Subgroups — C₂×#26028630 #36760441

Order 22784.

2814 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15668012	C₂	1	2	yes	maximal
2	#26028630	C₂	1	16	yes	maximal, direct product factor
3	C₂×#19151504	C₂	1	8	yes	maximal
4	C₂×#19151464	C₂	1	4	yes	maximal
5	C₂×C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₇₈	C₂²	1	4	yes	—
7	C₂×#15668012	—	2	16	no	—
8	C₂×#15668012	C₂²	1	16	yes	—
9	C₂²×C₄⋊Dic₈₉	C₂²	1	4	yes	—
10	C₂²×C₄⋊Dic₈₉	C₂²	1	4	yes	—
11	#19151504	C₂²	1	64	yes	—
12	#19151464	C₂²	1	32	yes	—
13	C₂×#15667919	C₂²	1	16	yes	—
14	C₂²×C₄²×C₈₉	C₂²	1	4	yes	—
15	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
17	C₄³×C₈₉	C₂²	1	8	yes	—
18	C₂²×Dic₁₇₈	—	2	16	no	—
19	C₂²×Dic₁₇₈	—	2	32	no	—
20	C₂²×Dic₁₇₈	C₂³	1	16	yes	—
21	C₂³×Dic₈₉	C₂³	1	8	yes	—
22	#15668012	—	2	64	no	—
23	C₂×C₄⋊Dic₈₉	—	2	16	no	—
24	C₂×C₄⋊Dic₈₉	—	2	16	no	—
25	C₂×C₄⋊Dic₈₉	C₂³	1	16	yes	—
26	C₂×C₄⋊Dic₈₉	C₂³	1	16	yes	—
27	#15667919	C₂³	1	64	yes	—
28	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
29	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
31	C₂×C₄²×C₈₉	—	2	16	no	—
32	C₂×C₄²×C₈₉	D₄	1	8	yes	—
33	C₂×C₄²×C₈₉	Q₈	1	16	yes	complemented
34	C₂×C₄²×C₈₉	C₂³	1	16	yes	—
35	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
36	C₂×C₄²×C₈₉	C₂³	1	8	yes	—
37	C₂×Dic₁₇₈	—	2	32	no	—
38	C₂×Dic₁₇₈	—	2	64	no	—
39	C₂×Dic₁₇₈	—	4	64	no	—
40	C₂²×Dic₈₉	—	2	16	no	—
41	C₂²×Dic₈₉	—	2	32	no	—
42	C₂²×Dic₈₉	C₂⁴	1	16	yes	—
43	C₄⋊Dic₈₉	—	2	32	no	—
44	C₄⋊Dic₈₉	—	2	32	no	—
45	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	—	2	16	no	—
47	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
48	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
49	C₂²×C₄×C₈₉	C₂×Q₈	1	16	yes	—
50	C₂²×C₄×C₈₉	C₂×Q₈	1	8	yes	—
51	C₂²×C₄×C₈₉	#41	1	16	yes	—
52	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
53	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
54	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
55	C₄²×C₈₉	—	2	32	no	—
56	C₄²×C₈₉	C₂×D₄	1	16	yes	—
57	C₄²×C₈₉	C₂×Q₈	1	32	yes	complemented
58	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
59	C₂³×C₈₉	C₂²×Q₈	1	4	yes	—
60	C₂³×C₈₉	C₂×#41	1	8	yes	—
61	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
62	C₂×C₄×C₈₉	—	2	32	no	—
63	C₂×C₄×C₈₉	—	2	16	no	—
64	C₂×C₄×C₈₉	#124	1	32	yes	—
65	C₂×C₄×C₈₉	C₄⋊Q₈	1	16	yes	—
66	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
67	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
68	C₂×C₄×C₈₉	C₂²×Q₈	1	16	yes	—
69	C₂×C₄×C₈₉	C₂²×Q₈	1	8	yes	—
70	C₂×C₄×C₈₉	C₂×#41	1	16	yes	—
71	C₂×Dic₈₉	—	2	16	no	—
72	C₂×Dic₈₉	—	2	32	no	—
73	C₂×Dic₈₉	—	4	32	no	—
74	Dic₁₇₈	—	4	64	no	—
75	C₂²×C₈₉	C₂×#124	1	8	yes	—
76	C₂²×C₈₉	C₂×C₄⋊Q₈	1	4	yes	—
77	C₂²×C₈₉	#533	1	16	yes	—
78	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
79	C₂²×C₈₉	C₂³×Q₈	1	2	yes	—
80	C₂²×C₈₉	C₂²×#41	1	4	yes	—
81	C₄×C₈₉	—	2	16	no	—
82	C₄×C₈₉	C₂×#124	1	16	yes	—
83	C₄×C₈₉	C₂×C₄⋊Q₈	1	8	yes	—
84	Dic₈₉	—	4	16	no	—
85	C₂×#2862	—	89	1	no	maximal, Sylow
86	C₂×C₈₉	#2862	1	8	yes	complemented
87	C₂×C₈₉	C₂²×#124	1	2	yes	—
88	C₂×C₈₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
89	C₂×C₈₉	C₂×#533	1	4	yes	—
90	C₂²×C₄⋊Q₈	—	89	2	no	—
91	#2862	—	89	16	no	—
92	C₂×#391	—	89	8	no	—
93	C₂×#382	—	89	4	no	—
94	C₂×C₄³	D₈₉	1	1	yes	characteristic
95	C₈₉	C₂×#2862	1	1	yes	characteristic, complemented, Sylow
96	C₂³×Q₈	—	89	4	no	—
97	C₂×C₄⋊Q₈	—	89	16	no	—
98	C₂×C₄⋊Q₈	—	178	16	no	—
99	C₂²×C₄⋊C₄	—	89	4	no	—
100	C₂²×C₄⋊C₄	—	89	4	no	—
101	C₂²×C₄²	C₂×D₈₉	1	4	yes	—
102	C₂²×C₄²	C₂×D₈₉	1	2	yes	—
103	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
104	#391	—	89	64	no	—
105	#382	—	89	32	no	—
106	C₂×#95	—	89	16	no	—
107	C₄³	C₂×D₈₉	1	8	yes	—
108	C₂²×Q₈	—	89	16	no	—
109	C₂²×Q₈	—	178	16	no	—
110	C₂²×Q₈	—	178	32	no	—
111	C₂³×C₄	—	89	8	no	—
112	C₂³×C₄	C₂²×D₈₉	1	4	yes	—
113	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
114	C₂³×C₄	C₂²×D₈₉	1	2	yes	—
115	C₄⋊Q₈	—	178	64	no	—
116	C₂×C₄⋊C₄	—	89	16	no	—
117	C₂×C₄⋊C₄	—	89	16	no	—
118	C₂×C₄⋊C₄	—	178	16	no	—
119	C₂×C₄⋊C₄	—	178	16	no	—
120	C₂×C₄²	—	2	16	no	—
121	C₂×C₄²	Dic₁₇₈	1	16	yes	complemented
122	C₂×C₄²	D₃₅₆	1	8	yes	—
123	C₂×C₄²	C₂²×D₈₉	1	16	yes	—
124	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
125	C₂×C₄²	C₂²×D₈₉	1	8	yes	—
126	#95	—	89	64	no	—
127	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
128	C₂×Q₈	—	356	64	no	—
129	C₂×Q₈	—	178	32	no	—
130	C₂×Q₈	—	178	64	no	—
131	C₂²×C₄	—	2	16	no	—
132	C₂²×C₄	—	89	16	no	—
133	C₂²×C₄	—	178	16	no	—
134	C₂²×C₄	—	178	32	no	—
135	C₂²×C₄	C₂×D₃₅₆	1	8	yes	—
136	C₂²×C₄	C₂×D₃₅₆	1	4	yes	—
137	C₂²×C₄	C₂×Dic₁₇₈	1	16	yes	—
138	C₂²×C₄	C₂×Dic₁₇₈	1	8	yes	—
139	C₂²×C₄	#13083574	1	16	yes	—
140	C₂²×C₄	C₂³×D₈₉	1	8	yes	—
141	C₂²×C₄	C₂³×D₈₉	1	2	yes	—
142	C₂²×C₄	C₂³×D₈₉	1	4	yes	—
143	C₄⋊C₄	—	178	32	no	—
144	C₄⋊C₄	—	178	32	no	—
145	C₄²	—	2	32	no	—
146	C₄²	C₂×D₃₅₆	1	16	yes	—
147	C₄²	C₂×Dic₁₇₈	1	32	yes	complemented
148	C₂³	C₂²×D₃₅₆	1	2	yes	—
149	C₂³	C₂²×Dic₁₇₈	1	4	yes	—
150	C₂³	C₂×#13083574	1	8	yes	—
151	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
152	Q₈	—	356	64	no	—
153	C₂×C₄	—	2	32	no	—
154	C₂×C₄	—	2	16	no	—
155	C₂×C₄	—	356	32	no	—
156	C₂×C₄	—	178	16	no	—
157	C₂×C₄	—	178	32	no	—
158	C₂×C₄	#15668000	1	32	yes	—
159	C₂×C₄	#15668012	1	16	yes	—
160	C₂×C₄	C₂²×D₃₅₆	1	4	yes	—
161	C₂×C₄	C₂²×D₃₅₆	1	8	yes	—
162	C₂×C₄	C₂²×Dic₁₇₈	1	16	yes	—
163	C₂×C₄	C₂²×Dic₁₇₈	1	8	yes	—
164	C₂×C₄	C₂×#13083574	1	16	yes	—
165	C₂²	C₂×#15668000	1	8	yes	—
166	C₂²	C₂×#15668012	1	4	yes	—
167	C₂²	#19152155	1	16	yes	—
168	C₂²	C₂³×D₃₅₆	1	1	yes	characteristic
169	C₂²	C₂³×Dic₁₇₈	1	2	yes	—
170	C₂²	C₂²×#13083574	1	4	yes	—
171	C₄	—	2	16	no	—
172	C₄	—	356	16	no	—
173	C₄	C₂×#15668000	1	16	yes	—
174	C₄	C₂×#15668012	1	8	yes	—
175	C₂	#26028630	1	8	yes	direct product factor
176	C₂	C₂²×#15668000	1	2	yes	—
177	C₂	C₂²×#15668012	1	1	yes	characteristic
178	C₂	C₂×#19152155	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₈₉
commutator, cosocle	C₂³×C₈₉	C₂⁵
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂×C₄³×C₈₉	C₂
socle, prime radical	C₂⁴×C₈₉	C₂⁴
hypercenter	C₂×C₄³	D₈₉
hypocenter	C₈₉	C₂×#2862
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #26028630	C₂	direct (N × H)
33. C₂×C₄²×C₈₉	Q₈	semidirect
57. C₄²×C₈₉	C₂×Q₈	semidirect
86. C₂×C₈₉	#2862	semidirect
95. C₈₉	C₂×#2862	semidirect
121. C₂×C₄²	Dic₁₇₈	semidirect
147. C₄²	C₂×Dic₁₇₈	semidirect
175. C₂	#26028630	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×#15668012	2
#26028630	2
C₂×#19151504	2
C₂×#19151464	2
C₂×C₄³×C₈₉	2
C₂×#2862	89

Subgroups — C2×#26028630 #36760441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26028630 #36760441