Subgroups — C₂×#398×C₁₃₇ #31777402

Order 17536.

692 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₄×C₁₃₇	C₂	1	2	yes	maximal
4	#398×C₁₃₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#95×C₁₃₇	C₂	1	2	yes	maximal
6	C₂×#95×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₁₃₇	C₂²	1	2	yes	—
9	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₃₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
13	C₂×C₄⋊C₄×C₁₃₇	—	2	8	no	—
14	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
15	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	8	yes	—
18	#95×C₁₃₇	C₂²	1	8	yes	—
19	#95×C₁₃₇	C₂²	1	4	yes	—
20	C₂⁴×C₁₃₇	C₂³	1	1	yes	characteristic
21	C₂²×C₄×C₁₃₇	—	2	2	no	—
22	C₂²×C₄×C₁₃₇	—	2	4	no	—
23	C₂²×C₄×C₁₃₇	—	2	2	no	—
24	C₂²×C₄×C₁₃₇	—	2	4	no	—
25	C₂²×C₄×C₁₃₇	—	2	4	no	—
26	C₂²×C₄×C₁₃₇	—	2	4	no	—
27	C₂²×C₄×C₁₃₇	—	2	4	no	—
28	C₂²×C₄×C₁₃₇	—	2	2	no	—
29	C₂²×C₄×C₁₃₇	—	2	4	no	—
30	C₂²×C₄×C₁₃₇	—	2	4	no	—
31	C₂²×C₄×C₁₃₇	D₄	1	4	yes	—
32	C₂²×C₄×C₁₃₇	D₄	1	4	yes	—
33	C₂²×C₄×C₁₃₇	Q₈	1	8	yes	—
34	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
36	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
39	C₄⋊C₄×C₁₃₇	—	2	8	no	—
40	C₄⋊C₄×C₁₃₇	—	2	16	no	—
41	C₄⋊C₄×C₁₃₇	—	2	8	no	—
42	C₂³×C₁₃₇	C₂×D₄	1	2	yes	—
43	C₂³×C₁₃₇	C₂×D₄	1	2	yes	—
44	C₂³×C₁₃₇	C₂×Q₈	1	4	yes	—
45	C₂³×C₁₃₇	#41	1	2	yes	—
46	C₂³×C₁₃₇	#41	1	4	yes	—
47	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
48	C₂×C₄×C₁₃₇	—	2	8	no	—
49	C₂×C₄×C₁₃₇	—	2	16	no	—
50	C₂×C₄×C₁₃₇	—	2	8	no	—
51	C₂×C₄×C₁₃₇	—	2	2	no	—
52	C₂×C₄×C₁₃₇	—	2	4	no	—
53	C₂×C₄×C₁₃₇	—	2	2	no	—
54	C₂×C₄×C₁₃₇	—	2	4	no	—
55	C₂×C₄×C₁₃₇	—	2	4	no	—
56	C₂×C₄×C₁₃₇	—	2	4	no	—
57	C₂×C₄×C₁₃₇	—	2	4	no	—
58	C₂×C₄×C₁₃₇	—	2	2	no	—
59	C₂×C₄×C₁₃₇	—	2	4	no	—
60	C₂×C₄×C₁₃₇	—	2	4	no	—
61	C₂×C₄×C₁₃₇	—	4	8	no	—
62	C₂×C₄×C₁₃₇	—	4	4	no	—
63	C₂×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
64	C₂×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
65	C₂×C₄×C₁₃₇	C₂×Q₈	1	8	yes	—
66	C₂²×C₁₃₇	#121	1	4	yes	—
67	C₂²×C₁₃₇	C₂²⋊Q₈	1	8	yes	—
68	C₂²×C₁₃₇	#123	1	4	yes	—
69	C₂²×C₁₃₇	#125	1	8	yes	—
70	C₂²×C₁₃₇	C₄⋊Q₈	1	4	yes	—
71	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
72	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₁₃₇	C₂²×Q₈	1	2	yes	—
74	C₂²×C₁₃₇	C₂×#41	1	1	yes	characteristic
75	C₂²×C₁₃₇	C₂×#41	1	2	yes	—
76	C₄×C₁₃₇	—	2	4	no	—
77	C₄×C₁₃₇	—	2	8	no	—
78	C₄×C₁₃₇	—	2	4	no	—
79	C₄×C₁₃₇	—	4	4	no	—
80	C₄×C₁₃₇	—	4	2	no	—
81	C₂×C₁₃₇	#398	1	8	yes	direct product factor
82	C₂×C₁₃₇	C₂×#121	1	1	yes	characteristic
83	C₂×C₁₃₇	C₂×C₂²⋊Q₈	1	2	yes	—
84	C₂×C₁₃₇	C₂×#123	1	1	yes	characteristic
85	C₂×C₁₃₇	C₂×#125	1	2	yes	—
86	C₂×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
87	C₁₃₇	C₂×#398	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×#398	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic
90	C₂²×C₄⋊C₄	C₂×C₁₃₇	1	2	yes	—
91	C₂²×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic
92	#398	C₂×C₁₃₇	1	8	yes	direct product factor
93	C₂×#95	C₂×C₁₃₇	1	2	yes	—
94	C₂×#95	C₂×C₁₃₇	1	1	yes	characteristic
95	C₂³×C₄	C₂²×C₁₃₇	1	2	yes	—
96	C₂³×C₄	C₂²×C₁₃₇	1	1	yes	characteristic
97	C₂³×C₄	C₂²×C₁₃₇	1	1	yes	characteristic
98	C₂³×C₄	C₂²×C₁₃₇	1	1	yes	characteristic
99	C₂³×C₄	C₂²×C₁₃₇	1	2	yes	—
100	C₂×C₄⋊C₄	—	2	4	no	—
101	C₂×C₄⋊C₄	—	2	8	no	—
102	C₂×C₄⋊C₄	—	2	4	no	—
103	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
104	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
105	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	8	yes	—
106	#95	C₂²×C₁₃₇	1	8	yes	—
107	#95	C₂²×C₁₃₇	1	4	yes	—
108	C₂⁴	C₂³×C₁₃₇	1	1	yes	characteristic
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	—	2	4	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	—	2	4	no	—
114	C₂²×C₄	—	2	4	no	—
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	4	no	—
118	C₂²×C₄	—	2	4	no	—
119	C₂²×C₄	D₄×C₁₃₇	1	4	yes	—
120	C₂²×C₄	D₄×C₁₃₇	1	4	yes	—
121	C₂²×C₄	Q₈×C₁₃₇	1	8	yes	—
122	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
123	C₂²×C₄	C₂³×C₁₃₇	1	4	yes	—
124	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
125	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
126	C₂²×C₄	C₂³×C₁₃₇	1	4	yes	—
127	C₄⋊C₄	—	2	8	no	—
128	C₄⋊C₄	—	2	16	no	—
129	C₄⋊C₄	—	2	8	no	—
130	C₂³	C₂×D₄×C₁₃₇	1	2	yes	—
131	C₂³	C₂×D₄×C₁₃₇	1	2	yes	—
132	C₂³	C₂×Q₈×C₁₃₇	1	4	yes	—
133	C₂³	#41×C₁₃₇	1	2	yes	—
134	C₂³	#41×C₁₃₇	1	4	yes	—
135	C₂³	C₂⁴×C₁₃₇	1	1	yes	characteristic
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	2	16	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	4	8	no	—
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
152	C₂×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
153	C₂×C₄	C₂×Q₈×C₁₃₇	1	8	yes	—
154	C₂²	#121×C₁₃₇	1	4	yes	—
155	C₂²	C₂²⋊Q₈×C₁₃₇	1	8	yes	—
156	C₂²	#123×C₁₃₇	1	4	yes	—
157	C₂²	#125×C₁₃₇	1	8	yes	—
158	C₂²	C₄⋊Q₈×C₁₃₇	1	4	yes	—
159	C₂²	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
160	C₂²	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
161	C₂²	C₂²×Q₈×C₁₃₇	1	2	yes	—
162	C₂²	C₂×#41×C₁₃₇	1	1	yes	characteristic
163	C₂²	C₂×#41×C₁₃₇	1	2	yes	—
164	C₄	—	2	4	no	—
165	C₄	—	2	8	no	—
166	C₄	—	2	4	no	—
167	C₄	—	4	4	no	—
168	C₄	—	4	2	no	—
169	C₂	#398×C₁₃₇	1	8	yes	direct product factor
170	C₂	C₂×#121×C₁₃₇	1	1	yes	characteristic
171	C₂	C₂×C₂²⋊Q₈×C₁₃₇	1	2	yes	—
172	C₂	C₂×#123×C₁₃₇	1	1	yes	characteristic
173	C₂	C₂×#125×C₁₃₇	1	2	yes	—
174	C₂	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. #398×C₁₃₇	C₂	direct (N × H)
81. C₂×C₁₃₇	#398	direct (N × H)
87. C₁₃₇	C₂×#398	direct (N × H)
88. C₂×#398	C₁₃₇	direct (N × H)
92. #398	C₂×C₁₃₇	direct (N × H)
169. C₂	#398×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
C₂²×C₄⋊C₄×C₁₃₇	2
#398×C₁₃₇	2
C₂×#95×C₁₃₇	2
C₂×#398	137

Subgroups — C2×#398×C137 #31777402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#398×C₁₃₇ #31777402