Subgroups — C₁₃₇×#21712 #46871098

Order 35072.

822 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₁₃₇	C₂	1	2	yes	maximal
2	C₄×C₄⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1984×C₁₃₇	C₂	1	2	yes	maximal
4	#1939×C₁₃₇	C₂	1	4	yes	maximal
5	#1842×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#1770×C₁₃₇	C₂	1	4	yes	maximal
7	C₄²×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₁₃₇	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
10	#446×C₁₃₇	—	2	8	no	—
11	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	4	yes	—
12	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	4	yes	—
14	C₂×#97×C₁₃₇	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
19	#391×C₁₃₇	C₄	1	2	yes	—
20	#389×C₁₃₇	C₄	1	4	yes	—
21	#384×C₁₃₇	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
23	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
24	#336×C₁₃₇	C₂²	1	8	yes	—
25	C₂²×Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
27	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
30	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
31	C₄×Q₈×C₁₃₇	—	2	4	no	—
32	C₄×Q₈×C₁₃₇	—	2	4	no	—
33	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃₇	C₂³	1	4	yes	—
41	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
43	C₄⋊C₈×C₁₃₇	—	2	8	no	—
44	C₄⋊C₈×C₁₃₇	—	2	4	no	—
45	C₄⋊C₈×C₁₃₇	—	2	8	no	—
46	#97×C₁₃₇	—	2	8	no	—
47	C₄×C₈×C₁₃₇	—	2	2	no	—
48	C₄×C₈×C₁₃₇	—	2	4	no	—
49	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
50	C₄×C₈×C₁₃₇	Q₈	1	8	yes	complemented
51	#95×C₁₃₇	C₂×C₄	1	8	yes	—
52	C₂×Q₈×C₁₃₇	—	2	4	no	—
53	C₂×Q₈×C₁₃₇	—	2	4	no	—
54	C₂×Q₈×C₁₃₇	—	2	4	no	—
55	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
62	C₂×C₈×C₁₃₇	—	2	2	no	—
63	C₂×C₈×C₁₃₇	—	2	2	no	—
64	C₂×C₈×C₁₃₇	—	2	2	no	—
65	C₂×C₈×C₁₃₇	—	2	4	no	—
66	C₂×C₈×C₁₃₇	—	2	8	no	—
67	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
70	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
71	C₄⋊C₄×C₁₃₇	—	2	4	no	—
72	C₄⋊C₄×C₁₃₇	—	2	8	no	—
73	C₄⋊C₄×C₁₃₇	—	2	8	no	—
74	C₄⋊C₄×C₁₃₇	—	2	8	no	—
75	C₄²×C₁₃₇	—	2	2	no	—
76	C₄²×C₁₃₇	—	2	4	no	—
77	C₄²×C₁₃₇	—	2	4	no	—
78	C₄²×C₁₃₇	—	2	4	no	—
79	C₄²×C₁₃₇	M₁₆	1	4	yes	—
80	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
81	C₄²×C₁₃₇	C₂×Q₈	1	4	yes	—
82	C₄²×C₁₃₇	#41	1	2	yes	—
83	C₄²×C₁₃₇	#41	1	4	yes	—
84	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
85	Q₈×C₁₃₇	—	4	8	no	—
86	C₂×C₄×C₁₃₇	—	2	2	no	—
87	C₂×C₄×C₁₃₇	—	2	2	no	—
88	C₂×C₄×C₁₃₇	—	2	2	no	—
89	C₂×C₄×C₁₃₇	—	2	4	no	—
90	C₂×C₄×C₁₃₇	—	2	4	no	—
91	C₂×C₄×C₁₃₇	—	2	4	no	—
92	C₂×C₄×C₁₃₇	—	2	8	no	—
93	C₂×C₄×C₁₃₇	—	2	4	no	—
94	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₁₃₇	C₄×Q₈	1	4	yes	—
97	C₂×C₄×C₁₃₇	C₄×Q₈	1	4	yes	—
98	C₂×C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₁₃₇	C₂×M₁₆	1	4	yes	—
100	C₂×C₄×C₁₃₇	#131	1	2	yes	—
101	C₂×C₄×C₁₃₇	#131	1	4	yes	—
102	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
104	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
105	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
106	C₈×C₁₃₇	—	2	2	no	—
107	C₈×C₁₃₇	—	2	4	no	—
108	C₈×C₁₃₇	—	4	4	no	—
109	C₈×C₁₃₇	C₄⋊Q₈	1	4	yes	complemented
110	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂²×C₁₃₇	C₂×C₄×Q₈	1	2	yes	—
112	C₂²×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
113	C₂²×C₁₃₇	C₂×#131	1	1	yes	characteristic
114	C₂²×C₁₃₇	C₂×#131	1	2	yes	—
115	C₄×C₁₃₇	—	2	2	no	—
116	C₄×C₁₃₇	—	2	4	no	—
117	C₄×C₁₃₇	—	4	4	no	—
118	C₄×C₁₃₇	—	4	4	no	—
119	C₄×C₁₃₇	#436	1	4	yes	—
120	C₄×C₁₃₇	#446	1	8	yes	complemented
121	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₄×C₁₃₇	#532	1	1	yes	characteristic
123	C₄×C₁₃₇	#533	1	2	yes	—
124	C₂×C₁₃₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
125	C₂×C₁₃₇	C₂×#436	1	1	yes	characteristic
126	C₂×C₁₃₇	#2931	1	1	yes	characteristic
127	C₂×C₁₃₇	C₂×#446	1	2	yes	—
128	C₂×C₁₃₇	#2960	1	2	yes	—
129	#21712	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₁₃₇	#21712	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×#446	C₂×C₁₃₇	1	2	yes	—
132	C₄×C₄⋊Q₈	C₂×C₁₃₇	1	1	yes	characteristic
133	#1984	C₂×C₁₃₇	1	2	yes	—
134	#1939	C₂×C₁₃₇	1	4	yes	—
135	#1842	C₂×C₁₃₇	1	1	yes	characteristic
136	#1770	C₂×C₁₃₇	1	4	yes	—
137	C₄²×C₈	C₂×C₁₃₇	1	1	yes	characteristic
138	C₂×C₄⋊Q₈	C₄×C₁₃₇	1	1	yes	characteristic
139	C₂×C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
140	#446	—	2	8	no	—
141	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
142	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
143	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
144	C₂×#97	C₂²×C₁₃₇	1	4	yes	—
145	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
146	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
149	#391	C₄×C₁₃₇	1	2	yes	—
150	#389	C₄×C₁₃₇	1	4	yes	—
151	#384	C₄×C₁₃₇	1	1	yes	characteristic
152	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
153	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
154	#336	C₂²×C₁₃₇	1	8	yes	—
155	C₂²×Q₈	C₂×C₄×C₁₃₇	1	2	yes	—
156	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
157	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
158	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
159	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
160	C₄⋊Q₈	—	2	4	no	—
161	C₄×Q₈	—	2	4	no	—
162	C₄×Q₈	—	2	4	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
164	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
165	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
166	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
167	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
168	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
169	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
170	C₂×C₄²	C₂³×C₁₃₇	1	4	yes	—
171	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
172	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
173	C₄⋊C₈	—	2	8	no	—
174	C₄⋊C₈	—	2	4	no	—
175	C₄⋊C₈	—	2	8	no	—
176	#97	—	2	8	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	—	2	4	no	—
179	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
180	C₄×C₈	Q₈×C₁₃₇	1	8	yes	complemented
181	#95	C₂×C₄×C₁₃₇	1	8	yes	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
186	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
187	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
188	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
189	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
191	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	8	no	—
197	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
198	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
199	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
200	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	8	no	—
203	C₄⋊C₄	—	2	8	no	—
204	C₄⋊C₄	—	2	8	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	4	no	—
207	C₄²	—	2	4	no	—
208	C₄²	—	2	4	no	—
209	C₄²	M₁₆×C₁₃₇	1	4	yes	—
210	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
211	C₄²	C₂×Q₈×C₁₃₇	1	4	yes	—
212	C₄²	#41×C₁₃₇	1	2	yes	—
213	C₄²	#41×C₁₃₇	1	4	yes	—
214	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
215	Q₈	—	4	8	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
225	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
226	C₂×C₄	C₄×Q₈×C₁₃₇	1	4	yes	—
227	C₂×C₄	C₄×Q₈×C₁₃₇	1	4	yes	—
228	C₂×C₄	C₂×M₁₆×C₁₃₇	1	2	yes	—
229	C₂×C₄	C₂×M₁₆×C₁₃₇	1	4	yes	—
230	C₂×C₄	#131×C₁₃₇	1	2	yes	—
231	C₂×C₄	#131×C₁₃₇	1	4	yes	—
232	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
233	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
234	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
235	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	4	no	—
238	C₈	—	4	4	no	—
239	C₈	C₄⋊Q₈×C₁₃₇	1	4	yes	complemented
240	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
241	C₂²	C₂×C₄×Q₈×C₁₃₇	1	2	yes	—
242	C₂²	C₂²×M₁₆×C₁₃₇	1	1	yes	characteristic
243	C₂²	C₂×#131×C₁₃₇	1	1	yes	characteristic
244	C₂²	C₂×#131×C₁₃₇	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	2	4	no	—
247	C₄	—	4	4	no	—
248	C₄	—	4	4	no	—
249	C₄	#436×C₁₃₇	1	4	yes	—
250	C₄	#446×C₁₃₇	1	8	yes	complemented
251	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
252	C₄	#532×C₁₃₇	1	1	yes	characteristic
253	C₄	#533×C₁₃₇	1	2	yes	—
254	C₂	C₄×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
255	C₂	C₂×#436×C₁₃₇	1	1	yes	characteristic
256	C₂	#2931×C₁₃₇	1	1	yes	characteristic
257	C₂	C₂×#446×C₁₃₇	1	2	yes	—
258	C₂	#2960×C₁₃₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
50. C₄×C₈×C₁₃₇	Q₈	semidirect
109. C₈×C₁₃₇	C₄⋊Q₈	semidirect
120. C₄×C₁₃₇	#446	semidirect
129. #21712	C₁₃₇	direct (N × H)
130. C₁₃₇	#21712	direct (N × H)
180. C₄×C₈	Q₈×C₁₃₇	semidirect
239. C₈	C₄⋊Q₈×C₁₃₇	semidirect
250. C₄	#446×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#446×C₁₃₇	2
C₄×C₄⋊Q₈×C₁₃₇	2
#1984×C₁₃₇	2
#1939×C₁₃₇	2
#1842×C₁₃₇	2
#1770×C₁₃₇	2
C₄²×C₈×C₁₃₇	2
#21712	137

Subgroups — C137×#21712 #46871098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21712 #46871098