Subgroups — #552×C₂₇₇ #31799686

Order 17728.

214 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#126×C₂₇₇	C₂	1	2	yes	maximal, complemented
2	#125×C₂₇₇	C₂	1	2	yes	maximal, complemented
3	#125×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#123×C₂₇₇	C₂	1	2	yes	maximal
5	C₂²⋊Q₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊Q₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Q₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₄×C₂₇₇	C₂	1	1	yes	maximal, characteristic
9	#117×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#117×C₂₇₇	C₂	1	2	yes	maximal, complemented
11	C₂×C₄⋊C₄×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₂₇₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₂₇₇	C₂²	1	2	yes	—
15	C₂²×C₄×C₂₇₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₂₇₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
20	C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
21	C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
23	C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
24	C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
25	C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
26	C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
27	C₂²⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₂₇₇	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₂₇₇	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
31	C₂²⋊C₄×C₂₇₇	C₂²	1	2	yes	—
32	C₄²×C₂₇₇	C₂²	1	2	yes	—
33	C₄²×C₂₇₇	C₂²	1	2	yes	—
34	C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
35	C₂³×C₂₇₇	C₂³	1	1	yes	characteristic
36	C₂³×C₂₇₇	C₂³	1	1	yes	characteristic
37	Q₈×C₂₇₇	—	2	2	no	—
38	D₄×C₂₇₇	—	2	2	no	—
39	C₂×C₄×C₂₇₇	—	2	4	no	—
40	C₂×C₄×C₂₇₇	—	2	2	no	—
41	C₂×C₄×C₂₇₇	—	2	2	no	—
42	C₂×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₂₇₇	C₂³	1	2	yes	—
45	C₂×C₄×C₂₇₇	C₂³	1	2	yes	—
46	C₂×C₄×C₂₇₇	C₂³	1	2	yes	—
47	C₂×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
49	C₂×C₄×C₂₇₇	C₂³	1	2	yes	—
50	C₂×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
51	C₂²×C₂₇₇	—	2	1	no	—
52	C₂²×C₂₇₇	—	2	1	no	—
53	C₂²×C₂₇₇	—	2	1	no	—
54	C₂²×C₂₇₇	—	2	1	no	—
55	C₂²×C₂₇₇	—	2	1	no	—
56	C₂²×C₂₇₇	#41	1	2	yes	—
57	C₂²×C₂₇₇	C₂⁴	1	1	yes	characteristic
58	C₄×C₂₇₇	—	2	2	no	—
59	C₄×C₂₇₇	—	2	2	no	—
60	C₄×C₂₇₇	—	2	1	no	—
61	C₄×C₂₇₇	—	2	1	no	—
62	C₄×C₂₇₇	—	2	1	no	—
63	C₄×C₂₇₇	—	2	2	no	—
64	C₄×C₂₇₇	—	2	1	no	—
65	C₄×C₂₇₇	—	2	2	no	—
66	C₄×C₂₇₇	#41	1	2	yes	—
67	C₂×C₂₇₇	—	2	2	no	—
68	C₂×C₂₇₇	—	4	1	no	—
69	C₂×C₂₇₇	C₂×#41	1	1	yes	characteristic
70	C₂×C₂₇₇	C₂×#41	1	1	yes	characteristic
71	C₂×C₂₇₇	2₋¹⁺⁴	1	1	yes	characteristic
72	C₂₇₇	#552	1	1	yes	characteristic, direct product factor, Sylow
73	#552	C₂₇₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	#126	C₂×C₂₇₇	1	2	yes	complemented
75	#125	C₂×C₂₇₇	1	2	yes	complemented
76	#125	C₂×C₂₇₇	1	1	yes	characteristic, complemented
77	#123	C₂×C₂₇₇	1	2	yes	—
78	C₂²⋊Q₈	C₂×C₂₇₇	1	1	yes	characteristic, complemented
79	C₂²⋊Q₈	C₂×C₂₇₇	1	1	yes	characteristic, complemented
80	C₄×Q₈	C₂×C₂₇₇	1	1	yes	characteristic, complemented
81	C₄×D₄	C₂×C₂₇₇	1	1	yes	characteristic
82	#117	C₂×C₂₇₇	1	1	yes	characteristic, complemented
83	#117	C₂×C₂₇₇	1	2	yes	complemented
84	C₂×C₄⋊C₄	C₂×C₂₇₇	1	1	yes	characteristic, complemented
85	C₂×Q₈	C₂²×C₂₇₇	1	1	yes	characteristic
86	C₂×D₄	C₂²×C₂₇₇	1	1	yes	characteristic
87	C₂²×C₄	C₂²×C₂₇₇	1	2	yes	—
88	C₂²×C₄	C₂²×C₂₇₇	1	1	yes	characteristic
89	C₂²×C₄	C₂²×C₂₇₇	1	1	yes	characteristic
90	C₄⋊C₄	C₂²×C₂₇₇	1	1	yes	characteristic
91	C₄⋊C₄	C₂²×C₂₇₇	1	1	yes	characteristic
92	C₄⋊C₄	C₂²×C₂₇₇	1	2	yes	—
93	C₄⋊C₄	C₂²×C₂₇₇	1	2	yes	—
94	C₄⋊C₄	C₂²×C₂₇₇	1	1	yes	characteristic
95	C₄⋊C₄	C₂²×C₂₇₇	1	2	yes	—
96	C₄⋊C₄	C₂²×C₂₇₇	1	1	yes	characteristic
97	C₄⋊C₄	C₂²×C₂₇₇	1	2	yes	—
98	C₄⋊C₄	C₂²×C₂₇₇	1	2	yes	—
99	C₄⋊C₄	C₂²×C₂₇₇	1	2	yes	—
100	C₂²⋊C₄	C₂²×C₂₇₇	1	1	yes	characteristic
101	C₂²⋊C₄	C₂²×C₂₇₇	1	2	yes	—
102	C₂²⋊C₄	C₂²×C₂₇₇	1	2	yes	—
103	C₂²⋊C₄	C₂²×C₂₇₇	1	1	yes	characteristic
104	C₂²⋊C₄	C₂²×C₂₇₇	1	2	yes	—
105	C₄²	C₂²×C₂₇₇	1	2	yes	—
106	C₄²	C₂²×C₂₇₇	1	2	yes	—
107	C₄²	C₂²×C₂₇₇	1	1	yes	characteristic
108	C₂³	C₂³×C₂₇₇	1	1	yes	characteristic
109	C₂³	C₂³×C₂₇₇	1	1	yes	characteristic
110	Q₈	—	2	2	no	—
111	D₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	C₂³×C₂₇₇	1	1	yes	characteristic
116	C₂×C₄	C₂³×C₂₇₇	1	1	yes	characteristic
117	C₂×C₄	C₂³×C₂₇₇	1	2	yes	—
118	C₂×C₄	C₂³×C₂₇₇	1	2	yes	—
119	C₂×C₄	C₂³×C₂₇₇	1	2	yes	—
120	C₂×C₄	C₂³×C₂₇₇	1	1	yes	characteristic
121	C₂×C₄	C₂³×C₂₇₇	1	1	yes	characteristic
122	C₂×C₄	C₂³×C₂₇₇	1	2	yes	—
123	C₂×C₄	C₂³×C₂₇₇	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	#41×C₂₇₇	1	2	yes	—
130	C₂²	C₂⁴×C₂₇₇	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	2	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	2	no	—
139	C₄	#41×C₂₇₇	1	2	yes	—
140	C₂	—	2	2	no	—
141	C₂	—	4	1	no	—
142	C₂	C₂×#41×C₂₇₇	1	1	yes	characteristic
143	C₂	C₂×#41×C₂₇₇	1	1	yes	characteristic
144	C₂	2₋¹⁺⁴×C₂₇₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₇₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₂₇₇
prime radical	C₂×D₄×C₂₇₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₇₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #126×C₂₇₇	C₂	semidirect
2. #125×C₂₇₇	C₂	semidirect
3. #125×C₂₇₇	C₂	semidirect
5. C₂²⋊Q₈×C₂₇₇	C₂	semidirect
6. C₂²⋊Q₈×C₂₇₇	C₂	semidirect
7. C₄×Q₈×C₂₇₇	C₂	semidirect
9. #117×C₂₇₇	C₂	semidirect
10. #117×C₂₇₇	C₂	semidirect
11. C₂×C₄⋊C₄×C₂₇₇	C₂	semidirect
72. C₂₇₇	#552	direct (N × H)
73. #552	C₂₇₇	direct (N × H)
74. #126	C₂×C₂₇₇	semidirect
75. #125	C₂×C₂₇₇	semidirect
76. #125	C₂×C₂₇₇	semidirect
78. C₂²⋊Q₈	C₂×C₂₇₇	semidirect
79. C₂²⋊Q₈	C₂×C₂₇₇	semidirect
80. C₄×Q₈	C₂×C₂₇₇	semidirect
82. #117	C₂×C₂₇₇	semidirect
83. #117	C₂×C₂₇₇	semidirect
84. C₂×C₄⋊C₄	C₂×C₂₇₇	semidirect

Maximal subgroups

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

subgroup	index
#126×C₂₇₇	2
#125×C₂₇₇	2
#123×C₂₇₇	2
C₂²⋊Q₈×C₂₇₇	2
C₄×Q₈×C₂₇₇	2
C₄×D₄×C₂₇₇	2
#117×C₂₇₇	2
C₂×C₄⋊C₄×C₂₇₇	2
#552	277

Subgroups — #552×C277 #31799686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #552×C₂₇₇ #31799686