Subgroups — #3355×C₁₃₇ #31778366

Order 17536.

360 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#553×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	#499×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#482×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#480×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#478×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#469×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	C₄×D₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#436×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#428×C₁₃₇	C₂	1	2	yes	maximal, complemented
10	C₂×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
12	#126×C₁₃₇	—	2	2	no	—
13	#125×C₁₃₇	C₂²	1	2	yes	—
14	#123×C₁₃₇	—	2	2	no	—
15	#121×C₁₃₇	—	2	2	no	—
16	#121×C₁₃₇	C₂²	1	2	yes	—
17	#121×C₁₃₇	C₂²	1	2	yes	—
18	C₄×D₄×C₁₃₇	—	2	2	no	—
19	C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
20	C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
21	#117×C₁₃₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
23	#107×C₁₃₇	C₂²	1	1	yes	characteristic
24	#107×C₁₃₇	C₂²	1	2	yes	—
25	#107×C₁₃₇	C₂²	1	1	yes	characteristic
26	#107×C₁₃₇	C₂²	1	1	yes	characteristic
27	#106×C₁₃₇	C₂²	1	2	yes	—
28	#106×C₁₃₇	C₂²	1	2	yes	—
29	C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
30	#102×C₁₃₇	C₂²	1	2	yes	—
31	#102×C₁₃₇	C₂²	1	2	yes	—
32	#102×C₁₃₇	C₂²	1	2	yes	—
33	C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
34	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
35	C₂×D₄×C₁₃₇	—	2	2	no	—
36	C₂×D₄×C₁₃₇	—	2	2	no	—
37	C₂×D₄×C₁₃₇	D₄	1	1	yes	characteristic
38	C₂×D₄×C₁₃₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃₇	—	2	2	no	—
40	C₂²×C₄×C₁₃₇	—	2	2	no	—
41	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
42	D₈×C₁₃₇	—	2	2	no	—
43	M₁₆×C₁₃₇	—	2	4	no	—
44	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
46	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₁₃₇	—	2	2	no	—
48	C₄⋊C₄×C₁₃₇	—	2	2	no	—
49	C₄⋊C₄×C₁₃₇	—	2	2	no	—
50	C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
51	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
52	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
53	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
54	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
55	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
56	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
57	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
58	C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
59	C₄²×C₁₃₇	—	2	2	no	—
60	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
61	C₂³×C₁₃₇	—	2	2	no	—
62	C₂³×C₁₃₇	C₂×D₄	1	2	yes	—
63	D₄×C₁₃₇	—	2	2	no	—
64	D₄×C₁₃₇	—	2	2	no	—
65	D₄×C₁₃₇	—	4	1	no	—
66	D₄×C₁₃₇	—	4	4	no	—
67	D₄×C₁₃₇	—	8	2	no	—
68	C₂×C₄×C₁₃₇	—	2	2	no	—
69	C₂×C₄×C₁₃₇	—	2	2	no	—
70	C₂×C₄×C₁₃₇	—	2	2	no	—
71	C₂×C₄×C₁₃₇	—	2	2	no	—
72	C₂×C₄×C₁₃₇	—	2	2	no	—
73	C₂×C₄×C₁₃₇	—	8	2	no	—
74	C₂×C₄×C₁₃₇	—	8	2	no	—
75	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
79	C₈×C₁₃₇	—	2	2	no	—
80	C₈×C₁₃₇	—	2	1	no	—
81	C₈×C₁₃₇	#41	1	2	yes	—
82	C₂²×C₁₃₇	—	2	2	no	—
83	C₂²×C₁₃₇	—	2	2	no	—
84	C₂²×C₁₃₇	—	2	2	no	—
85	C₂²×C₁₃₇	—	4	2	no	—
86	C₂²×C₁₃₇	—	4	2	no	—
87	C₂²×C₁₃₇	—	4	2	no	—
88	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₁₃₇	—	2	2	no	—
90	C₄×C₁₃₇	—	2	1	no	—
91	C₄×C₁₃₇	—	2	1	no	—
92	C₄×C₁₃₇	—	4	2	no	—
93	C₄×C₁₃₇	—	4	2	no	—
94	C₄×C₁₃₇	—	4	2	no	—
95	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
96	C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
97	C₂×C₁₃₇	—	4	2	no	—
98	C₂×C₁₃₇	—	8	2	no	—
99	C₂×C₁₃₇	#547	1	1	yes	characteristic
100	C₂×C₁₃₇	#575	1	1	yes	characteristic
101	C₂×C₁₃₇	#576	1	1	yes	characteristic
102	C₁₃₇	#3355	1	1	yes	characteristic, direct product factor, Sylow
103	#3355	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	#553	C₂×C₁₃₇	1	2	yes	complemented
105	#499	C₂×C₁₃₇	1	1	yes	characteristic, complemented
106	#482	C₂×C₁₃₇	1	2	yes	complemented
107	#480	C₂×C₁₃₇	1	2	yes	complemented
108	#478	C₂×C₁₃₇	1	2	yes	complemented
109	#469	C₂×C₁₃₇	1	2	yes	complemented
110	C₄×D₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
111	#436	C₂×C₁₃₇	1	1	yes	characteristic, complemented
112	#428	C₂×C₁₃₇	1	2	yes	complemented
113	C₂×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
114	C₂×M₁₆	C₂²×C₁₃₇	1	2	yes	—
115	#126	—	2	2	no	—
116	#125	C₂²×C₁₃₇	1	2	yes	—
117	#123	—	2	2	no	—
118	#121	—	2	2	no	—
119	#121	C₂²×C₁₃₇	1	2	yes	—
120	#121	C₂²×C₁₃₇	1	2	yes	—
121	C₄×D₄	—	2	2	no	—
122	C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
123	C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
124	#117	C₂²×C₁₃₇	1	2	yes	—
125	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
126	#107	C₂²×C₁₃₇	1	1	yes	characteristic
127	#107	C₂²×C₁₃₇	1	2	yes	—
128	#107	C₂²×C₁₃₇	1	1	yes	characteristic
129	#107	C₂²×C₁₃₇	1	1	yes	characteristic
130	#106	C₂²×C₁₃₇	1	2	yes	—
131	#106	C₂²×C₁₃₇	1	2	yes	—
132	C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
133	#102	C₂²×C₁₃₇	1	2	yes	—
134	#102	C₂²×C₁₃₇	1	2	yes	—
135	#102	C₂²×C₁₃₇	1	2	yes	—
136	C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
137	C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	D₄×C₁₃₇	1	1	yes	characteristic
141	C₂×D₄	C₂³×C₁₃₇	1	2	yes	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	C₂³×C₁₃₇	1	2	yes	—
145	D₈	—	2	2	no	—
146	M₁₆	—	2	4	no	—
147	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
148	C₂×C₈	C₂³×C₁₃₇	1	2	yes	—
149	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
154	C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
155	C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
156	C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₂²⋊C₄	—	2	2	no	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	D₄×C₁₃₇	1	2	yes	—
162	C₄²	—	2	2	no	—
163	C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
164	C₂³	—	2	2	no	—
165	C₂³	C₂×D₄×C₁₃₇	1	2	yes	—
166	D₄	—	2	2	no	—
167	D₄	—	2	2	no	—
168	D₄	—	4	1	no	—
169	D₄	—	4	4	no	—
170	D₄	—	8	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
179	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
180	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
181	C₂×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	2	1	no	—
184	C₈	#41×C₁₃₇	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	4	2	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
199	C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
200	C₂	—	4	2	no	—
201	C₂	—	8	2	no	—
202	C₂	#547×C₁₃₇	1	1	yes	characteristic
203	C₂	#575×C₁₃₇	1	1	yes	characteristic
204	C₂	#576×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₄×D₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #553×C₁₃₇	C₂	semidirect
2. #499×C₁₃₇	C₂	semidirect
3. #482×C₁₃₇	C₂	semidirect
4. #480×C₁₃₇	C₂	semidirect
5. #478×C₁₃₇	C₂	semidirect
6. #469×C₁₃₇	C₂	semidirect
7. C₄×D₈×C₁₃₇	C₂	semidirect
8. #436×C₁₃₇	C₂	semidirect
9. #428×C₁₃₇	C₂	semidirect
102. C₁₃₇	#3355	direct (N × H)
103. #3355	C₁₃₇	direct (N × H)
104. #553	C₂×C₁₃₇	semidirect
105. #499	C₂×C₁₃₇	semidirect
106. #482	C₂×C₁₃₇	semidirect
107. #480	C₂×C₁₃₇	semidirect
108. #478	C₂×C₁₃₇	semidirect
109. #469	C₂×C₁₃₇	semidirect
110. C₄×D₈	C₂×C₁₃₇	semidirect
111. #436	C₂×C₁₃₇	semidirect
112. #428	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#553×C₁₃₇	2
#499×C₁₃₇	2
#482×C₁₃₇	2
#480×C₁₃₇	2
#478×C₁₃₇	2
#469×C₁₃₇	2
C₄×D₈×C₁₃₇	2
#436×C₁₃₇	2
#428×C₁₃₇	2
#3355	137

Subgroups — #3355×C137 #31778366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3355×C₁₃₇ #31778366