Subgroups — #2981×C₁₃₇ #31777992

Order 17536.

402 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#532×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#436×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#435×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	C₈×D₄×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#432×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#431×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#408×C₁₃₇	C₂	1	4	yes	maximal, complemented
8	C₂×#41×C₁₃₇	C₂²	1	2	yes	—
9	C₂×M₁₆×C₁₃₇	C₂²	1	4	yes	—
10	C₂²×C₈×C₁₃₇	C₂²	1	4	yes	—
11	C₄⋊Q₈×C₁₃₇	C₄	1	1	yes	characteristic
12	#127×C₁₃₇	C₄	1	1	yes	characteristic
13	#124×C₁₃₇	C₄	1	2	yes	—
14	#121×C₁₃₇	C₄	1	2	yes	—
15	#121×C₁₃₇	C₄	1	2	yes	—
16	C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
17	C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
20	C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
21	C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
22	C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
23	C₂²⋊C₈×C₁₃₇	C₂²	1	4	yes	—
24	C₂²⋊C₈×C₁₃₇	C₂²	1	4	yes	—
25	#97×C₁₃₇	C₂²	1	2	yes	—
26	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
28	#41×C₁₃₇	—	4	4	no	—
29	C₂×Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₁₃₇	C₂×C₄	1	2	yes	—
31	C₂×D₄×C₁₃₇	C₂×C₄	1	2	yes	—
32	C₂×D₄×C₁₃₇	C₂×C₄	1	4	yes	—
33	C₂×D₄×C₁₃₇	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
38	M₁₆×C₁₃₇	—	4	4	no	—
39	C₂×C₈×C₁₃₇	—	4	4	no	—
40	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
41	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
42	C₂×C₈×C₁₃₇	C₂³	1	4	yes	—
43	C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
45	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
46	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
47	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
50	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
51	C₂³×C₁₃₇	C₂²×C₄	1	4	yes	—
52	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
53	Q₈×C₁₃₇	—	4	2	no	—
54	D₄×C₁₃₇	—	2	4	no	—
55	D₄×C₁₃₇	—	4	2	no	—
56	D₄×C₁₃₇	—	4	2	no	—
57	D₄×C₁₃₇	—	4	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	—	2	2	no	—
62	C₂×C₄×C₁₃₇	—	4	1	no	—
63	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
68	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
70	C₈×C₁₃₇	—	2	2	no	—
71	C₈×C₁₃₇	—	2	4	no	—
72	C₈×C₁₃₇	—	2	2	no	—
73	C₂²×C₁₃₇	—	2	1	no	—
74	C₂²×C₁₃₇	—	2	4	no	—
75	C₂²×C₁₃₇	—	2	1	no	—
76	C₂²×C₁₃₇	—	2	4	no	—
77	C₂²×C₁₃₇	—	2	1	no	—
78	C₂²×C₁₃₇	—	2	4	no	—
79	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
80	C₄×C₁₃₇	—	2	1	no	—
81	C₄×C₁₃₇	—	2	4	no	—
82	C₄×C₁₃₇	—	2	1	no	—
83	C₄×C₁₃₇	—	2	1	no	—
84	C₄×C₁₃₇	—	2	1	no	—
85	C₄×C₁₃₇	#131	1	2	yes	—
86	C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
87	C₂×C₁₃₇	—	4	1	no	—
88	C₂×C₁₃₇	—	4	4	no	—
89	C₂×C₁₃₇	#518	1	1	yes	characteristic
90	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
91	C₂×C₁₃₇	#568	1	1	yes	characteristic
92	C₁₃₇	#2981	1	1	yes	characteristic, direct product factor, Sylow
93	#2981	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	#532	C₂×C₁₃₇	1	1	yes	characteristic
95	#436	C₂×C₁₃₇	1	2	yes	complemented
96	#435	C₂×C₁₃₇	1	4	yes	complemented
97	C₈×D₄	C₂×C₁₃₇	1	2	yes	complemented
98	#432	C₂×C₁₃₇	1	1	yes	characteristic, complemented
99	#431	C₂×C₁₃₇	1	1	yes	characteristic, complemented
100	#408	C₂×C₁₃₇	1	4	yes	complemented
101	C₂×#41	C₂²×C₁₃₇	1	2	yes	—
102	C₂×M₁₆	C₂²×C₁₃₇	1	4	yes	—
103	C₂²×C₈	C₂²×C₁₃₇	1	4	yes	—
104	C₄⋊Q₈	C₄×C₁₃₇	1	1	yes	characteristic
105	#127	C₄×C₁₃₇	1	1	yes	characteristic
106	#124	C₄×C₁₃₇	1	2	yes	—
107	#121	C₄×C₁₃₇	1	2	yes	—
108	#121	C₄×C₁₃₇	1	2	yes	—
109	C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
110	C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
111	C₂×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
112	C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
113	C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
114	C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
115	C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
116	C₂²⋊C₈	C₂²×C₁₃₇	1	4	yes	—
117	C₂²⋊C₈	C₂²×C₁₃₇	1	4	yes	—
118	#97	C₂²×C₁₃₇	1	2	yes	—
119	C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
120	C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
121	#41	—	4	4	no	—
122	C₂×Q₈	C₂×C₄×C₁₃₇	1	2	yes	—
123	C₂×D₄	C₂×C₄×C₁₃₇	1	2	yes	—
124	C₂×D₄	C₂×C₄×C₁₃₇	1	2	yes	—
125	C₂×D₄	C₂×C₄×C₁₃₇	1	4	yes	—
126	C₂×D₄	C₂×C₄×C₁₃₇	1	2	yes	—
127	C₂²×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
129	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
130	C₂²×C₄	C₂³×C₁₃₇	1	4	yes	—
131	M₁₆	—	4	4	no	—
132	C₂×C₈	—	4	4	no	—
133	C₂×C₈	C₂³×C₁₃₇	1	2	yes	—
134	C₂×C₈	C₂³×C₁₃₇	1	2	yes	—
135	C₂×C₈	C₂³×C₁₃₇	1	4	yes	—
136	C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
137	C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
138	C₂²⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
139	C₂²⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
140	C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
141	C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
142	C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
143	C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
144	C₂³	C₂²×C₄×C₁₃₇	1	4	yes	—
145	C₂³	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
146	Q₈	—	4	2	no	—
147	D₄	—	2	4	no	—
148	D₄	—	4	2	no	—
149	D₄	—	4	4	no	—
150	D₄	—	4	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
157	C₂×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
158	C₂×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
159	C₂×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
160	C₂×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
161	C₂×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
162	C₂×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
163	C₈	—	2	2	no	—
164	C₈	—	2	4	no	—
165	C₈	—	2	2	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	4	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	4	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	2	4	no	—
172	C₂²	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	4	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	#131×C₁₃₇	1	2	yes	—
179	C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
180	C₂	—	4	1	no	—
181	C₂	—	4	4	no	—
182	C₂	#518×C₁₃₇	1	1	yes	characteristic
183	C₂	C₂×#131×C₁₃₇	1	1	yes	characteristic
184	C₂	#568×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃₇
socle	C₂²×C₁₃₇	C₂³×C₄
prime radical	#532×C₁₃₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #436×C₁₃₇	C₂	semidirect
3. #435×C₁₃₇	C₂	semidirect
4. C₈×D₄×C₁₃₇	C₂	semidirect
5. #432×C₁₃₇	C₂	semidirect
6. #431×C₁₃₇	C₂	semidirect
7. #408×C₁₃₇	C₂	semidirect
92. C₁₃₇	#2981	direct (N × H)
93. #2981	C₁₃₇	direct (N × H)
95. #436	C₂×C₁₃₇	semidirect
96. #435	C₂×C₁₃₇	semidirect
97. C₈×D₄	C₂×C₁₃₇	semidirect
98. #432	C₂×C₁₃₇	semidirect
99. #431	C₂×C₁₃₇	semidirect
100. #408	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#532×C₁₃₇	2
#436×C₁₃₇	2
#435×C₁₃₇	2
C₈×D₄×C₁₃₇	2
#432×C₁₃₇	2
#431×C₁₃₇	2
#408×C₁₃₇	2
#2981	137

Subgroups — #2981×C137 #31777992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2981×C₁₃₇ #31777992