Subgroups — C₁₃₇×#28702 #46878088

Order 35072.

598 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3166×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×#499×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2960×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2090×C₁₃₇	C₂	1	4	yes	maximal
5	#1998×C₁₃₇	C₂	1	2	yes	maximal
6	#1952×C₁₃₇	C₂	1	4	yes	maximal
7	C₄×#108×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#533×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
10	#501×C₁₃₇	—	2	4	no	—
11	#499×C₁₃₇	—	2	4	no	—
12	#499×C₁₃₇	—	2	2	no	—
13	#499×C₁₃₇	—	2	2	no	—
14	#446×C₁₃₇	—	2	2	no	—
15	C₈×Q₈×C₁₃₇	—	2	2	no	—
16	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
17	#428×C₁₃₇	C₂²	1	8	yes	—
18	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
19	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
20	#424×C₁₃₇	C₂²	1	2	yes	—
21	#423×C₁₃₇	C₂²	1	1	yes	characteristic
22	#405×C₁₃₇	C₂²	1	2	yes	—
23	C₄×M₁₆×C₁₃₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
25	#343×C₁₃₇	C₂²	1	8	yes	—
26	#341×C₁₃₇	C₂²	1	2	yes	—
27	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
28	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
29	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₁₃₇	D₄	1	2	yes	—
32	#125×C₁₃₇	—	2	2	no	—
33	#125×C₁₃₇	—	2	4	no	—
34	#125×C₁₃₇	—	2	2	no	—
35	#125×C₁₃₇	D₄	1	2	yes	—
36	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
37	C₄×Q₈×C₁₃₇	—	2	2	no	—
38	#117×C₁₃₇	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
41	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
42	#108×C₁₃₇	Q₈	1	8	yes	complemented
43	#107×C₁₃₇	—	2	2	no	—
44	#107×C₁₃₇	—	2	2	no	—
45	#107×C₁₃₇	—	2	8	no	—
46	#106×C₁₃₇	—	2	4	no	—
47	#106×C₁₃₇	—	2	8	no	—
48	C₄⋊C₈×C₁₃₇	—	2	4	no	—
49	C₄⋊C₈×C₁₃₇	—	2	2	no	—
50	#97×C₁₃₇	—	2	2	no	—
51	#97×C₁₃₇	—	2	2	no	—
52	C₄×C₈×C₁₃₇	—	2	2	no	—
53	C₄×C₈×C₁₃₇	—	2	2	no	—
54	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
55	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
56	C₂×Q₈×C₁₃₇	—	2	2	no	—
57	C₂²×C₄×C₁₃₇	—	2	4	no	—
58	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
59	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
60	M₁₆×C₁₃₇	—	2	4	no	—
61	M₁₆×C₁₃₇	—	2	4	no	—
62	M₁₆×C₁₃₇	C₂×Q₈	1	8	yes	—
63	C₂×C₈×C₁₃₇	—	2	4	no	—
64	C₂×C₈×C₁₃₇	—	2	2	no	—
65	C₂×C₈×C₁₃₇	—	2	2	no	—
66	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
70	C₄⋊C₄×C₁₃₇	—	2	4	no	—
71	C₄⋊C₄×C₁₃₇	—	2	2	no	—
72	C₄⋊C₄×C₁₃₇	—	2	4	no	—
73	C₄⋊C₄×C₁₃₇	—	2	4	no	—
74	C₄⋊C₄×C₁₃₇	—	4	4	no	—
75	C₄⋊C₄×C₁₃₇	—	4	4	no	—
76	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
77	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
78	C₄²×C₁₃₇	—	2	2	no	—
79	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
81	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
82	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
83	Q₈×C₁₃₇	—	4	4	no	—
84	C₂×C₄×C₁₃₇	—	2	4	no	—
85	C₂×C₄×C₁₃₇	—	4	2	no	—
86	C₂×C₄×C₁₃₇	—	4	4	no	—
87	C₂×C₄×C₁₃₇	—	8	4	no	—
88	C₂×C₄×C₁₃₇	#121	1	2	yes	—
89	C₂×C₄×C₁₃₇	#121	1	2	yes	—
90	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
93	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
94	C₈×C₁₃₇	—	2	2	no	—
95	C₈×C₁₃₇	—	2	8	no	—
96	C₈×C₁₃₇	—	4	2	no	—
97	C₈×C₁₃₇	C₄⋊Q₈	1	2	yes	—
98	C₈×C₁₃₇	C₄⋊Q₈	1	2	yes	—
99	C₂²×C₁₃₇	—	2	2	no	—
100	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
101	C₂²×C₁₃₇	#557	1	2	yes	—
102	C₄×C₁₃₇	—	2	2	no	—
103	C₄×C₁₃₇	—	2	2	no	—
104	C₄×C₁₃₇	—	4	4	no	—
105	C₄×C₁₃₇	—	8	4	no	—
106	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
107	C₄×C₁₃₇	#536	1	1	yes	characteristic
108	C₄×C₁₃₇	D₄×Q₈	1	2	yes	—
109	C₂×C₁₃₇	—	2	2	no	—
110	C₂×C₁₃₇	#2546	1	1	yes	characteristic
111	C₂×C₁₃₇	#3067	1	2	yes	—
112	#28702	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₁₃₇	#28702	1	1	yes	characteristic, direct product factor, Sylow
114	#3166	C₂×C₁₃₇	1	2	yes	—
115	C₂×#499	C₂×C₁₃₇	1	1	yes	characteristic
116	#2960	C₂×C₁₃₇	1	1	yes	characteristic
117	#2090	C₂×C₁₃₇	1	4	yes	—
118	#1998	C₂×C₁₃₇	1	2	yes	—
119	#1952	C₂×C₁₃₇	1	4	yes	—
120	C₄×#108	C₂×C₁₃₇	1	1	yes	characteristic
121	#533	C₂²×C₁₃₇	1	1	yes	characteristic
122	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
123	#501	—	2	4	no	—
124	#499	—	2	4	no	—
125	#499	—	2	2	no	—
126	#499	—	2	2	no	—
127	#446	—	2	2	no	—
128	C₈×Q₈	—	2	2	no	—
129	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
130	#428	C₂²×C₁₃₇	1	8	yes	—
131	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
132	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
133	#424	C₂²×C₁₃₇	1	2	yes	—
134	#423	C₂²×C₁₃₇	1	1	yes	characteristic
135	#405	C₂²×C₁₃₇	1	2	yes	—
136	C₄×M₁₆	C₂²×C₁₃₇	1	2	yes	—
137	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
138	#343	C₂²×C₁₃₇	1	8	yes	—
139	#341	C₂²×C₁₃₇	1	2	yes	—
140	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
141	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
142	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
144	C₄⋊Q₈	D₄×C₁₃₇	1	2	yes	—
145	#125	—	2	2	no	—
146	#125	—	2	4	no	—
147	#125	—	2	2	no	—
148	#125	D₄×C₁₃₇	1	2	yes	—
149	C₂²⋊Q₈	—	2	2	no	—
150	C₄×Q₈	—	2	2	no	—
151	#117	C₂³×C₁₃₇	1	2	yes	—
152	C₂×C₄⋊C₄	—	2	4	no	—
153	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
154	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
155	#108	Q₈×C₁₃₇	1	8	yes	complemented
156	#107	—	2	2	no	—
157	#107	—	2	2	no	—
158	#107	—	2	8	no	—
159	#106	—	2	4	no	—
160	#106	—	2	8	no	—
161	C₄⋊C₈	—	2	4	no	—
162	C₄⋊C₈	—	2	2	no	—
163	#97	—	2	2	no	—
164	#97	—	2	2	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
168	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
169	C₂×Q₈	—	2	2	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
172	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
173	M₁₆	—	2	4	no	—
174	M₁₆	—	2	4	no	—
175	M₁₆	C₂×Q₈×C₁₃₇	1	8	yes	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
180	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
181	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
182	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
193	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
194	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
195	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
196	Q₈	—	4	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	8	4	no	—
201	C₂×C₄	#121×C₁₃₇	1	2	yes	—
202	C₂×C₄	#121×C₁₃₇	1	2	yes	—
203	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
204	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
205	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
206	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
207	C₈	—	2	2	no	—
208	C₈	—	2	8	no	—
209	C₈	—	4	2	no	—
210	C₈	C₄⋊Q₈×C₁₃₇	1	2	yes	—
211	C₈	C₄⋊Q₈×C₁₃₇	1	2	yes	—
212	C₂²	—	2	2	no	—
213	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
214	C₂²	#557×C₁₃₇	1	2	yes	—
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	4	4	no	—
218	C₄	—	8	4	no	—
219	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
220	C₄	#536×C₁₃₇	1	1	yes	characteristic
221	C₄	D₄×Q₈×C₁₃₇	1	2	yes	—
222	C₂	—	2	2	no	—
223	C₂	#2546×C₁₃₇	1	1	yes	characteristic
224	C₂	#3067×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂³×C₁₃₇	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
42. #108×C₁₃₇	Q₈	semidirect
112. #28702	C₁₃₇	direct (N × H)
113. C₁₃₇	#28702	direct (N × H)
155. #108	Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3166×C₁₃₇	2
C₂×#499×C₁₃₇	2
#2960×C₁₃₇	2
#2090×C₁₃₇	2
#1998×C₁₃₇	2
#1952×C₁₃₇	2
C₄×#108×C₁₃₇	2
#28702	137

Subgroups — C137×#28702 #46878088

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28702 #46878088