Subgroups — C₁₃₇×#7868 #46857254

Order 35072.

662 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2920×C₁₃₇	C₂	1	2	yes	maximal
2	C₄×#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1276×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₁₃₇	C₄	1	2	yes	—
5	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
6	#432×C₁₃₇	—	2	2	no	—
7	#432×C₁₃₇	C₄	1	4	yes	complemented
8	#431×C₁₃₇	—	2	2	no	—
9	#431×C₁₃₇	C₂²	1	4	yes	—
10	#423×C₁₃₇	C₄	1	4	yes	complemented
11	#407×C₁₃₇	—	2	2	no	—
12	C₄×M₁₆×C₁₃₇	—	2	2	no	—
13	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
14	C₄×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
15	#376×C₁₃₇	C₄	1	2	yes	—
16	C₄³×C₁₃₇	C₂²	1	2	yes	—
17	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₃₇	D₄	1	2	yes	—
19	C₂×M₁₆×C₁₃₇	—	2	4	no	—
20	#117×C₁₃₇	—	2	8	no	—
21	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
22	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
23	C₂×C₄²×C₁₃₇	—	2	4	no	—
24	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
29	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃₇	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₁₃₇	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₁₃₇	—	2	4	no	—
35	C₄⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	—
36	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
37	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
38	#97×C₁₃₇	—	2	4	no	—
39	C₄×C₈×C₁₃₇	—	2	4	no	—
40	#95×C₁₃₇	—	2	4	no	—
41	C₂⁴×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₇	—	2	4	no	—
43	C₂²×C₄×C₁₃₇	—	2	4	no	—
44	C₂²×C₄×C₁₃₇	—	4	4	no	—
45	C₂²×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
57	M₁₆×C₁₃₇	—	4	8	no	—
58	C₂×C₈×C₁₃₇	—	2	8	no	—
59	C₄⋊C₄×C₁₃₇	—	2	8	no	—
60	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
61	C₄²×C₁₃₇	—	2	4	no	—
62	C₄²×C₁₃₇	—	4	8	no	—
63	C₄²×C₁₃₇	C₄²	1	2	yes	—
64	C₄²×C₁₃₇	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₁₃₇	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
68	C₄²×C₁₃₇	C₄⋊C₄	1	2	yes	—
69	C₄²×C₁₃₇	C₄⋊C₄	1	2	yes	—
70	C₄²×C₁₃₇	M₁₆	1	8	yes	complemented
71	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
72	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
74	C₂³×C₁₃₇	—	2	2	no	—
75	C₂³×C₁₃₇	—	4	2	no	—
76	C₂³×C₁₃₇	#95	1	1	yes	characteristic
77	C₂³×C₁₃₇	#95	1	1	yes	characteristic
78	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃₇	—	2	2	no	—
80	C₂×C₄×C₁₃₇	—	2	2	no	—
81	C₂×C₄×C₁₃₇	—	2	2	no	—
82	C₂×C₄×C₁₃₇	—	2	4	no	—
83	C₂×C₄×C₁₃₇	—	2	4	no	—
84	C₂×C₄×C₁₃₇	—	2	4	no	—
85	C₂×C₄×C₁₃₇	—	2	2	no	—
86	C₂×C₄×C₁₃₇	—	2	2	no	—
87	C₂×C₄×C₁₃₇	—	2	8	no	—
88	C₂×C₄×C₁₃₇	—	4	2	no	—
89	C₂×C₄×C₁₃₇	—	4	2	no	—
90	C₂×C₄×C₁₃₇	—	4	2	no	—
91	C₂×C₄×C₁₃₇	—	4	8	no	—
92	C₂×C₄×C₁₃₇	#95	1	2	yes	—
93	C₂×C₄×C₁₃₇	#95	1	2	yes	—
94	C₂×C₄×C₁₃₇	#95	1	1	yes	characteristic
95	C₂×C₄×C₁₃₇	#95	1	1	yes	characteristic
96	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃₇	C₂×M₁₆	1	4	yes	—
101	C₈×C₁₃₇	—	4	8	no	—
102	C₂²×C₁₃₇	—	2	2	no	—
103	C₂²×C₁₃₇	—	2	1	no	—
104	C₂²×C₁₃₇	—	2	1	no	—
105	C₂²×C₁₃₇	—	2	2	no	—
106	C₂²×C₁₃₇	—	2	2	no	—
107	C₂²×C₁₃₇	—	4	4	no	—
108	C₂²×C₁₃₇	C₂×#95	1	1	yes	characteristic
109	C₂²×C₁₃₇	#407	1	2	yes	—
110	C₄×C₁₃₇	—	2	1	no	—
111	C₄×C₁₃₇	—	2	1	no	—
112	C₄×C₁₃₇	—	2	2	no	—
113	C₄×C₁₃₇	—	2	2	no	—
114	C₄×C₁₃₇	—	2	2	no	—
115	C₄×C₁₃₇	—	4	8	no	—
116	C₄×C₁₃₇	C₄×M₁₆	1	2	yes	—
117	C₄×C₁₃₇	#421	1	4	yes	—
118	C₄×C₁₃₇	#423	1	2	yes	—
119	C₂×C₁₃₇	—	2	2	no	—
120	C₂×C₁₃₇	—	4	1	no	—
121	C₂×C₁₃₇	—	4	1	no	—
122	C₂×C₁₃₇	#1728	1	1	yes	characteristic
123	C₂×C₁₃₇	#1733	1	2	yes	—
124	#7868	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₁₃₇	#7868	1	1	yes	characteristic, direct product factor, Sylow
126	#2920	C₂×C₁₃₇	1	2	yes	—
127	C₄×#117	C₂×C₁₃₇	1	1	yes	characteristic
128	#1276	C₂×C₁₃₇	1	4	yes	complemented
129	C₂×#117	C₄×C₁₃₇	1	2	yes	—
130	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
131	#432	—	2	2	no	—
132	#432	C₄×C₁₃₇	1	4	yes	complemented
133	#431	—	2	2	no	—
134	#431	C₂²×C₁₃₇	1	4	yes	—
135	#423	C₄×C₁₃₇	1	4	yes	complemented
136	#407	—	2	2	no	—
137	C₄×M₁₆	—	2	2	no	—
138	C₄×C₄⋊C₄	—	2	2	no	—
139	C₄×C₂²⋊C₄	—	2	2	no	—
140	#376	C₄×C₁₃₇	1	2	yes	—
141	C₄³	C₂²×C₁₃₇	1	2	yes	—
142	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
143	C₂³×C₄	D₄×C₁₃₇	1	2	yes	—
144	C₂×M₁₆	—	2	4	no	—
145	#117	—	2	8	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₂²⋊C₄	—	2	2	no	—
148	C₂×C₄²	—	2	4	no	—
149	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
150	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
151	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
152	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
153	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
154	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
155	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
156	C₂×C₄²	Q₈×C₁₃₇	1	1	yes	characteristic
157	C₂×C₄²	Q₈×C₁₃₇	1	1	yes	characteristic
158	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
159	C₄⋊C₈	—	2	4	no	—
160	C₄⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	—
161	C₂²⋊C₈	—	2	4	no	—
162	C₂²⋊C₈	—	2	4	no	—
163	#97	—	2	4	no	—
164	C₄×C₈	—	2	4	no	—
165	#95	—	2	4	no	—
166	C₂⁴	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	C₄²×C₁₃₇	1	1	yes	characteristic
171	C₂²×C₄	C₄²×C₁₃₇	1	1	yes	characteristic
172	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
173	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
174	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
175	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
176	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
177	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
179	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
180	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
182	M₁₆	—	4	8	no	—
183	C₂×C₈	—	2	8	no	—
184	C₄⋊C₄	—	2	8	no	—
185	C₂²⋊C₄	—	2	8	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	4	8	no	—
188	C₄²	C₄²×C₁₃₇	1	2	yes	—
189	C₄²	C₂²⋊C₄×C₁₃₇	1	2	yes	—
190	C₄²	C₂²⋊C₄×C₁₃₇	1	2	yes	—
191	C₄²	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
192	C₄²	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
193	C₄²	C₄⋊C₄×C₁₃₇	1	2	yes	—
194	C₄²	C₄⋊C₄×C₁₃₇	1	2	yes	—
195	C₄²	M₁₆×C₁₃₇	1	8	yes	complemented
196	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
197	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
198	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
199	C₂³	—	2	2	no	—
200	C₂³	—	4	2	no	—
201	C₂³	#95×C₁₃₇	1	1	yes	characteristic
202	C₂³	#95×C₁₃₇	1	1	yes	characteristic
203	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	#95×C₁₃₇	1	2	yes	—
218	C₂×C₄	#95×C₁₃₇	1	2	yes	—
219	C₂×C₄	#95×C₁₃₇	1	1	yes	characteristic
220	C₂×C₄	#95×C₁₃₇	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
222	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	2	yes	—
223	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
224	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
225	C₂×C₄	C₂×M₁₆×C₁₃₇	1	4	yes	—
226	C₈	—	4	8	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	4	no	—
233	C₂²	C₂×#95×C₁₃₇	1	1	yes	characteristic
234	C₂²	#407×C₁₃₇	1	2	yes	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	4	8	no	—
241	C₄	C₄×M₁₆×C₁₃₇	1	2	yes	—
242	C₄	#421×C₁₃₇	1	4	yes	—
243	C₄	#423×C₁₃₇	1	2	yes	—
244	C₂	—	2	2	no	—
245	C₂	—	4	1	no	—
246	C₂	—	4	1	no	—
247	C₂	#1728×C₁₃₇	1	1	yes	characteristic
248	C₂	#1733×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₃₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	C₂×#95
prime radical	C₂⁴×C₁₃₇	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₃₇ ◁ C₂²×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1276×C₁₃₇	C₂	semidirect
7. #432×C₁₃₇	C₄	semidirect
10. #423×C₁₃₇	C₄	semidirect
70. C₄²×C₁₃₇	M₁₆	semidirect
124. #7868	C₁₃₇	direct (N × H)
125. C₁₃₇	#7868	direct (N × H)
128. #1276	C₂×C₁₃₇	semidirect
132. #432	C₄×C₁₃₇	semidirect
135. #423	C₄×C₁₃₇	semidirect
195. C₄²	M₁₆×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2920×C₁₃₇	2
C₄×#117×C₁₃₇	2
#1276×C₁₃₇	2
#7868	137

Subgroups — C137×#7868 #46857254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#7868 #46857254