Subgroups — C₂×#26022124 #36707726

Order 22784.

670 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₅₆	C₂	1	2	yes	maximal
3	#26022124	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151426	C₂	1	1	yes	maximal, characteristic
5	C₂×#19151403	C₂	1	2	yes	maximal
6	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₅₆	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₇₈	—	2	2	no	—
9	C₂×Dic₇₁₂	C₄	1	8	yes	complemented
10	C₂×Dic₇₁₂	C₂²	1	4	yes	—
11	C₄×Dic₃₅₆	—	2	4	no	—
12	C₄×Dic₃₅₆	C₂²	1	8	yes	—
13	C₂×#15667946	C₂²	1	2	yes	—
14	C₂×#15667937	—	2	2	no	—
15	#19151426	C₂²	1	4	yes	—
16	#19151403	C₂²	1	8	yes	—
17	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	#346×C₈₉	C₂²	1	4	yes	—
21	C₂²×Dic₁₇₈	—	2	2	no	—
22	C₂×Dic₃₅₆	—	2	2	no	—
23	C₂×Dic₃₅₆	—	2	4	no	—
24	C₂×Dic₃₅₆	C₂×C₄	1	8	yes	—
25	C₂×Dic₃₅₆	C₂³	1	4	yes	—
26	C₄×Dic₁₇₈	—	2	8	no	—
27	C₄×Dic₁₇₈	—	4	4	no	—
28	C₂×C₄⋊Dic₈₉	—	2	2	no	—
29	C₂×#13083553	—	4	2	no	—
30	C₂×C₄×Dic₈₉	—	4	2	no	—
31	Dic₇₁₂	C₂×C₄	1	16	yes	complemented
32	#15667946	—	2	2	no	—
33	#15667946	C₂³	1	4	yes	—
34	#15667937	—	2	8	no	—
35	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₈₉	—	2	2	no	—
39	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
40	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
41	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
42	C₄×C₈×C₈₉	D₄	1	2	yes	—
43	C₄×C₈×C₈₉	C₂³	1	2	yes	—
44	C₂×Dic₁₇₈	—	2	8	no	—
45	C₂×Dic₁₇₈	—	2	4	no	—
46	C₂×Dic₁₇₈	—	4	2	no	—
47	C₂×Dic₁₇₈	—	4	4	no	—
48	C₂²×Dic₈₉	—	4	2	no	—
49	C₂²×Dic₈₉	—	4	2	no	—
50	Dic₃₅₆	—	2	8	no	—
51	Dic₃₅₆	—	2	4	no	—
52	Dic₃₅₆	C₂²×C₄	1	8	yes	—
53	C₄⋊Dic₈₉	—	2	4	no	—
54	C₄⋊Dic₈₉	—	4	2	no	—
55	#13083553	—	4	8	no	—
56	C₄×Dic₈₉	—	4	8	no	—
57	C₂²×C₄×C₈₉	D₈	1	2	yes	—
58	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	—	2	2	no	—
60	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₈₉	#41	1	2	yes	—
64	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
65	C₄²×C₈₉	D₈	1	2	yes	—
66	C₄²×C₈₉	C₂×D₄	1	2	yes	—
67	C₁₆×C₈₉	—	2	2	no	—
68	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
69	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₈₉	—	2	4	no	—
71	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
73	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₈₉	#135	1	2	yes	—
75	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
76	C₂×Dic₈₉	—	4	4	no	—
77	C₂×Dic₈₉	—	4	8	no	—
78	C₂×Dic₈₉	—	4	4	no	—
79	C₂×Dic₈₉	—	8	4	no	—
80	Dic₁₇₈	—	2	8	no	—
81	Dic₁₇₈	—	4	4	no	—
82	Dic₁₇₈	—	4	8	no	—
83	C₈×C₈₉	—	2	2	no	—
84	C₈×C₈₉	C₄×D₄	1	2	yes	—
85	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
86	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
87	C₂²×C₈₉	C₄×D₈	1	2	yes	—
88	C₂²×C₈₉	#510	1	4	yes	—
89	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
90	C₄×C₈₉	—	2	4	no	—
91	C₄×C₈₉	C₄×D₈	1	2	yes	—
92	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
93	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
94	Dic₈₉	—	4	8	no	—
95	Dic₈₉	—	8	4	no	—
96	C₂×#2176	—	89	1	no	maximal, Sylow
97	C₂×C₈₉	#2176	1	4	yes	complemented
98	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
99	C₂×C₈₉	C₂×#510	1	2	yes	—
100	C₂²×Q₃₂	—	89	1	no	—
101	C₂×C₄×Q₁₆	—	89	2	no	—
102	#2176	—	89	8	no	—
103	C₂×#367	—	89	1	no	—
104	C₂×#358	—	89	2	no	—
105	C₂×#346	D₈₉	1	1	yes	characteristic
106	C₈₉	C₂×#2176	1	1	yes	characteristic, complemented, Sylow
107	C₂²×Q₁₆	—	89	2	no	—
108	C₂×C₄×Q₈	—	178	2	no	—
109	C₂×Q₃₂	—	89	8	no	—
110	C₂×Q₃₂	—	89	4	no	—
111	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
112	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
113	C₄×Q₁₆	—	89	8	no	—
114	C₄×Q₁₆	—	178	4	no	—
115	C₂×#107	—	89	2	no	—
116	C₂×#103	—	178	2	no	—
117	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
118	#367	—	89	4	no	—
119	#358	—	89	8	no	—
120	#346	C₂×D₈₉	1	4	yes	—
121	C₂²×Q₈	—	178	2	no	—
122	C₂×Q₁₆	—	89	8	no	—
123	C₂×Q₁₆	—	89	4	no	—
124	C₂×Q₁₆	—	178	2	no	—
125	C₂×Q₁₆	—	178	4	no	—
126	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
128	C₄×Q₈	—	356	4	no	—
129	C₄×Q₈	—	178	8	no	—
130	C₂×C₄⋊C₄	—	356	2	no	—
131	C₂×C₄⋊C₄	—	178	2	no	—
132	C₂×C₄²	—	356	2	no	—
133	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
134	Q₃₂	—	89	16	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	C₄×D₈₉	1	4	yes	—
137	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
138	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
139	#107	—	89	4	no	—
140	#107	—	178	2	no	—
141	#103	—	178	8	no	—
142	C₄×C₈	D₃₅₆	1	2	yes	—
143	C₄×C₈	C₂²×D₈₉	1	2	yes	—
144	C₂×Q₈	—	356	2	no	—
145	C₂×Q₈	—	356	4	no	—
146	C₂×Q₈	—	178	8	no	—
147	C₂×Q₈	—	178	4	no	—
148	C₂²×C₄	—	356	2	no	—
149	C₂²×C₄	—	356	2	no	—
150	C₂²×C₄	D₇₁₂	1	2	yes	—
151	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
152	Q₁₆	—	89	8	no	—
153	Q₁₆	—	178	8	no	—
154	Q₁₆	—	178	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
157	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
158	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
159	C₂×C₈	#13083574	1	2	yes	—
160	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
161	C₄⋊C₄	—	356	8	no	—
162	C₄⋊C₄	—	356	2	no	—
163	C₄⋊C₄	—	178	4	no	—
164	C₄²	—	356	8	no	—
165	C₄²	D₇₁₂	1	2	yes	—
166	C₄²	C₂×D₃₅₆	1	2	yes	—
167	C₁₆	—	2	2	no	—
168	C₁₆	C₂×C₄×D₈₉	1	4	yes	—
169	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
170	Q₈	—	356	8	no	—
171	Q₈	—	356	4	no	—
172	Q₈	—	178	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	356	8	no	—
175	C₂×C₄	—	356	4	no	—
176	C₂×C₄	—	356	4	no	—
177	C₂×C₄	—	712	4	no	—
178	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
179	C₂×C₄	C₂×D₇₁₂	1	4	yes	—
180	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
181	C₂×C₄	#15668034	1	2	yes	—
182	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	C₄×D₃₅₆	1	2	yes	—
185	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
186	C₈	C₂×#13083574	1	1	yes	characteristic
187	C₂²	C₄×D₇₁₂	1	2	yes	—
188	C₂²	#19152041	1	4	yes	—
189	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
190	C₄	—	2	4	no	—
191	C₄	—	356	8	no	—
192	C₄	—	712	4	no	—
193	C₄	C₄×D₇₁₂	1	2	yes	—
194	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
195	C₄	C₂×#15668034	1	1	yes	characteristic
196	C₂	#26022124	1	4	yes	direct product factor
197	C₂	C₂×C₄×D₇₁₂	1	1	yes	characteristic
198	C₂	C₂×#19152041	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#346×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	C₂×#2176
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #26022124	C₂	direct (N × H)
9. C₂×Dic₇₁₂	C₄	semidirect
31. Dic₇₁₂	C₂×C₄	semidirect
97. C₂×C₈₉	#2176	semidirect
106. C₈₉	C₂×#2176	semidirect
196. C₂	#26022124	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₇₁₂	2
C₂×C₄×Dic₃₅₆	2
#26022124	2
C₂×#19151426	2
C₂×#19151403	2
C₂×#346×C₈₉	2
C₂×#2176	89

Subgroups — C2×#26022124 #36707726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022124 #36707726