Subgroups — #2568×C₁₃₇ #31777579

Order 17536.

468 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#400×C₁₃₇	C₂	1	4	yes	maximal
4	#391×C₁₃₇	C₂	1	4	yes	maximal
5	#391×C₁₃₇	C₂	1	2	yes	maximal
6	#382×C₁₃₇	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₁₃₇	C₂²	1	2	yes	—
9	C₂²×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
11	C₄×Q₈×C₁₃₇	—	2	4	no	—
12	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
20	#95×C₁₃₇	C₂²	1	2	yes	—
21	#95×C₁₃₇	C₂²	1	8	yes	—
22	#95×C₁₃₇	C₂²	1	8	yes	—
23	C₂×Q₈×C₁₃₇	—	2	2	no	—
24	C₂×Q₈×C₁₃₇	—	2	4	no	—
25	C₂×Q₈×C₁₃₇	—	2	2	no	—
26	C₂×Q₈×C₁₃₇	—	2	8	no	—
27	C₂×Q₈×C₁₃₇	Q₈	1	4	yes	complemented
28	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
31	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₃₇	—	2	4	no	—
36	C₄⋊C₄×C₁₃₇	—	2	2	no	—
37	C₄⋊C₄×C₁₃₇	—	2	4	no	—
38	C₄⋊C₄×C₁₃₇	—	2	4	no	—
39	C₄⋊C₄×C₁₃₇	—	2	4	no	—
40	C₄²×C₁₃₇	—	2	4	no	—
41	C₄²×C₁₃₇	—	2	4	no	—
42	C₄²×C₁₃₇	D₄	1	4	yes	—
43	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
44	Q₈×C₁₃₇	—	2	8	no	—
45	Q₈×C₁₃₇	—	4	8	no	—
46	C₂×C₄×C₁₃₇	—	2	2	no	—
47	C₂×C₄×C₁₃₇	—	2	2	no	—
48	C₂×C₄×C₁₃₇	—	2	2	no	—
49	C₂×C₄×C₁₃₇	—	2	2	no	—
50	C₂×C₄×C₁₃₇	—	2	4	no	—
51	C₂×C₄×C₁₃₇	—	2	4	no	—
52	C₂×C₄×C₁₃₇	—	2	4	no	—
53	C₂×C₄×C₁₃₇	—	2	8	no	—
54	C₂×C₄×C₁₃₇	—	2	4	no	—
55	C₂×C₄×C₁₃₇	—	2	4	no	—
56	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
60	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₁₃₇	#41	1	2	yes	—
62	C₂×C₄×C₁₃₇	#41	1	4	yes	—
63	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
64	C₂²×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
65	C₂²×C₁₃₇	C₂×#41	1	1	yes	characteristic
66	C₂²×C₁₃₇	C₂×#41	1	2	yes	—
67	C₂²×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
68	C₄×C₁₃₇	—	2	4	no	—
69	C₄×C₁₃₇	—	2	2	no	—
70	C₄×C₁₃₇	—	4	4	no	—
71	C₄×C₁₃₇	—	4	2	no	—
72	C₄×C₁₃₇	—	4	4	no	—
73	C₄×C₁₃₇	C₂²⋊Q₈	1	4	yes	—
74	C₄×C₁₃₇	#124	1	4	yes	—
75	C₂×C₁₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₁₃₇	C₂×#124	1	1	yes	characteristic
77	C₂×C₁₃₇	#536	1	1	yes	characteristic
78	C₂×C₁₃₇	#556	1	2	yes	—
79	C₂×C₁₃₇	#557	1	2	yes	—
80	C₁₃₇	#2568	1	1	yes	characteristic, direct product factor, Sylow
81	#2568	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₄⋊Q₈	C₂×C₁₃₇	1	1	yes	characteristic
83	C₂×C₄×Q₈	C₂×C₁₃₇	1	1	yes	characteristic
84	#400	C₂×C₁₃₇	1	4	yes	—
85	#391	C₂×C₁₃₇	1	4	yes	—
86	#391	C₂×C₁₃₇	1	2	yes	—
87	#382	C₂×C₁₃₇	1	2	yes	—
88	C₄×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic
89	C₂²×Q₈	C₂²×C₁₃₇	1	2	yes	—
90	C₂²×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
91	C₄⋊Q₈	—	2	4	no	—
92	C₄×Q₈	—	2	4	no	—
93	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
95	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
96	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
97	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
98	C₂×C₄²	C₂²×C₁₃₇	1	2	yes	—
99	C₂×C₄²	C₂²×C₁₃₇	1	2	yes	—
100	C₂×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
101	#95	C₂²×C₁₃₇	1	2	yes	—
102	#95	C₂²×C₁₃₇	1	8	yes	—
103	#95	C₂²×C₁₃₇	1	8	yes	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	—	2	8	no	—
108	C₂×Q₈	Q₈×C₁₃₇	1	4	yes	complemented
109	C₂²×C₄	C₂³×C₁₃₇	1	4	yes	—
110	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
111	C₂²×C₄	C₂³×C₁₃₇	1	4	yes	—
112	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
113	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
114	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
115	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄²	—	2	4	no	—
122	C₄²	—	2	4	no	—
123	C₄²	D₄×C₁₃₇	1	4	yes	—
124	C₂³	C₂⁴×C₁₃₇	1	1	yes	characteristic
125	Q₈	—	2	8	no	—
126	Q₈	—	4	8	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
138	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
139	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
140	C₂×C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
141	C₂×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
142	C₂×C₄	#41×C₁₃₇	1	2	yes	—
143	C₂×C₄	#41×C₁₃₇	1	4	yes	—
144	C₂²	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
145	C₂²	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
146	C₂²	C₂×#41×C₁₃₇	1	1	yes	characteristic
147	C₂²	C₂×#41×C₁₃₇	1	2	yes	—
148	C₂²	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	4	no	—
154	C₄	C₂²⋊Q₈×C₁₃₇	1	4	yes	—
155	C₄	#124×C₁₃₇	1	4	yes	—
156	C₂	C₂×C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
157	C₂	C₂×#124×C₁₃₇	1	1	yes	characteristic
158	C₂	#536×C₁₃₇	1	1	yes	characteristic
159	C₂	#556×C₁₃₇	1	2	yes	—
160	C₂	#557×C₁₃₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
27. C₂×Q₈×C₁₃₇	Q₈	semidirect
80. C₁₃₇	#2568	direct (N × H)
81. #2568	C₁₃₇	direct (N × H)
108. C₂×Q₈	Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈×C₁₃₇	2
C₂×C₄×Q₈×C₁₃₇	2
#400×C₁₃₇	2
#391×C₁₃₇	2
#382×C₁₃₇	2
C₄×C₄⋊C₄×C₁₃₇	2
#2568	137

Subgroups — #2568×C137 #31777579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2568×C₁₃₇ #31777579