Subgroups — C₂×#541×C₁₃₇ #31778464

Order 17536.

1004 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#541×C₁₃₇	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₄⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#126×C₁₃₇	C₂	1	4	yes	maximal
4	C₂×#125×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#125×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	C₂×C₂²⋊Q₈×C₁₃₇	C₂	1	2	yes	maximal
7	C₂×C₄×Q₈×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₂×#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₁₃₇	C₂²	1	2	yes	—
10	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈×C₁₃₇	C₂²	1	8	yes	complemented
12	#126×C₁₃₇	C₂²	1	32	yes	—
13	#125×C₁₃₇	C₂²	1	8	yes	complemented
14	#125×C₁₃₇	C₂²	1	32	yes	complemented
15	C₂²⋊Q₈×C₁₃₇	C₂²	1	16	yes	—
16	C₄×Q₈×C₁₃₇	C₂²	1	16	yes	complemented
17	#117×C₁₃₇	C₂²	1	8	yes	—
18	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
23	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	4	yes	—
26	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₁₃₇	C₂²	1	4	yes	—
29	C₂⁴×C₁₃₇	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₁₃₇	—	2	8	no	—
31	C₂×Q₈×C₁₃₇	C₂³	1	8	yes	—
32	C₂²×C₄×C₁₃₇	—	2	4	no	—
33	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
34	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
36	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
37	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
38	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₁₃₇	C₂³	1	8	yes	—
41	C₄⋊C₄×C₁₃₇	C₂³	1	8	yes	—
42	C₄⋊C₄×C₁₃₇	C₂³	1	16	yes	—
43	C₄⋊C₄×C₁₃₇	C₂³	1	16	yes	—
44	C₄⋊C₄×C₁₃₇	C₂³	1	16	yes	—
45	C₄⋊C₄×C₁₃₇	C₂³	1	16	yes	—
46	C₂²⋊C₄×C₁₃₇	C₂³	1	8	yes	—
47	C₂²⋊C₄×C₁₃₇	C₂³	1	16	yes	—
48	C₄²×C₁₃₇	C₂³	1	4	yes	—
49	C₄²×C₁₃₇	C₂³	1	4	yes	—
50	C₄²×C₁₃₇	C₂³	1	16	yes	—
51	C₂³×C₁₃₇	—	2	2	no	—
52	C₂³×C₁₃₇	—	2	4	no	—
53	C₂³×C₁₃₇	C₂⁴	1	2	yes	—
54	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
55	Q₈×C₁₃₇	—	2	16	no	—
56	C₂×C₄×C₁₃₇	—	2	2	no	—
57	C₂×C₄×C₁₃₇	—	2	8	no	—
58	C₂×C₄×C₁₃₇	—	2	4	no	—
59	C₂×C₄×C₁₃₇	—	2	4	no	—
60	C₂×C₄×C₁₃₇	—	2	8	no	—
61	C₂×C₄×C₁₃₇	—	2	8	no	—
62	C₂×C₄×C₁₃₇	#41	1	4	yes	—
63	C₂×C₄×C₁₃₇	C₂⁴	1	4	yes	—
64	C₂×C₄×C₁₃₇	C₂⁴	1	4	yes	—
65	C₂×C₄×C₁₃₇	C₂⁴	1	8	yes	—
66	C₂×C₄×C₁₃₇	C₂⁴	1	8	yes	—
67	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
68	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
69	C₂²×C₁₃₇	—	2	2	no	—
70	C₂²×C₁₃₇	—	2	4	no	—
71	C₂²×C₁₃₇	—	4	4	no	—
72	C₂²×C₁₃₇	C₂×#41	1	2	yes	—
73	C₂²×C₁₃₇	2₋¹⁺⁴	1	4	yes	—
74	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
75	C₄×C₁₃₇	—	2	2	no	—
76	C₄×C₁₃₇	—	2	8	no	—
77	C₄×C₁₃₇	—	2	4	no	—
78	C₄×C₁₃₇	—	2	4	no	—
79	C₄×C₁₃₇	—	2	8	no	—
80	C₄×C₁₃₇	C₂×#41	1	4	yes	—
81	C₂×C₁₃₇	—	4	2	no	—
82	C₂×C₁₃₇	#541	1	4	yes	direct product factor
83	C₂×C₁₃₇	C₂²×#41	1	1	yes	characteristic
84	C₂×C₁₃₇	C₂×2₋¹⁺⁴	1	2	yes	—
85	C₁₃₇	C₂×#541	1	1	yes	characteristic, direct product factor, Sylow
86	C₂×#541	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	#541	C₂×C₁₃₇	1	16	yes	direct product factor
88	C₂×C₄⋊Q₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
89	C₂×#126	C₂×C₁₃₇	1	4	yes	—
90	C₂×#125	C₂×C₁₃₇	1	1	yes	characteristic, complemented
91	C₂×#125	C₂×C₁₃₇	1	4	yes	complemented
92	C₂×C₂²⋊Q₈	C₂×C₁₃₇	1	2	yes	—
93	C₂×C₄×Q₈	C₂×C₁₃₇	1	2	yes	complemented
94	C₂×#117	C₂×C₁₃₇	1	1	yes	characteristic
95	C₂²×Q₈	C₂²×C₁₃₇	1	2	yes	—
96	C₂³×C₄	C₂²×C₁₃₇	1	1	yes	characteristic
97	C₄⋊Q₈	C₂²×C₁₃₇	1	8	yes	complemented
98	#126	C₂²×C₁₃₇	1	32	yes	—
99	#125	C₂²×C₁₃₇	1	8	yes	complemented
100	#125	C₂²×C₁₃₇	1	32	yes	complemented
101	C₂²⋊Q₈	C₂²×C₁₃₇	1	16	yes	—
102	C₄×Q₈	C₂²×C₁₃₇	1	16	yes	complemented
103	#117	C₂²×C₁₃₇	1	8	yes	—
104	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
105	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
106	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
107	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
108	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
109	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
110	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
111	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	4	yes	—
112	C₂×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
113	C₂×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
114	C₂×C₄²	C₂²×C₁₃₇	1	4	yes	—
115	C₂⁴	C₂³×C₁₃₇	1	1	yes	characteristic
116	C₂×Q₈	—	2	8	no	—
117	C₂×Q₈	C₂³×C₁₃₇	1	8	yes	—
118	C₂²×C₄	—	2	4	no	—
119	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
120	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
121	C₂²×C₄	C₂³×C₁₃₇	1	4	yes	—
122	C₂²×C₄	C₂³×C₁₃₇	1	4	yes	—
123	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
124	C₂²×C₄	C₂³×C₁₃₇	1	4	yes	—
125	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
126	C₄⋊C₄	C₂³×C₁₃₇	1	8	yes	—
127	C₄⋊C₄	C₂³×C₁₃₇	1	8	yes	—
128	C₄⋊C₄	C₂³×C₁₃₇	1	16	yes	—
129	C₄⋊C₄	C₂³×C₁₃₇	1	16	yes	—
130	C₄⋊C₄	C₂³×C₁₃₇	1	16	yes	—
131	C₄⋊C₄	C₂³×C₁₃₇	1	16	yes	—
132	C₂²⋊C₄	C₂³×C₁₃₇	1	8	yes	—
133	C₂²⋊C₄	C₂³×C₁₃₇	1	16	yes	—
134	C₄²	C₂³×C₁₃₇	1	4	yes	—
135	C₄²	C₂³×C₁₃₇	1	4	yes	—
136	C₄²	C₂³×C₁₃₇	1	16	yes	—
137	C₂³	—	2	2	no	—
138	C₂³	—	2	4	no	—
139	C₂³	C₂⁴×C₁₃₇	1	2	yes	—
140	C₂³	C₂⁴×C₁₃₇	1	1	yes	characteristic
141	Q₈	—	2	16	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	8	no	—
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	#41×C₁₃₇	1	4	yes	—
149	C₂×C₄	C₂⁴×C₁₃₇	1	4	yes	—
150	C₂×C₄	C₂⁴×C₁₃₇	1	4	yes	—
151	C₂×C₄	C₂⁴×C₁₃₇	1	8	yes	—
152	C₂×C₄	C₂⁴×C₁₃₇	1	8	yes	—
153	C₂×C₄	C₂⁴×C₁₃₇	1	2	yes	—
154	C₂×C₄	C₂⁴×C₁₃₇	1	2	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	4	no	—
157	C₂²	—	4	4	no	—
158	C₂²	C₂×#41×C₁₃₇	1	2	yes	—
159	C₂²	2₋¹⁺⁴×C₁₃₇	1	4	yes	—
160	C₂²	C₂⁵×C₁₃₇	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	8	no	—
163	C₄	—	2	4	no	—
164	C₄	—	2	4	no	—
165	C₄	—	2	8	no	—
166	C₄	C₂×#41×C₁₃₇	1	4	yes	—
167	C₂	—	4	2	no	—
168	C₂	#541×C₁₃₇	1	4	yes	direct product factor
169	C₂	C₂²×#41×C₁₃₇	1	1	yes	characteristic
170	C₂	C₂×2₋¹⁺⁴×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₃₇
prime radical	C₂⁴×C₁₃₇	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #541×C₁₃₇	C₂	direct (N × H)
2. C₂×C₄⋊Q₈×C₁₃₇	C₂	semidirect
4. C₂×#125×C₁₃₇	C₂	semidirect
5. C₂×#125×C₁₃₇	C₂	semidirect
7. C₂×C₄×Q₈×C₁₃₇	C₂	semidirect
11. C₄⋊Q₈×C₁₃₇	C₂²	semidirect
13. #125×C₁₃₇	C₂²	semidirect
14. #125×C₁₃₇	C₂²	semidirect
16. C₄×Q₈×C₁₃₇	C₂²	semidirect
82. C₂×C₁₃₇	#541	direct (N × H)
85. C₁₃₇	C₂×#541	direct (N × H)
86. C₂×#541	C₁₃₇	direct (N × H)
87. #541	C₂×C₁₃₇	direct (N × H)
88. C₂×C₄⋊Q₈	C₂×C₁₃₇	semidirect
90. C₂×#125	C₂×C₁₃₇	semidirect
91. C₂×#125	C₂×C₁₃₇	semidirect
93. C₂×C₄×Q₈	C₂×C₁₃₇	semidirect
97. C₄⋊Q₈	C₂²×C₁₃₇	semidirect
99. #125	C₂²×C₁₃₇	semidirect
100. #125	C₂²×C₁₃₇	semidirect
102. C₄×Q₈	C₂²×C₁₃₇	semidirect
168. C₂	#541×C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
#541×C₁₃₇	2
C₂×C₄⋊Q₈×C₁₃₇	2
C₂×#126×C₁₃₇	2
C₂×#125×C₁₃₇	2
C₂×C₂²⋊Q₈×C₁₃₇	2
C₂×C₄×Q₈×C₁₃₇	2
C₂×#117×C₁₃₇	2
C₂×#541	137

Subgroups — C2×#541×C137 #31778464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#541×C₁₃₇ #31778464