Subgroups — C₁₃₇×#9792 #46859178

Order 35072.

358 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1568×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1528×C₁₃₇	C₂	1	2	yes	maximal
4	#1470×C₁₃₇	C₂	1	2	yes	maximal
5	#1451×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
7	#499×C₁₃₇	—	2	1	no	—
8	#499×C₁₃₇	—	2	1	no	—
9	#499×C₁₃₇	—	2	1	no	—
10	#499×C₁₃₇	—	2	1	no	—
11	#432×C₁₃₇	C₂²	1	2	yes	—
12	#431×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#107×C₁₃₇	C₄	1	1	yes	characteristic
14	C₂×#107×C₁₃₇	C₄	1	1	yes	characteristic
15	C₂×#106×C₁₃₇	C₄	1	2	yes	—
16	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	#335×C₁₃₇	—	2	1	no	—
19	#334×C₁₃₇	—	2	1	no	—
20	#330×C₁₃₇	—	2	4	no	—
21	#324×C₁₃₇	—	2	4	no	—
22	#322×C₁₃₇	—	2	1	no	—
23	#322×C₁₃₇	—	2	1	no	—
24	C₂×M₁₆×C₁₃₇	—	2	1	no	—
25	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	#125×C₁₃₇	—	2	2	no	—
28	#125×C₁₃₇	—	2	2	no	—
29	#125×C₁₃₇	—	2	2	no	—
30	#125×C₁₃₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
36	#107×C₁₃₇	—	2	1	no	—
37	#107×C₁₃₇	—	2	1	no	—
38	#107×C₁₃₇	—	2	1	no	—
39	#107×C₁₃₇	—	2	1	no	—
40	#106×C₁₃₇	—	2	2	no	—
41	#106×C₁₃₇	—	2	2	no	—
42	C₄⋊C₈×C₁₃₇	—	2	2	no	—
43	C₄⋊C₈×C₁₃₇	—	2	1	no	—
44	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
45	C₂²⋊C₈×C₁₃₇	D₄	1	2	yes	—
46	#97×C₁₃₇	—	2	2	no	—
47	#97×C₁₃₇	D₄	1	2	yes	—
48	C₄×C₈×C₁₃₇	—	2	1	no	—
49	C₄×C₈×C₁₃₇	—	2	1	no	—
50	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₇	—	2	2	no	—
55	C₂²×C₄×C₁₃₇	—	2	2	no	—
56	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
59	M₁₆×C₁₃₇	—	4	2	no	—
60	C₂×C₈×C₁₃₇	—	2	2	no	—
61	C₂×C₈×C₁₃₇	—	2	2	no	—
62	C₂×C₈×C₁₃₇	—	2	2	no	—
63	C₂×C₈×C₁₃₇	—	2	2	no	—
64	C₂×C₈×C₁₃₇	—	4	2	no	—
65	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₁₃₇	—	2	2	no	—
70	C₄⋊C₄×C₁₃₇	—	2	2	no	—
71	C₄⋊C₄×C₁₃₇	—	2	2	no	—
72	C₄⋊C₄×C₁₃₇	—	2	2	no	—
73	C₄⋊C₄×C₁₃₇	—	4	2	no	—
74	C₄⋊C₄×C₁₃₇	—	4	2	no	—
75	C₄⋊C₄×C₁₃₇	—	4	2	no	—
76	C₄⋊C₄×C₁₃₇	—	4	2	no	—
77	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₃₇	#41	1	1	yes	characteristic
81	C₂³×C₁₃₇	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃₇	—	2	2	no	—
83	C₂×C₄×C₁₃₇	—	2	1	no	—
84	C₂×C₄×C₁₃₇	—	2	1	no	—
85	C₂×C₄×C₁₃₇	—	4	1	no	—
86	C₂×C₄×C₁₃₇	—	4	2	no	—
87	C₂×C₄×C₁₃₇	—	4	2	no	—
88	C₂×C₄×C₁₃₇	—	4	4	no	—
89	C₂×C₄×C₁₃₇	—	8	2	no	—
90	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
93	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
94	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
95	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
96	C₈×C₁₃₇	—	2	1	no	—
97	C₈×C₁₃₇	—	2	1	no	—
98	C₈×C₁₃₇	—	4	2	no	—
99	C₈×C₁₃₇	—	4	2	no	—
100	C₈×C₁₃₇	—	8	2	no	—
101	C₈×C₁₃₇	#101	1	2	yes	—
102	C₈×C₁₃₇	#101	1	2	yes	—
103	C₂²×C₁₃₇	—	2	1	no	—
104	C₂²×C₁₃₇	—	2	1	no	—
105	C₂²×C₁₃₇	#390	1	1	yes	characteristic
106	C₂²×C₁₃₇	#444	1	2	yes	—
107	C₄×C₁₃₇	—	2	1	no	—
108	C₄×C₁₃₇	—	2	1	no	—
109	C₄×C₁₃₇	—	2	1	no	—
110	C₄×C₁₃₇	—	2	1	no	—
111	C₄×C₁₃₇	—	2	1	no	—
112	C₄×C₁₃₇	—	8	2	no	—
113	C₄×C₁₃₇	—	8	2	no	—
114	C₄×C₁₃₇	C₂×#101	1	1	yes	characteristic
115	C₄×C₁₃₇	C₂×#101	1	1	yes	characteristic
116	C₂×C₁₃₇	—	2	2	no	—
117	C₂×C₁₃₇	#1966	1	1	yes	characteristic
118	C₂×C₁₃₇	#1975	1	1	yes	characteristic
119	C₂×C₁₃₇	#1976	1	1	yes	characteristic
120	#9792	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₁₃₇	#9792	1	1	yes	characteristic, direct product factor, Sylow
122	C₂×#499	C₂×C₁₃₇	1	1	yes	characteristic
123	#1568	C₂×C₁₃₇	1	1	yes	characteristic
124	#1528	C₂×C₁₃₇	1	2	yes	—
125	#1470	C₂×C₁₃₇	1	2	yes	—
126	#1451	C₂×C₁₃₇	1	1	yes	characteristic
127	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
128	#499	—	2	1	no	—
129	#499	—	2	1	no	—
130	#499	—	2	1	no	—
131	#499	—	2	1	no	—
132	#432	C₂²×C₁₃₇	1	2	yes	—
133	#431	C₂²×C₁₃₇	1	1	yes	characteristic
134	C₂×#107	C₄×C₁₃₇	1	1	yes	characteristic
135	C₂×#107	C₄×C₁₃₇	1	1	yes	characteristic
136	C₂×#106	C₄×C₁₃₇	1	2	yes	—
137	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
139	#335	—	2	1	no	—
140	#334	—	2	1	no	—
141	#330	—	2	4	no	—
142	#324	—	2	4	no	—
143	#322	—	2	1	no	—
144	#322	—	2	1	no	—
145	C₂×M₁₆	—	2	1	no	—
146	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
147	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
148	#125	—	2	2	no	—
149	#125	—	2	2	no	—
150	#125	—	2	2	no	—
151	#125	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
155	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
156	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
157	#107	—	2	1	no	—
158	#107	—	2	1	no	—
159	#107	—	2	1	no	—
160	#107	—	2	1	no	—
161	#106	—	2	2	no	—
162	#106	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	2	1	no	—
165	C₂²⋊C₈	—	2	2	no	—
166	C₂²⋊C₈	D₄×C₁₃₇	1	2	yes	—
167	#97	—	2	2	no	—
168	#97	D₄×C₁₃₇	1	2	yes	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	—	2	1	no	—
171	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
172	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
173	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
174	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
179	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
180	M₁₆	—	4	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
187	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
188	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
199	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
200	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
201	C₄²	#41×C₁₃₇	1	1	yes	characteristic
202	C₂³	C₄×D₄×C₁₃₇	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
212	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
213	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
214	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
215	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
216	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	4	2	no	—
220	C₈	—	4	2	no	—
221	C₈	—	8	2	no	—
222	C₈	#101×C₁₃₇	1	2	yes	—
223	C₈	#101×C₁₃₇	1	2	yes	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	#390×C₁₃₇	1	1	yes	characteristic
227	C₂²	#444×C₁₃₇	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	8	2	no	—
234	C₄	—	8	2	no	—
235	C₄	C₂×#101×C₁₃₇	1	1	yes	characteristic
236	C₄	C₂×#101×C₁₃₇	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	#1966×C₁₃₇	1	1	yes	characteristic
239	C₂	#1975×C₁₃₇	1	1	yes	characteristic
240	C₂	#1976×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#390
commutator	C₂²×C₄	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
prime radical	C₂³×C₁₃₇	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ 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
120. #9792	C₁₃₇	direct (N × H)
121. C₁₃₇	#9792	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#499×C₁₃₇	2
#1568×C₁₃₇	2
#1528×C₁₃₇	2
#1470×C₁₃₇	2
#1451×C₁₃₇	2
#9792	137

Subgroups — C137×#9792 #46859178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#9792 #46859178