₈	—	178	2	no	—
161	C₂×C₈	—	178	6	no	—
162	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
163	C₂×C₈	C₂²×Dic₈₉	1	3	yes	—
164	C₂×C₈	C₂²×Dic₈₉	1	3	yes	—
165	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
166	C₄⋊C₄	C₄⋊Dic₈₉	1	3	yes	—
167	C₄⋊C₄	C₄⋊Dic₈₉	1	3	yes	—
168	C₂²⋊C₄	C₄⋊Dic₈₉	1	6	yes	—
169	C₄²	C₂²×Dic₈₉	1	3	yes	—
170	C₄²	C₂×D₃₅₆	1	3	yes	—
171	C₄²	C₂×Dic₁₇₈	1	3	yes	—
172	C₄²	C₂³×D₈₉	1	1	yes	characteristic
173	C₂³	C₂×C₄⋊Dic₈₉	1	3	yes	—
174	Q₈	#15667946	1	4	yes	—
175	D₄	#15667946	1	12	yes	—
176	C₂×C₄	—	2	3	no	—
177	C₂×C₄	—	2	3	no	—
178	C₂×C₄	—	4	3	no	—
179	C₂×C₄	C₂×C₄⋊Dic₈₉	1	3	yes	—
180	C₂×C₄	C₂×C₄⋊Dic₈₉	1	3	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₈₉	1	3	yes	—
182	C₂×C₄	C₂×D₇₁₂	1	6	yes	—
183	C₂×C₄	C₂×Dic₃₅₆	1	6	yes	—
184	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
185	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
186	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
187	C₈	—	2	3	no	—
188	C₈	—	2	1	no	—
189	C₈	—	2	3	no	—
190	C₈	—	356	2	no	—
191	C₈	—	356	6	no	—
192	C₈	#15668019	1	2	yes	complemented
193	C₂²	—	2	3	no	—
194	C₂²	—	2	3	no	—
195	C₂²	C₂×#15667946	1	6	yes	—
196	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
197	C₄	—	2	3	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	3	no	—
200	C₄	C₂×#15667946	1	6	yes	—
201	C₄	C₂×#15668019	1	1	yes	characteristic
202	C₄	#19152365	1	1	yes	characteristic
203	C₄	C₂²×D₇₁₂	1	1	yes	characteristic
204	C₄	C₂²×Dic₃₅₆	1	1	yes	characteristic
205	C₂	—	2	6	no	—
206	C₂	#26028820	1	1	yes	characteristic
207	C₂	C₂²×#15667946	1	1	yes	characteristic
208	C₂	#26028863	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈×#41×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₈₉	D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₈×#41	D₈₉
hypocenter	C₈₉	#17828
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #26019027	C₂	semidirect
3. #26019011	C₂	semidirect
4. C₂×#19151351	C₂	semidirect
85. C₈×C₈₉	#131	semidirect
110. C₈₉	#17828	semidirect
192. C₈	#15668019	semidirect

Maximal subgroups

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

subgroup	index
#26028820	2
#26019027	2
#26019011	2
C₂×#19151351	2
C₈×#41×C₈₉	2
#17828	89