Subgroups — C₂×#26028649 #36760525

Order 22784.

3774 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15668011	C₂	1	6	yes	maximal
2	#26028649	C₂	1	16	yes	maximal, direct product factor
3	C₂×#19151468	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151467	C₂	1	6	yes	maximal
5	C₂×C₄²×Dic₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#384×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₇₈	C₂²	1	4	yes	—
8	C₂×#15668011	—	2	48	no	—
9	C₂×#15668011	C₂²	1	48	yes	—
10	C₂²×C₄⋊Dic₈₉	C₂²	1	6	yes	—
11	C₂²×#13083553	C₂²	1	12	yes	—
12	C₂²×C₄×Dic₈₉	C₂²	1	3	yes	—
13	C₂²×C₄×Dic₈₉	C₂²	1	3	yes	—
14	#19151468	C₂²	1	8	yes	—
15	#19151467	C₂²	1	48	yes	—
16	C₄²×Dic₈₉	C₂²	1	8	yes	—
17	C₂²×C₄⋊C₄×C₈₉	C₂²	1	6	yes	—
18	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
19	#384×C₈₉	C₂²	1	8	yes	—
20	C₂²×Dic₁₇₈	—	2	48	no	—
21	C₂²×Dic₁₇₈	C₂³	1	16	yes	—
22	C₂³×Dic₈₉	C₂³	1	4	yes	—
23	C₂³×Dic₈₉	C₂³	1	1	yes	characteristic
24	C₂³×Dic₈₉	C₂³	1	3	yes	—
25	#15668011	—	2	192	no	—
26	C₂×C₄⋊Dic₈₉	—	2	24	no	—
27	C₂×C₄⋊Dic₈₉	C₂³	1	24	yes	—
28	C₂×#13083553	—	2	48	no	—
29	C₂×#13083553	C₂³	1	48	yes	—
30	C₂×C₄×Dic₈₉	D₄	1	24	yes	—
31	C₂×C₄×Dic₈₉	Q₈	1	24	yes	complemented
32	C₂×C₄×Dic₈₉	C₂³	1	12	yes	—
33	C₂×C₄×Dic₈₉	C₂³	1	12	yes	—
34	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
35	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
36	C₂×C₄⋊C₄×C₈₉	—	2	24	no	—
37	C₂×C₄⋊C₄×C₈₉	C₂³	1	24	yes	—
38	C₂×C₄²×C₈₉	Q₈	1	8	yes	complemented
39	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
40	C₂×Dic₁₇₈	—	2	96	no	—
41	C₂×Dic₁₇₈	—	4	64	no	—
42	C₂²×Dic₈₉	—	2	24	no	—
43	C₂²×Dic₈₉	C₂×D₄	1	12	yes	—
44	C₂²×Dic₈₉	C₂×D₄	1	12	yes	—
45	C₂²×Dic₈₉	C₂×Q₈	1	24	yes	—
46	C₂²×Dic₈₉	C₂⁴	1	8	yes	—
47	C₂²×Dic₈₉	C₂⁴	1	2	yes	—
48	C₂²×Dic₈₉	C₂⁴	1	6	yes	—
49	C₄⋊Dic₈₉	—	2	48	no	—
50	#13083553	—	2	96	no	—
51	C₄×Dic₈₉	C₂×D₄	1	48	yes	—
52	C₄×Dic₈₉	C₂×Q₈	1	48	yes	complemented
53	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	—	2	24	no	—
55	C₂²×C₄×C₈₉	C₂×D₄	1	12	yes	—
56	C₂²×C₄×C₈₉	C₂×Q₈	1	12	yes	—
57	C₂²×C₄×C₈₉	C₂×Q₈	1	12	yes	—
58	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
59	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
60	C₄⋊C₄×C₈₉	—	2	48	no	—
61	C₄²×C₈₉	C₂×Q₈	1	16	yes	complemented
62	C₂³×C₈₉	C₂²×D₄	1	6	yes	—
63	C₂³×C₈₉	C₂²×Q₈	1	6	yes	—
64	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
65	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
66	C₂×C₄×C₈₉	—	2	24	no	—
67	C₂×C₄×C₈₉	—	4	16	no	—
68	C₂×C₄×C₈₉	C₄⋊Q₈	1	48	yes	complemented
69	C₂×C₄×C₈₉	C₂²×D₄	1	12	yes	—
70	C₂×C₄×C₈₉	C₂²×Q₈	1	12	yes	—
71	C₂×C₄×C₈₉	C₂²×Q₈	1	12	yes	—
72	C₂×Dic₈₉	—	2	24	no	—
73	C₂×Dic₈₉	—	4	16	no	—
74	C₂×Dic₈₉	#127	1	16	yes	—
75	C₂×Dic₈₉	C₄⋊Q₈	1	48	yes	complemented
76	C₂×Dic₈₉	C₂²×D₄	1	12	yes	—
77	C₂×Dic₈₉	C₂²×D₄	1	12	yes	—
78	C₂×Dic₈₉	C₂²×Q₈	1	24	yes	—
79	Dic₁₇₈	—	4	64	no	—
80	C₂²×C₈₉	C₂×#127	1	4	yes	—
81	C₂²×C₈₉	C₂×C₄⋊Q₈	1	12	yes	—
82	C₂²×C₈₉	C₂×C₄⋊Q₈	1	12	yes	—
83	C₂²×C₈₉	C₂³×D₄	1	3	yes	—
84	C₂²×C₈₉	C₂³×Q₈	1	3	yes	—
85	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
86	C₄×C₈₉	—	4	8	no	—
87	C₄×C₈₉	C₂×C₄⋊Q₈	1	24	yes	complemented
88	Dic₈₉	—	4	8	no	—
89	Dic₈₉	C₂×#127	1	8	yes	—
90	Dic₈₉	C₂×C₄⋊Q₈	1	24	yes	complemented
91	C₂×#2868	—	89	1	no	maximal, Sylow
92	C₂×C₈₉	#2868	1	8	yes	complemented
93	C₂×C₈₉	C₂²×#127	1	1	yes	characteristic
94	C₂×C₈₉	C₂²×C₄⋊Q₈	1	3	yes	—
95	C₂×C₈₉	C₂²×C₄⋊Q₈	1	3	yes	—
96	C₂²×C₄⋊Q₈	—	89	6	no	—
97	#2868	—	89	16	no	—
98	C₂×#384	—	89	1	no	—
99	C₂×#384	—	89	6	no	—
100	C₂×#384	D₈₉	1	1	yes	characteristic
101	C₂×C₄³	—	89	1	no	—
102	C₈₉	C₂×#2868	1	1	yes	characteristic, complemented, Sylow
103	C₂³×Q₈	—	89	4	no	—
104	C₂×C₄⋊Q₈	—	89	48	no	—
105	C₂×C₄⋊Q₈	—	178	48	no	—
106	C₂²×C₄⋊C₄	—	89	12	no	—
107	C₂²×C₄⋊C₄	—	89	6	no	—
108	C₂²×C₄⋊C₄	C₂×D₈₉	1	6	yes	—
109	C₂²×C₄²	—	89	3	no	—
110	C₂²×C₄²	—	89	3	no	—
111	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
112	#384	—	89	48	no	—
113	#384	—	89	8	no	—
114	#384	C₂×D₈₉	1	8	yes	—
115	C₄³	—	89	8	no	—
116	C₂²×Q₈	—	89	16	no	—
117	C₂²×Q₈	—	178	48	no	—
118	C₂³×C₄	—	89	3	no	—
119	C₂³×C₄	—	89	1	no	—
120	C₂³×C₄	—	89	4	no	—
121	C₂³×C₄	C₂²×D₈₉	1	4	yes	—
122	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
123	C₄⋊Q₈	—	178	192	no	—
124	C₂×C₄⋊C₄	—	2	24	no	—
125	C₂×C₄⋊C₄	—	89	48	no	—
126	C₂×C₄⋊C₄	—	89	24	no	—
127	C₂×C₄⋊C₄	—	178	48	no	—
128	C₂×C₄⋊C₄	—	178	24	no	—
129	C₂×C₄⋊C₄	C₂²×D₈₉	1	24	yes	—
130	C₂×C₄²	—	89	24	no	—
131	C₂×C₄²	—	89	24	no	—
132	C₂×C₄²	—	89	12	no	—
133	C₂×C₄²	—	89	12	no	—
134	C₂×C₄²	Dic₁₇₈	1	8	yes	complemented
135	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
136	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
137	C₂×Q₈	—	356	64	no	—
138	C₂×Q₈	—	178	96	no	—
139	C₂²×C₄	—	2	24	no	—
140	C₂²×C₄	—	89	12	no	—
141	C₂²×C₄	—	89	12	no	—
142	C₂²×C₄	—	89	24	no	—
143	C₂²×C₄	—	89	6	no	—
144	C₂²×C₄	—	89	2	no	—
145	C₂²×C₄	—	89	8	no	—
146	C₂²×C₄	—	178	24	no	—
147	C₂²×C₄	D₄×D₈₉	1	12	yes	—
148	C₂²×C₄	Q₈×D₈₉	1	12	yes	—
149	C₂²×C₄	C₂×Dic₁₇₈	1	12	yes	—
150	C₂²×C₄	C₂³×D₈₉	1	8	yes	—
151	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
152	C₄⋊C₄	—	2	48	no	—
153	C₄⋊C₄	—	178	96	no	—
154	C₄⋊C₄	—	178	48	no	—
155	C₄²	—	89	48	no	—
156	C₄²	—	89	48	no	—
157	C₄²	C₂×Dic₁₇₈	1	16	yes	complemented
158	C₂³	C₂×D₄×D₈₉	1	6	yes	—
159	C₂³	C₂×Q₈×D₈₉	1	6	yes	—
160	C₂³	C₂²×Dic₁₇₈	1	2	yes	—
161	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
162	Q₈	—	356	64	no	—
163	C₂×C₄	—	2	24	no	—
164	C₂×C₄	—	4	16	no	—
165	C₂×C₄	—	89	16	no	—
166	C₂×C₄	—	89	48	no	—
167	C₂×C₄	—	89	12	no	—
168	C₂×C₄	—	89	12	no	—
169	C₂×C₄	—	89	24	no	—
170	C₂×C₄	—	356	16	no	—
171	C₂×C₄	—	178	24	no	—
172	C₂×C₄	#15668011	1	48	yes	complemented
173	C₂×C₄	C₂×D₄×D₈₉	1	12	yes	—
174	C₂×C₄	C₂×Q₈×D₈₉	1	12	yes	—
175	C₂×C₄	C₂²×Dic₁₇₈	1	12	yes	—
176	C₂²	#127×D₈₉	1	4	yes	—
177	C₂²	C₄⋊Q₈×D₈₉	1	12	yes	—
178	C₂²	C₂×#15668011	1	12	yes	—
179	C₂²	C₂²×D₄×D₈₉	1	3	yes	—
180	C₂²	C₂²×Q₈×D₈₉	1	3	yes	—
181	C₂²	C₂³×Dic₁₇₈	1	1	yes	characteristic
182	C₄	—	4	8	no	—
183	C₄	—	89	8	no	—
184	C₄	—	89	24	no	—
185	C₄	—	356	8	no	—
186	C₄	C₂×#15668011	1	24	yes	complemented
187	C₂	#26028649	1	8	yes	direct product factor
188	C₂	C₂×#127×D₈₉	1	1	yes	characteristic
189	C₂	C₂×C₄⋊Q₈×D₈₉	1	3	yes	—
190	C₂	C₂²×#15668011	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₈₉
commutator, cosocle	C₂³×C₈₉	C₂⁵
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂×#384×C₈₉	C₂
socle, prime radical	C₂⁴×C₈₉	C₂⁴
hypercenter	C₂×#384	D₈₉
hypocenter	C₈₉	C₂×#2868
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
2. #26028649	C₂	direct (N × H)
31. C₂×C₄×Dic₈₉	Q₈	semidirect
38. C₂×C₄²×C₈₉	Q₈	semidirect
52. C₄×Dic₈₉	C₂×Q₈	semidirect
61. C₄²×C₈₉	C₂×Q₈	semidirect
68. C₂×C₄×C₈₉	C₄⋊Q₈	semidirect
75. C₂×Dic₈₉	C₄⋊Q₈	semidirect
87. C₄×C₈₉	C₂×C₄⋊Q₈	semidirect
90. Dic₈₉	C₂×C₄⋊Q₈	semidirect
92. C₂×C₈₉	#2868	semidirect
102. C₈₉	C₂×#2868	semidirect
134. C₂×C₄²	Dic₁₇₈	semidirect
157. C₄²	C₂×Dic₁₇₈	semidirect
172. C₂×C₄	#15668011	semidirect
186. C₄	C₂×#15668011	semidirect
187. C₂	#26028649	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×#15668011	2
#26028649	2
C₂×#19151468	2
C₂×#19151467	2
C₂×C₄²×Dic₈₉	2
C₂×#384×C₈₉	2
C₂×#2868	89

Subgroups — C2×#26028649 #36760525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26028649 #36760525