Subgroups — C₁₃₇×#21300 #46870686

Order 35072.

1198 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#423×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1984×C₁₃₇	C₂	1	8	yes	maximal, complemented
5	#1729×C₁₃₇	C₂	1	4	yes	maximal
6	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
7	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
8	#533×C₁₃₇	—	2	4	no	—
9	#533×C₁₃₇	—	2	4	no	—
10	C₂×C₄⋊Q₈×C₁₃₇	C₄	1	2	yes	—
11	C₂×#125×C₁₃₇	C₄	1	2	yes	—
12	C₂×C₂²⋊Q₈×C₁₃₇	C₄	1	4	yes	—
13	C₂×C₄×Q₈×C₁₃₇	C₂²	1	4	yes	—
14	C₂×#117×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	C₂×C₄×C₈×C₁₃₇	—	2	4	no	—
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	4	yes	—
20	#336×C₁₃₇	C₂²	1	16	yes	—
21	C₂²×Q₈×C₁₃₇	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₁₃₇	—	2	4	no	—
25	C₂×M₁₆×C₁₃₇	—	2	8	no	—
26	C₂²×C₈×C₁₃₇	—	2	4	no	—
27	C₂²×C₈×C₁₃₇	—	2	4	no	—
28	C₂²×C₈×C₁₃₇	Q₈	1	8	yes	complemented
29	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
30	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
31	C₄⋊Q₈×C₁₃₇	—	2	8	no	—
32	#125×C₁₃₇	—	2	8	no	—
33	C₂²⋊Q₈×C₁₃₇	—	2	16	no	—
34	C₄×Q₈×C₁₃₇	—	2	8	no	—
35	C₄×Q₈×C₁₃₇	—	2	8	no	—
36	#117×C₁₃₇	—	2	4	no	—
37	#117×C₁₃₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	—
40	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃₇	D₄	1	4	yes	—
45	C₂×C₄²×C₁₃₇	D₄	1	4	yes	—
46	C₂×C₄²×C₁₃₇	C₂³	1	4	yes	—
47	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
48	C₄⋊C₈×C₁₃₇	—	2	8	no	—
49	C₄×C₈×C₁₃₇	—	2	8	no	—
50	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×Q₈×C₁₃₇	—	2	8	no	—
52	C₂×Q₈×C₁₃₇	—	2	16	no	—
53	C₂²×C₄×C₁₃₇	—	2	8	no	—
54	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	8	yes	—
55	C₂²×C₄×C₁₃₇	C₂²×C₄	1	8	yes	—
56	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
62	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
63	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
64	M₁₆×C₁₃₇	—	4	8	no	—
65	C₂×C₈×C₁₃₇	—	2	16	no	—
66	C₂×C₈×C₁₃₇	—	2	4	no	—
67	C₂×C₈×C₁₃₇	—	2	4	no	—
68	C₂×C₈×C₁₃₇	—	2	4	no	—
69	C₂×C₈×C₁₃₇	—	2	8	no	—
70	C₂×C₈×C₁₃₇	C₂×Q₈	1	8	yes	—
71	C₄⋊C₄×C₁₃₇	—	2	8	no	—
72	C₄⋊C₄×C₁₃₇	—	2	16	no	—
73	C₄⋊C₄×C₁₃₇	—	2	8	no	—
74	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
75	C₄²×C₁₃₇	—	2	4	no	—
76	C₄²×C₁₃₇	—	2	4	no	—
77	C₄²×C₁₃₇	C₂²⋊C₄	1	4	yes	—
78	C₄²×C₁₃₇	C₂²⋊C₄	1	4	yes	—
79	C₄²×C₁₃₇	C₂×D₄	1	4	yes	—
80	C₄²×C₁₃₇	C₂×D₄	1	4	yes	—
81	C₂³×C₁₃₇	—	2	4	no	—
82	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂³×C₁₃₇	C₄×Q₈	1	4	yes	—
84	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
85	Q₈×C₁₃₇	—	4	16	no	—
86	C₂×C₄×C₁₃₇	—	2	4	no	—
87	C₂×C₄×C₁₃₇	—	2	16	no	—
88	C₂×C₄×C₁₃₇	—	2	8	no	—
89	C₂×C₄×C₁₃₇	—	2	16	no	—
90	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
91	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	2	yes	—
92	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
93	C₂×C₄×C₁₃₇	C₄×Q₈	1	4	yes	—
94	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	8	yes	—
95	C₂×C₄×C₁₃₇	#124	1	4	yes	—
96	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	4	yes	—
97	C₂×C₄×C₁₃₇	#131	1	8	yes	—
98	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
101	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
102	C₈×C₁₃₇	—	2	8	no	—
103	C₈×C₁₃₇	—	4	4	no	—
104	C₂²×C₁₃₇	—	2	4	no	—
105	C₂²×C₁₃₇	—	2	8	no	—
106	C₂²×C₁₃₇	#391	1	4	yes	—
107	C₂²×C₁₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
108	C₂²×C₁₃₇	C₂×C₄×Q₈	1	2	yes	—
109	C₂²×C₁₃₇	C₂×#131	1	4	yes	—
110	C₄×C₁₃₇	—	2	4	no	—
111	C₄×C₁₃₇	—	4	8	no	—
112	C₄×C₁₃₇	#391	1	4	yes	—
113	C₄×C₁₃₇	#408	1	8	yes	—
114	C₄×C₁₃₇	C₂×C₂²⋊Q₈	1	2	yes	—
115	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
116	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₁₃₇	—	2	4	no	—
118	C₂×C₁₃₇	C₂×#391	1	1	yes	characteristic
119	C₂×C₁₃₇	C₂×#408	1	2	yes	—
120	C₂×C₁₃₇	#2960	1	4	yes	—
121	#21300	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₁₃₇	#21300	1	1	yes	characteristic, direct product factor, Sylow
123	C₂×#533	C₂×C₁₃₇	1	1	yes	characteristic
124	C₂×#423	C₂×C₁₃₇	1	1	yes	characteristic
125	C₂²×C₄×C₈	C₂×C₁₃₇	1	1	yes	characteristic
126	#1984	C₂×C₁₃₇	1	8	yes	complemented
127	#1729	C₂×C₁₃₇	1	4	yes	—
128	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
129	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
130	#533	—	2	4	no	—
131	#533	—	2	4	no	—
132	C₂×C₄⋊Q₈	C₄×C₁₃₇	1	2	yes	—
133	C₂×#125	C₄×C₁₃₇	1	2	yes	—
134	C₂×C₂²⋊Q₈	C₄×C₁₃₇	1	4	yes	—
135	C₂×C₄×Q₈	C₂²×C₁₃₇	1	4	yes	—
136	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
137	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
138	#423	—	2	4	no	—
139	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
140	C₂×C₄×C₈	—	2	4	no	—
141	C₂×C₄×C₈	C₂²×C₁₃₇	1	4	yes	—
142	#336	C₂²×C₁₃₇	1	16	yes	—
143	C₂²×Q₈	C₂×C₄×C₁₃₇	1	4	yes	—
144	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
145	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
146	C₂×M₁₆	—	2	4	no	—
147	C₂×M₁₆	—	2	8	no	—
148	C₂²×C₈	—	2	4	no	—
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	Q₈×C₁₃₇	1	8	yes	complemented
151	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
152	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
153	C₄⋊Q₈	—	2	8	no	—
154	#125	—	2	8	no	—
155	C₂²⋊Q₈	—	2	16	no	—
156	C₄×Q₈	—	2	8	no	—
157	C₄×Q₈	—	2	8	no	—
158	#117	—	2	4	no	—
159	#117	—	2	4	no	—
160	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
161	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	8	yes	—
162	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
163	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
164	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
165	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
166	C₂×C₄²	D₄×C₁₃₇	1	4	yes	—
167	C₂×C₄²	D₄×C₁₃₇	1	4	yes	—
168	C₂×C₄²	C₂³×C₁₃₇	1	4	yes	—
169	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
170	C₄⋊C₈	—	2	8	no	—
171	C₄×C₈	—	2	8	no	—
172	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
173	C₂×Q₈	—	2	8	no	—
174	C₂×Q₈	—	2	16	no	—
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	8	yes	—
177	C₂²×C₄	C₂²×C₄×C₁₃₇	1	8	yes	—
178	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
179	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
180	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
181	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
183	C₂²×C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
184	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
185	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
186	M₁₆	—	4	8	no	—
187	C₂×C₈	—	2	16	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	C₂×Q₈×C₁₃₇	1	8	yes	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	—	2	16	no	—
195	C₄⋊C₄	—	2	8	no	—
196	C₂²⋊C₄	—	2	8	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	4	no	—
199	C₄²	C₂²⋊C₄×C₁₃₇	1	4	yes	—
200	C₄²	C₂²⋊C₄×C₁₃₇	1	4	yes	—
201	C₄²	C₂×D₄×C₁₃₇	1	4	yes	—
202	C₄²	C₂×D₄×C₁₃₇	1	4	yes	—
203	C₂³	—	2	4	no	—
204	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	2	yes	—
205	C₂³	C₄×Q₈×C₁₃₇	1	4	yes	—
206	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
207	Q₈	—	4	16	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	16	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	2	16	no	—
212	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
213	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	2	yes	—
214	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
215	C₂×C₄	C₄×Q₈×C₁₃₇	1	4	yes	—
216	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	8	yes	—
217	C₂×C₄	#124×C₁₃₇	1	4	yes	—
218	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	4	yes	—
219	C₂×C₄	#131×C₁₃₇	1	8	yes	—
220	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
222	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
223	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
224	C₈	—	2	8	no	—
225	C₈	—	4	4	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	2	8	no	—
228	C₂²	#391×C₁₃₇	1	4	yes	—
229	C₂²	C₂²×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
230	C₂²	C₂×C₄×Q₈×C₁₃₇	1	2	yes	—
231	C₂²	C₂×#131×C₁₃₇	1	4	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	4	8	no	—
234	C₄	#391×C₁₃₇	1	4	yes	—
235	C₄	#408×C₁₃₇	1	8	yes	—
236	C₄	C₂×C₂²⋊Q₈×C₁₃₇	1	2	yes	—
237	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
238	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
239	C₂	—	2	4	no	—
240	C₂	C₂×#391×C₁₃₇	1	1	yes	characteristic
241	C₂	C₂×#408×C₁₃₇	1	2	yes	—
242	C₂	#2960×C₁₃₇	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂³×C₁₃₇	C₂³×C₄
prime radical	C₂⁴×C₁₃₇	C₂²×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
4. #1984×C₁₃₇	C₂	semidirect
28. C₂²×C₈×C₁₃₇	Q₈	semidirect
121. #21300	C₁₃₇	direct (N × H)
122. C₁₃₇	#21300	direct (N × H)
126. #1984	C₂×C₁₃₇	semidirect
150. C₂²×C₈	Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#533×C₁₃₇	2
C₂×#423×C₁₃₇	2
C₂²×C₄×C₈×C₁₃₇	2
#1984×C₁₃₇	2
#1729×C₁₃₇	2
#21300	137

Subgroups — C137×#21300 #46870686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21300 #46870686