Subgroups — C₁₃₇×#21314 #46870700

Order 35072.

1174 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#423×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1989×C₁₃₇	C₂	1	8	yes	maximal, complemented
5	#1736×C₁₃₇	C₂	1	4	yes	maximal
6	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
7	C₂×#125×C₁₃₇	—	2	1	no	—
8	C₂×#125×C₁₃₇	—	2	1	no	—
9	C₂×#125×C₁₃₇	—	2	4	no	—
10	C₂×#125×C₁₃₇	C₄	1	8	yes	complemented
11	C₂×#125×C₁₃₇	C₂²	1	4	yes	—
12	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₁₃₇	—	2	2	no	—
14	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
16	#423×C₁₃₇	—	2	4	no	—
17	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	4	yes	—
18	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
19	C₄×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
20	#376×C₁₃₇	—	2	1	no	—
21	C₄³×C₁₃₇	—	2	1	no	—
22	#340×C₁₃₇	C₂²	1	16	yes	—
23	C₂³×C₄×C₁₃₇	—	2	2	no	—
24	C₂³×C₄×C₁₃₇	D₄	1	2	yes	—
25	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₁₃₇	—	2	4	no	—
27	C₂×M₁₆×C₁₃₇	—	2	8	no	—
28	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
29	#125×C₁₃₇	—	2	8	no	—
30	#125×C₁₃₇	—	2	8	no	—
31	#125×C₁₃₇	—	2	8	no	—
32	#125×C₁₃₇	C₂×C₄	1	16	yes	—
33	#117×C₁₃₇	—	2	4	no	—
34	#117×C₁₃₇	Q₈	1	8	yes	—
35	C₂×C₄⋊C₄×C₁₃₇	—	2	16	no	—
36	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
37	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
43	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
44	C₂×C₄²×C₁₃₇	—	2	2	no	—
45	C₂×C₄²×C₁₃₇	—	2	4	no	—
46	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₁₃₇	D₄	1	4	yes	—
50	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
51	C₄⋊C₈×C₁₃₇	—	2	8	no	—
52	#95×C₁₃₇	—	2	4	no	—
53	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₇	—	2	8	no	—
55	C₂²×C₄×C₁₃₇	—	2	4	no	—
56	C₂²×C₄×C₁₃₇	—	2	4	no	—
57	C₂²×C₄×C₁₃₇	—	4	4	no	—
58	C₂²×C₄×C₁₃₇	—	4	8	no	—
59	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	8	yes	—
60	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
67	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
68	M₁₆×C₁₃₇	—	4	8	no	—
69	C₂×C₈×C₁₃₇	—	2	4	no	—
70	C₂×C₈×C₁₃₇	—	2	8	no	—
71	C₄⋊C₄×C₁₃₇	—	2	16	no	—
72	C₄⋊C₄×C₁₃₇	—	2	16	no	—
73	C₄⋊C₄×C₁₃₇	—	2	8	no	—
74	C₄⋊C₄×C₁₃₇	—	2	4	no	—
75	C₄⋊C₄×C₁₃₇	—	2	4	no	—
76	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	8	yes	—
77	C₄⋊C₄×C₁₃₇	C₂×Q₈	1	8	yes	—
78	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
79	C₄²×C₁₃₇	—	2	8	no	—
80	C₄²×C₁₃₇	—	4	8	no	—
81	C₄²×C₁₃₇	C₂²⋊C₄	1	4	yes	—
82	C₄²×C₁₃₇	C₂²⋊C₄	1	4	yes	—
83	C₄²×C₁₃₇	C₂²×C₄	1	4	yes	—
84	C₄²×C₁₃₇	C₂×D₄	1	4	yes	—
85	C₂³×C₁₃₇	—	2	4	no	—
86	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	2	yes	—
87	C₂³×C₁₃₇	C₂²⋊Q₈	1	4	yes	—
88	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃₇	—	2	4	no	—
90	C₂×C₄×C₁₃₇	—	2	4	no	—
91	C₂×C₄×C₁₃₇	—	2	8	no	—
92	C₂×C₄×C₁₃₇	—	2	8	no	—
93	C₂×C₄×C₁₃₇	—	4	8	no	—
94	C₂×C₄×C₁₃₇	—	4	16	no	—
95	C₂×C₄×C₁₃₇	—	4	8	no	—
96	C₂×C₄×C₁₃₇	—	4	8	no	—
97	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
98	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
99	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂×C₄×C₁₃₇	C₄×Q₈	1	8	yes	—
101	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	4	yes	—
102	C₂×C₄×C₁₃₇	#124	1	4	yes	—
103	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	4	yes	—
104	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
107	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
108	C₈×C₁₃₇	—	4	4	no	—
109	C₂²×C₁₃₇	—	2	8	no	—
110	C₂²×C₁₃₇	—	2	4	no	—
111	C₂²×C₁₃₇	#391	1	4	yes	—
112	C₂²×C₁₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×C₁₃₇	C₂×C₂²⋊Q₈	1	2	yes	—
114	C₂²×C₁₃₇	#575	1	4	yes	—
115	C₄×C₁₃₇	—	2	4	no	—
116	C₄×C₁₃₇	—	2	4	no	—
117	C₄×C₁₃₇	—	4	8	no	—
118	C₄×C₁₃₇	—	4	8	no	—
119	C₄×C₁₃₇	#391	1	4	yes	—
120	C₄×C₁₃₇	C₂×C₄×Q₈	1	2	yes	—
121	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
122	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₂×C₁₃₇	—	2	4	no	—
124	C₂×C₁₃₇	C₂×#391	1	1	yes	characteristic
125	C₂×C₁₃₇	#2896	1	2	yes	—
126	C₂×C₁₃₇	#3080	1	4	yes	—
127	#21314	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₁₃₇	#21314	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²×#125	C₂×C₁₃₇	1	1	yes	characteristic
130	C₂×#423	C₂×C₁₃₇	1	1	yes	characteristic
131	C₄×#117	C₂×C₁₃₇	1	1	yes	characteristic
132	#1989	C₂×C₁₃₇	1	8	yes	complemented
133	#1736	C₂×C₁₃₇	1	4	yes	—
134	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
135	C₂×#125	—	2	1	no	—
136	C₂×#125	—	2	1	no	—
137	C₂×#125	—	2	4	no	—
138	C₂×#125	C₄×C₁₃₇	1	8	yes	complemented
139	C₂×#125	C₂²×C₁₃₇	1	4	yes	—
140	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
141	C₂²×C₄⋊C₄	—	2	2	no	—
142	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
143	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
144	#423	—	2	4	no	—
145	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
146	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
147	C₄×C₂²⋊C₄	—	2	2	no	—
148	#376	—	2	1	no	—
149	C₄³	—	2	1	no	—
150	#340	C₂²×C₁₃₇	1	16	yes	—
151	C₂³×C₄	—	2	2	no	—
152	C₂³×C₄	D₄×C₁₃₇	1	2	yes	—
153	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
154	C₂×M₁₆	—	2	4	no	—
155	C₂×M₁₆	—	2	8	no	—
156	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
157	#125	—	2	8	no	—
158	#125	—	2	8	no	—
159	#125	—	2	8	no	—
160	#125	C₂×C₄×C₁₃₇	1	16	yes	—
161	#117	—	2	4	no	—
162	#117	Q₈×C₁₃₇	1	8	yes	—
163	C₂×C₄⋊C₄	—	2	16	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	4	no	—
168	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	8	yes	—
169	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
170	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
171	C₂×C₂²⋊C₄	—	2	2	no	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	—	2	4	no	—
174	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
175	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
176	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
177	C₂×C₄²	D₄×C₁₃₇	1	4	yes	—
178	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
179	C₄⋊C₈	—	2	8	no	—
180	#95	—	2	4	no	—
181	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
182	C₂²×C₄	—	2	8	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	4	4	no	—
186	C₂²×C₄	—	4	8	no	—
187	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	8	yes	—
188	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
189	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
191	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
193	C₂²×C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
194	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
195	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
196	M₁₆	—	4	8	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	8	no	—
199	C₄⋊C₄	—	2	16	no	—
200	C₄⋊C₄	—	2	16	no	—
201	C₄⋊C₄	—	2	8	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	8	yes	—
205	C₄⋊C₄	C₂×Q₈×C₁₃₇	1	8	yes	—
206	C₂²⋊C₄	—	2	8	no	—
207	C₄²	—	2	8	no	—
208	C₄²	—	4	8	no	—
209	C₄²	C₂²⋊C₄×C₁₃₇	1	4	yes	—
210	C₄²	C₂²⋊C₄×C₁₃₇	1	4	yes	—
211	C₄²	C₂²×C₄×C₁₃₇	1	4	yes	—
212	C₄²	C₂×D₄×C₁₃₇	1	4	yes	—
213	C₂³	—	2	4	no	—
214	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	2	yes	—
215	C₂³	C₂²⋊Q₈×C₁₃₇	1	4	yes	—
216	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	4	8	no	—
222	C₂×C₄	—	4	16	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
226	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
227	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	2	yes	—
228	C₂×C₄	C₄×Q₈×C₁₃₇	1	8	yes	—
229	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	4	yes	—
230	C₂×C₄	#124×C₁₃₇	1	4	yes	—
231	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	4	yes	—
232	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
233	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
234	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
235	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
236	C₈	—	4	4	no	—
237	C₂²	—	2	8	no	—
238	C₂²	—	2	4	no	—
239	C₂²	#391×C₁₃₇	1	4	yes	—
240	C₂²	C₂²×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
241	C₂²	C₂×C₂²⋊Q₈×C₁₃₇	1	2	yes	—
242	C₂²	#575×C₁₃₇	1	4	yes	—
243	C₄	—	2	4	no	—
244	C₄	—	2	4	no	—
245	C₄	—	4	8	no	—
246	C₄	—	4	8	no	—
247	C₄	#391×C₁₃₇	1	4	yes	—
248	C₄	C₂×C₄×Q₈×C₁₃₇	1	2	yes	—
249	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
250	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
251	C₂	—	2	4	no	—
252	C₂	C₂×#391×C₁₃₇	1	1	yes	characteristic
253	C₂	#2896×C₁₃₇	1	2	yes	—
254	C₂	#3080×C₁₃₇	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. #1989×C₁₃₇	C₂	semidirect
10. C₂×#125×C₁₃₇	C₄	semidirect
127. #21314	C₁₃₇	direct (N × H)
128. C₁₃₇	#21314	direct (N × H)
132. #1989	C₂×C₁₃₇	semidirect
138. C₂×#125	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#125×C₁₃₇	2
C₂×#423×C₁₃₇	2
C₄×#117×C₁₃₇	2
#1989×C₁₃₇	2
#1736×C₁₃₇	2
#21314	137

Subgroups — C137×#21314 #46870700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21314 #46870700