Subgroups — C₁₃₇×#8641 #46858027

Order 35072.

336 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2960×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1567×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1566×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1561×C₁₃₇	C₂	1	2	yes	maximal
5	#1309×C₁₃₇	C₂	1	2	yes	maximal
6	#533×C₁₃₇	C₂²	1	1	yes	characteristic
7	#446×C₁₃₇	—	2	1	no	—
8	#446×C₁₃₇	—	2	1	no	—
9	C₈×Q₈×C₁₃₇	—	2	1	no	—
10	C₈×Q₈×C₁₃₇	—	2	1	no	—
11	#424×C₁₃₇	C₄	1	2	yes	—
12	#423×C₁₃₇	C₂²	1	2	yes	—
13	#423×C₁₃₇	C₂²	1	2	yes	—
14	#423×C₁₃₇	C₂²	1	1	yes	characteristic
15	#405×C₁₃₇	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	#335×C₁₃₇	—	2	1	no	—
18	#335×C₁₃₇	—	2	1	no	—
19	#334×C₁₃₇	—	2	1	no	—
20	#334×C₁₃₇	—	2	1	no	—
21	#332×C₁₃₇	—	2	2	no	—
22	#332×C₁₃₇	—	2	2	no	—
23	C₂×M₁₆×C₁₃₇	—	2	2	no	—
24	C₂×M₁₆×C₁₃₇	—	2	2	no	—
25	C₂×M₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
29	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
30	#125×C₁₃₇	Q₈	1	2	yes	—
31	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
32	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
33	C₄×Q₈×C₁₃₇	—	2	1	no	—
34	C₄×Q₈×C₁₃₇	—	2	1	no	—
35	#117×C₁₃₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₁₃₇	—	2	2	no	—
38	C₄⋊C₈×C₁₃₇	—	2	2	no	—
39	C₄⋊C₈×C₁₃₇	—	2	1	no	—
40	C₄⋊C₈×C₁₃₇	—	2	1	no	—
41	C₄⋊C₈×C₁₃₇	—	2	2	no	—
42	C₄⋊C₈×C₁₃₇	—	2	2	no	—
43	C₄⋊C₈×C₁₃₇	—	2	2	no	—
44	C₄⋊C₈×C₁₃₇	—	2	2	no	—
45	#97×C₁₃₇	—	2	2	no	—
46	C₄×C₈×C₁₃₇	—	2	2	no	—
47	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
51	C₂×Q₈×C₁₃₇	—	2	1	no	—
52	C₂×Q₈×C₁₃₇	—	2	1	no	—
53	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
56	M₁₆×C₁₃₇	—	4	2	no	—
57	M₁₆×C₁₃₇	—	8	2	no	—
58	M₁₆×C₁₃₇	—	8	2	no	—
59	C₂×C₈×C₁₃₇	—	2	2	no	—
60	C₂×C₈×C₁₃₇	—	2	2	no	—
61	C₂×C₈×C₁₃₇	—	4	1	no	—
62	C₂×C₈×C₁₃₇	—	4	2	no	—
63	C₂×C₈×C₁₃₇	—	4	2	no	—
64	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
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	C₄⋊C₄×C₁₃₇	—	2	1	no	—
69	C₄⋊C₄×C₁₃₇	—	2	1	no	—
70	C₄⋊C₄×C₁₃₇	—	2	2	no	—
71	C₄⋊C₄×C₁₃₇	—	2	2	no	—
72	C₄⋊C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
73	C₄⋊C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
74	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
75	C₄²×C₁₃₇	—	2	2	no	—
76	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
80	C₂³×C₁₃₇	#117	1	1	yes	characteristic
81	Q₈×C₁₃₇	—	4	2	no	—
82	Q₈×C₁₃₇	—	4	1	no	—
83	Q₈×C₁₃₇	—	4	1	no	—
84	C₂×C₄×C₁₃₇	—	2	1	no	—
85	C₂×C₄×C₁₃₇	—	2	1	no	—
86	C₂×C₄×C₁₃₇	—	2	2	no	—
87	C₂×C₄×C₁₃₇	—	2	1	no	—
88	C₂×C₄×C₁₃₇	—	2	1	no	—
89	C₂×C₄×C₁₃₇	—	2	2	no	—
90	C₂×C₄×C₁₃₇	—	2	2	no	—
91	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
94	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
95	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
97	C₈×C₁₃₇	—	2	1	no	—
98	C₈×C₁₃₇	—	2	1	no	—
99	C₈×C₁₃₇	—	2	1	no	—
100	C₈×C₁₃₇	—	4	2	no	—
101	C₈×C₁₃₇	—	8	2	no	—
102	C₈×C₁₃₇	—	8	2	no	—
103	C₈×C₁₃₇	#101	1	2	yes	—
104	C₂²×C₁₃₇	—	2	1	no	—
105	C₂²×C₁₃₇	—	2	1	no	—
106	C₂²×C₁₃₇	—	2	1	no	—
107	C₂²×C₁₃₇	#380	1	1	yes	characteristic
108	C₄×C₁₃₇	—	2	1	no	—
109	C₄×C₁₃₇	—	2	1	no	—
110	C₄×C₁₃₇	—	2	1	no	—
111	C₄×C₁₃₇	—	4	2	no	—
112	C₄×C₁₃₇	—	4	2	no	—
113	C₄×C₁₃₇	C₂×#101	1	1	yes	characteristic
114	C₄×C₁₃₇	#413	1	1	yes	characteristic
115	C₄×C₁₃₇	#471	1	2	yes	—
116	C₄×C₁₃₇	#472	1	1	yes	characteristic
117	C₄×C₁₃₇	#473	1	1	yes	characteristic
118	C₂×C₁₃₇	—	4	1	no	—
119	C₂×C₁₃₇	#1801	1	1	yes	characteristic
120	C₂×C₁₃₇	#1815	1	1	yes	characteristic
121	C₂×C₁₃₇	#1816	1	1	yes	characteristic
122	#8641	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₁₃₇	#8641	1	1	yes	characteristic, direct product factor, Sylow
124	#2960	C₂×C₁₃₇	1	1	yes	characteristic
125	#1567	C₂×C₁₃₇	1	1	yes	characteristic
126	#1566	C₂×C₁₃₇	1	1	yes	characteristic
127	#1561	C₂×C₁₃₇	1	2	yes	—
128	#1309	C₂×C₁₃₇	1	2	yes	—
129	#533	C₂²×C₁₃₇	1	1	yes	characteristic
130	#446	—	2	1	no	—
131	#446	—	2	1	no	—
132	C₈×Q₈	—	2	1	no	—
133	C₈×Q₈	—	2	1	no	—
134	#424	C₄×C₁₃₇	1	2	yes	—
135	#423	C₂²×C₁₃₇	1	2	yes	—
136	#423	C₂²×C₁₃₇	1	2	yes	—
137	#423	C₂²×C₁₃₇	1	1	yes	characteristic
138	#405	C₄×C₁₃₇	1	2	yes	—
139	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
140	#335	—	2	1	no	—
141	#335	—	2	1	no	—
142	#334	—	2	1	no	—
143	#334	—	2	1	no	—
144	#332	—	2	2	no	—
145	#332	—	2	2	no	—
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
149	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
150	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
151	C₄⋊Q₈	D₄×C₁₃₇	1	1	yes	characteristic
152	C₄⋊Q₈	D₄×C₁₃₇	1	1	yes	characteristic
153	#125	Q₈×C₁₃₇	1	2	yes	—
154	C₂²⋊Q₈	—	2	1	no	—
155	C₂²⋊Q₈	—	2	1	no	—
156	C₄×Q₈	—	2	1	no	—
157	C₄×Q₈	—	2	1	no	—
158	#117	C₂×C₄×C₁₃₇	1	2	yes	—
159	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	2	1	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	2	2	no	—
168	#97	—	2	2	no	—
169	C₄×C₈	—	2	2	no	—
170	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
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	C₂×Q₈	—	2	1	no	—
175	C₂×Q₈	—	2	1	no	—
176	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
177	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
178	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
179	M₁₆	—	4	2	no	—
180	M₁₆	—	8	2	no	—
181	M₁₆	—	8	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	2	no	—
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	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
196	C₄⋊C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
197	C₂²⋊C₄	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
200	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
201	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
202	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
203	C₂³	#117×C₁₃₇	1	1	yes	characteristic
204	Q₈	—	4	2	no	—
205	Q₈	—	4	1	no	—
206	Q₈	—	4	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
216	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
217	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
218	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
219	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	4	2	no	—
224	C₈	—	8	2	no	—
225	C₈	—	8	2	no	—
226	C₈	#101×C₁₃₇	1	2	yes	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	#380×C₁₃₇	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	2	no	—
236	C₄	C₂×#101×C₁₃₇	1	1	yes	characteristic
237	C₄	#413×C₁₃₇	1	1	yes	characteristic
238	C₄	#471×C₁₃₇	1	2	yes	—
239	C₄	#472×C₁₃₇	1	1	yes	characteristic
240	C₄	#473×C₁₃₇	1	1	yes	characteristic
241	C₂	—	4	1	no	—
242	C₂	#1801×C₁₃₇	1	1	yes	characteristic
243	C₂	#1815×C₁₃₇	1	1	yes	characteristic
244	C₂	#1816×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#380
commutator	C₂²×C₄	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
prime radical	C₂³×C₁₃₇	#117
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
122. #8641	C₁₃₇	direct (N × H)
123. C₁₃₇	#8641	direct (N × H)

Maximal subgroups

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

subgroup	index
#2960×C₁₃₇	2
#1567×C₁₃₇	2
#1566×C₁₃₇	2
#1561×C₁₃₇	2
#1309×C₁₃₇	2
#8641	137

Subgroups — C137×#8641 #46858027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8641 #46858027