Subgroups — C₁₃₇×#11386 #46860772

Order 35072.

1050 conjugacy classes of subgroups.

The subgroup lattice has 292 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×#339×C₁₃₇	C₂	1	4	yes	maximal
3	#1728×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₄×#95×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
6	#533×C₁₃₇	—	2	8	no	—
7	C₂×C₄⋊Q₈×C₁₃₇	C₄	1	2	yes	complemented
8	C₂×#125×C₁₃₇	C₄	1	2	yes	complemented
9	C₂×C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
10	C₂×C₄×Q₈×C₁₃₇	—	2	2	no	—
11	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
12	C₂²×C₄²×C₁₃₇	C₂²	1	2	yes	—
13	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#95×C₁₃₇	—	2	2	no	—
15	#339×C₁₃₇	—	2	8	no	—
16	#336×C₁₃₇	—	2	2	no	—
17	C₂²×Q₈×C₁₃₇	—	2	2	no	—
18	C₂³×C₄×C₁₃₇	—	2	2	no	—
19	C₂³×C₄×C₁₃₇	D₄	1	2	yes	—
20	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₁₃₇	—	2	2	no	—
22	C₂×M₁₆×C₁₃₇	—	2	4	no	—
23	C₂×M₁₆×C₁₃₇	—	2	4	no	—
24	C₂×M₁₆×C₁₃₇	Q₈	1	4	yes	complemented
25	C₂²×C₈×C₁₃₇	—	2	2	no	—
26	C₄⋊Q₈×C₁₃₇	—	2	8	no	—
27	#125×C₁₃₇	—	2	8	no	—
28	C₂²⋊Q₈×C₁₃₇	—	2	16	no	—
29	C₄×Q₈×C₁₃₇	—	2	16	no	—
30	#117×C₁₃₇	—	2	4	no	—
31	#117×C₁₃₇	D₄	1	8	yes	—
32	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
33	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
36	C₂×C₄²×C₁₃₇	—	2	4	no	—
37	C₂×C₄²×C₁₃₇	—	2	2	no	—
38	C₂×C₄²×C₁₃₇	—	2	4	no	—
39	C₂×C₄²×C₁₃₇	—	2	4	no	—
40	C₂×C₄²×C₁₃₇	—	2	2	no	—
41	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
44	C₂×C₄²×C₁₃₇	Q₈	1	4	yes	complemented
45	#95×C₁₃₇	—	2	8	no	—
46	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
47	C₂×Q₈×C₁₃₇	—	2	8	no	—
48	C₂×Q₈×C₁₃₇	—	4	4	no	—
49	C₂×Q₈×C₁₃₇	—	4	4	no	—
50	C₂²×C₄×C₁₃₇	—	2	8	no	—
51	C₂²×C₄×C₁₃₇	—	2	4	no	—
52	C₂²×C₄×C₁₃₇	—	2	4	no	—
53	C₂²×C₄×C₁₃₇	—	2	4	no	—
54	C₂²×C₄×C₁₃₇	—	2	4	no	—
55	C₂²×C₄×C₁₃₇	—	4	2	no	—
56	C₂²×C₄×C₁₃₇	—	4	4	no	—
57	C₂²×C₄×C₁₃₇	—	4	2	no	—
58	C₂²×C₄×C₁₃₇	—	4	2	no	—
59	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
67	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
68	M₁₆×C₁₃₇	—	2	8	no	—
69	M₁₆×C₁₃₇	—	4	4	no	—
70	C₂×C₈×C₁₃₇	—	2	4	no	—
71	C₂×C₈×C₁₃₇	—	4	2	no	—
72	C₂×C₈×C₁₃₇	—	4	2	no	—
73	C₄⋊C₄×C₁₃₇	—	2	16	no	—
74	C₄⋊C₄×C₁₃₇	—	2	8	no	—
75	C₄⋊C₄×C₁₃₇	—	2	4	no	—
76	C₄⋊C₄×C₁₃₇	C₂²⋊C₄	1	8	yes	—
77	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
78	C₄²×C₁₃₇	—	2	2	no	—
79	C₄²×C₁₃₇	—	2	2	no	—
80	C₄²×C₁₃₇	—	2	4	no	—
81	C₄²×C₁₃₇	—	2	8	no	—
82	C₄²×C₁₃₇	—	2	8	no	—
83	C₄²×C₁₃₇	—	4	4	no	—
84	C₂³×C₁₃₇	—	2	2	no	—
85	C₂³×C₁₃₇	—	2	2	no	—
86	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂³×C₁₃₇	#120	1	2	yes	—
88	C₂³×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
89	C₂³×C₁₃₇	#124	1	1	yes	characteristic
90	C₂³×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
91	Q₈×C₁₃₇	—	4	16	no	—
92	C₂×C₄×C₁₃₇	—	2	2	no	—
93	C₂×C₄×C₁₃₇	—	2	2	no	—
94	C₂×C₄×C₁₃₇	—	2	8	no	—
95	C₂×C₄×C₁₃₇	—	2	2	no	—
96	C₂×C₄×C₁₃₇	—	2	8	no	—
97	C₂×C₄×C₁₃₇	—	2	2	no	—
98	C₂×C₄×C₁₃₇	—	2	2	no	—
99	C₂×C₄×C₁₃₇	—	4	8	no	—
100	C₂×C₄×C₁₃₇	—	4	8	no	—
101	C₂×C₄×C₁₃₇	—	4	4	no	—
102	C₂×C₄×C₁₃₇	—	4	8	no	—
103	C₂×C₄×C₁₃₇	—	4	4	no	—
104	C₂×C₄×C₁₃₇	—	4	4	no	—
105	C₂×C₄×C₁₃₇	—	4	4	no	—
106	C₂×C₄×C₁₃₇	#104	1	4	yes	—
107	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂×C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
109	C₂×C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₁₃₇	#120	1	2	yes	—
111	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	4	yes	—
112	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	4	yes	—
113	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₁₃₇	#124	1	4	yes	—
115	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
116	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	4	yes	—
117	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
118	C₈×C₁₃₇	—	4	4	no	—
119	C₂²×C₁₃₇	—	2	2	no	—
120	C₂²×C₁₃₇	—	2	2	no	—
121	C₂²×C₁₃₇	—	2	2	no	—
122	C₂²×C₁₃₇	—	2	2	no	—
123	C₂²×C₁₃₇	—	2	4	no	—
124	C₂²×C₁₃₇	#379	1	1	yes	characteristic
125	C₂²×C₁₃₇	#391	1	1	yes	characteristic
126	C₂²×C₁₃₇	#391	1	1	yes	characteristic
127	C₂²×C₁₃₇	#393	1	2	yes	—
128	C₂²×C₁₃₇	#395	1	2	yes	—
129	C₂²×C₁₃₇	C₂×#104	1	2	yes	—
130	C₂²×C₁₃₇	#421	1	2	yes	—
131	C₄×C₁₃₇	—	2	4	no	—
132	C₄×C₁₃₇	—	4	8	no	—
133	C₄×C₁₃₇	—	4	2	no	—
134	C₄×C₁₃₇	—	4	8	no	—
135	C₄×C₁₃₇	#391	1	4	yes	—
136	C₄×C₁₃₇	#393	1	2	yes	—
137	C₄×C₁₃₇	#395	1	2	yes	—
138	C₂×C₁₃₇	—	2	2	no	—
139	C₂×C₁₃₇	—	2	2	no	—
140	C₂×C₁₃₇	#1446	1	1	yes	characteristic
141	C₂×C₁₃₇	#1789	1	2	yes	—
142	C₂×C₁₃₇	#1986	1	2	yes	—
143	C₂×C₁₃₇	#1994	1	1	yes	characteristic
144	C₂×C₁₃₇	#1995	1	1	yes	characteristic
145	#11386	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₁₃₇	#11386	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×#533	C₂×C₁₃₇	1	1	yes	characteristic
148	C₂×#339	C₂×C₁₃₇	1	4	yes	—
149	#1728	C₂×C₁₃₇	1	1	yes	characteristic
150	C₄×#95	C₂×C₁₃₇	1	1	yes	characteristic
151	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
152	#533	—	2	8	no	—
153	C₂×C₄⋊Q₈	C₄×C₁₃₇	1	2	yes	complemented
154	C₂×#125	C₄×C₁₃₇	1	2	yes	complemented
155	C₂×C₂²⋊Q₈	—	2	2	no	—
156	C₂×C₄×Q₈	—	2	2	no	—
157	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
158	C₂²×C₄²	C₂²×C₁₃₇	1	2	yes	—
159	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
160	C₂×#95	—	2	2	no	—
161	#339	—	2	8	no	—
162	#336	—	2	2	no	—
163	C₂²×Q₈	—	2	2	no	—
164	C₂³×C₄	—	2	2	no	—
165	C₂³×C₄	D₄×C₁₃₇	1	2	yes	—
166	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	4	no	—
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	Q₈×C₁₃₇	1	4	yes	complemented
171	C₂²×C₈	—	2	2	no	—
172	C₄⋊Q₈	—	2	8	no	—
173	#125	—	2	8	no	—
174	C₂²⋊Q₈	—	2	16	no	—
175	C₄×Q₈	—	2	16	no	—
176	#117	—	2	4	no	—
177	#117	D₄×C₁₃₇	1	8	yes	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
181	C₂×C₂²⋊C₄	—	2	2	no	—
182	C₂×C₄²	—	2	2	no	—
183	C₂×C₄²	—	2	4	no	—
184	C₂×C₄²	—	2	4	no	—
185	C₂×C₄²	—	2	4	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
188	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
189	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
190	C₂×C₄²	Q₈×C₁₃₇	1	4	yes	complemented
191	#95	—	2	8	no	—
192	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
193	C₂×Q₈	—	2	8	no	—
194	C₂×Q₈	—	4	4	no	—
195	C₂×Q₈	—	4	4	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	4	4	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
206	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
207	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
209	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
211	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
212	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
213	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
214	M₁₆	—	2	8	no	—
215	M₁₆	—	4	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₄⋊C₄	—	2	16	no	—
220	C₄⋊C₄	—	2	8	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	C₂²⋊C₄×C₁₃₇	1	8	yes	—
223	C₂²⋊C₄	—	2	8	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	4	no	—
227	C₄²	—	2	8	no	—
228	C₄²	—	2	8	no	—
229	C₄²	—	4	4	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
233	C₂³	#120×C₁₃₇	1	2	yes	—
234	C₂³	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
235	C₂³	#124×C₁₃₇	1	1	yes	characteristic
236	C₂³	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
237	Q₈	—	4	16	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	8	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	#104×C₁₃₇	1	4	yes	—
253	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	2	yes	—
254	C₂×C₄	C₄×Q₈×C₁₃₇	1	2	yes	—
255	C₂×C₄	C₄×Q₈×C₁₃₇	1	2	yes	—
256	C₂×C₄	#120×C₁₃₇	1	2	yes	—
257	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	4	yes	—
258	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	4	yes	—
259	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
260	C₂×C₄	#124×C₁₃₇	1	4	yes	—
261	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
262	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	4	yes	—
263	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
264	C₈	—	4	4	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	#379×C₁₃₇	1	1	yes	characteristic
271	C₂²	#391×C₁₃₇	1	1	yes	characteristic
272	C₂²	#391×C₁₃₇	1	1	yes	characteristic
273	C₂²	#393×C₁₃₇	1	2	yes	—
274	C₂²	#395×C₁₃₇	1	2	yes	—
275	C₂²	C₂×#104×C₁₃₇	1	2	yes	—
276	C₂²	#421×C₁₃₇	1	2	yes	—
277	C₄	—	2	4	no	—
278	C₄	—	4	8	no	—
279	C₄	—	4	2	no	—
280	C₄	—	4	8	no	—
281	C₄	#391×C₁₃₇	1	4	yes	—
282	C₄	#393×C₁₃₇	1	2	yes	—
283	C₄	#395×C₁₃₇	1	2	yes	—
284	C₂	—	2	2	no	—
285	C₂	—	2	2	no	—
286	C₂	#1446×C₁₃₇	1	1	yes	characteristic
287	C₂	#1789×C₁₃₇	1	2	yes	—
288	C₂	#1986×C₁₃₇	1	2	yes	—
289	C₂	#1994×C₁₃₇	1	1	yes	characteristic
290	C₂	#1995×C₁₃₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
7. C₂×C₄⋊Q₈×C₁₃₇	C₄	semidirect
8. C₂×#125×C₁₃₇	C₄	semidirect
24. C₂×M₁₆×C₁₃₇	Q₈	semidirect
44. C₂×C₄²×C₁₃₇	Q₈	semidirect
145. #11386	C₁₃₇	direct (N × H)
146. C₁₃₇	#11386	direct (N × H)
153. C₂×C₄⋊Q₈	C₄×C₁₃₇	semidirect
154. C₂×#125	C₄×C₁₃₇	semidirect
170. C₂×M₁₆	Q₈×C₁₃₇	semidirect
190. C₂×C₄²	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×#339×C₁₃₇	2
#1728×C₁₃₇	2
C₄×#95×C₁₃₇	2
#11386	137

Subgroups — C137×#11386 #46860772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#11386 #46860772