Subgroups — C₁₃₇×#19630 #46869016

Order 35072.

858 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3226×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	#3169×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#500×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#3152×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1722×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1713×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1706×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#137×C₁₃₇	—	2	4	no	—
10	#532×C₁₃₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
12	#500×C₁₃₇	—	2	2	no	—
13	#500×C₁₃₇	C₂²	1	2	yes	—
14	#500×C₁₃₇	C₂²	1	2	yes	—
15	#500×C₁₃₇	C₂²	1	1	yes	characteristic
16	#500×C₁₃₇	C₂²	1	4	yes	—
17	#497×C₁₃₇	—	2	4	no	—
18	#494×C₁₃₇	C₂²	1	4	yes	—
19	#493×C₁₃₇	C₂²	1	1	yes	characteristic
20	#492×C₁₃₇	C₂²	1	4	yes	—
21	#486×C₁₃₇	C₂²	1	2	yes	—
22	#486×C₁₃₇	C₂²	1	4	yes	—
23	#480×C₁₃₇	—	2	4	no	—
24	#474×C₁₃₇	—	2	4	no	—
25	#466×C₁₃₇	—	2	2	no	—
26	C₄×D₈×C₁₃₇	—	2	2	no	—
27	#431×C₁₃₇	C₂²	1	1	yes	characteristic
28	C₂×#107×C₁₃₇	—	2	2	no	—
29	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
31	#322×C₁₃₇	C₂²	1	2	yes	—
32	C₈²×C₁₃₇	C₂²	1	2	yes	—
33	C₂×#41×C₁₃₇	—	2	2	no	—
34	C₂²×Q₈×C₁₃₇	—	2	2	no	—
35	#137×C₁₃₇	—	4	16	no	—
36	C₂×Q₁₆×C₁₃₇	—	2	8	no	—
37	C₂×SD₁₆×C₁₃₇	—	2	8	no	—
38	C₂×D₈×C₁₃₇	—	2	2	no	—
39	C₂×M₁₆×C₁₃₇	D₄	1	2	yes	—
40	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
41	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
42	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
43	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
44	C₄⋊Q₈×C₁₃₇	C₂³	1	4	yes	—
45	C₄⋊Q₈×C₁₃₇	C₂³	1	2	yes	—
46	#127×C₁₃₇	C₂³	1	2	yes	—
47	#124×C₁₃₇	—	2	2	no	—
48	#121×C₁₃₇	—	2	2	no	—
49	#121×C₁₃₇	—	2	2	no	—
50	C₄×D₄×C₁₃₇	—	2	2	no	—
51	C₄×D₄×C₁₃₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
54	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
55	#107×C₁₃₇	—	2	4	no	—
56	#107×C₁₃₇	—	2	4	no	—
57	#107×C₁₃₇	—	2	2	no	—
58	#107×C₁₃₇	—	2	8	no	—
59	C₄⋊C₈×C₁₃₇	D₄	1	2	yes	—
60	#102×C₁₃₇	—	2	4	no	—
61	#102×C₁₃₇	—	2	8	no	—
62	C₂²⋊C₈×C₁₃₇	D₄	1	2	yes	—
63	#97×C₁₃₇	D₄	1	2	yes	—
64	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
65	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₃₇	C₂³	1	2	yes	—
68	C₄×C₈×C₁₃₇	C₂³	1	2	yes	—
69	#41×C₁₃₇	—	4	8	no	—
70	C₂×Q₈×C₁₃₇	—	2	2	no	—
71	C₂×Q₈×C₁₃₇	—	2	4	no	—
72	C₂×Q₈×C₁₃₇	—	4	2	no	—
73	C₂×Q₈×C₁₃₇	—	4	8	no	—
74	C₂×D₄×C₁₃₇	—	2	2	no	—
75	C₂×D₄×C₁₃₇	—	2	2	no	—
76	C₂×D₄×C₁₃₇	—	2	2	no	—
77	C₂×D₄×C₁₃₇	—	4	2	no	—
78	C₂²×C₄×C₁₃₇	—	4	2	no	—
79	C₂²×C₄×C₁₃₇	—	4	2	no	—
80	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
83	Q₁₆×C₁₃₇	—	4	16	no	—
84	SD₁₆×C₁₃₇	—	4	16	no	—
85	D₈×C₁₃₇	—	4	4	no	—
86	M₁₆×C₁₃₇	D₈	1	8	yes	complemented
87	C₂×C₈×C₁₃₇	—	2	4	no	—
88	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
90	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
91	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
92	C₄⋊C₄×C₁₃₇	—	2	4	no	—
93	C₄⋊C₄×C₁₃₇	—	2	2	no	—
94	C₄⋊C₄×C₁₃₇	—	2	4	no	—
95	C₄⋊C₄×C₁₃₇	—	2	2	no	—
96	C₄⋊C₄×C₁₃₇	—	4	2	no	—
97	C₄⋊C₄×C₁₃₇	—	4	2	no	—
98	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
99	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
100	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
104	C₂³×C₁₃₇	—	4	2	no	—
105	C₂³×C₁₃₇	#127	1	1	yes	characteristic
106	Q₈×C₁₃₇	—	4	8	no	—
107	Q₈×C₁₃₇	—	4	4	no	—
108	Q₈×C₁₃₇	—	8	4	no	—
109	D₄×C₁₃₇	—	4	4	no	—
110	D₄×C₁₃₇	—	4	4	no	—
111	D₄×C₁₃₇	—	4	4	no	—
112	D₄×C₁₃₇	—	8	4	no	—
113	C₂×C₄×C₁₃₇	—	2	2	no	—
114	C₂×C₄×C₁₃₇	—	2	2	no	—
115	C₂×C₄×C₁₃₇	—	4	2	no	—
116	C₂×C₄×C₁₃₇	—	4	4	no	—
117	C₂×C₄×C₁₃₇	—	8	4	no	—
118	C₂×C₄×C₁₃₇	—	8	4	no	—
119	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
120	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
121	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
122	C₂×C₄×C₁₃₇	C₂×D₈	1	4	yes	—
123	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₁₃₇	—	2	2	no	—
127	C₈×C₁₃₇	—	2	4	no	—
128	C₈×C₁₃₇	C₂×D₈	1	8	yes	—
129	C₈×C₁₃₇	#137	1	4	yes	complemented
130	C₂²×C₁₃₇	—	2	1	no	—
131	C₂²×C₁₃₇	—	2	1	no	—
132	C₂²×C₁₃₇	—	8	2	no	—
133	C₂²×C₁₃₇	—	8	2	no	—
134	C₂²×C₁₃₇	—	8	2	no	—
135	C₂²×C₁₃₇	#493	1	2	yes	—
136	C₂²×C₁₃₇	C₂×#127	1	1	yes	characteristic
137	C₄×C₁₃₇	—	2	1	no	—
138	C₄×C₁₃₇	—	2	1	no	—
139	C₄×C₁₃₇	—	8	2	no	—
140	C₄×C₁₃₇	—	8	4	no	—
141	C₄×C₁₃₇	#493	1	2	yes	—
142	C₄×C₁₃₇	C₂²×D₈	1	2	yes	—
143	C₄×C₁₃₇	C₂×#137	1	2	yes	—
144	C₄×C₁₃₇	#578	1	2	yes	—
145	C₂×C₁₃₇	—	2	2	no	—
146	C₂×C₁₃₇	—	16	2	no	—
147	C₂×C₁₃₇	C₂×#493	1	1	yes	characteristic
148	C₂×C₁₃₇	#3152	1	1	yes	characteristic
149	C₂×C₁₃₇	#3156	1	1	yes	characteristic
150	#19630	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₁₃₇	#19630	1	1	yes	characteristic, direct product factor, Sylow
152	#3226	C₂×C₁₃₇	1	2	yes	complemented
153	#3169	C₂×C₁₃₇	1	1	yes	characteristic
154	C₂×#500	C₂×C₁₃₇	1	1	yes	characteristic, complemented
155	#3152	C₂×C₁₃₇	1	2	yes	complemented
156	#1722	C₂×C₁₃₇	1	4	yes	complemented
157	#1713	C₂×C₁₃₇	1	2	yes	complemented
158	#1706	C₂×C₁₃₇	1	2	yes	complemented
159	C₈×M₁₆	C₂×C₁₃₇	1	1	yes	characteristic, complemented
160	C₂×#137	—	2	4	no	—
161	#532	C₂²×C₁₃₇	1	2	yes	—
162	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
163	#500	—	2	2	no	—
164	#500	C₂²×C₁₃₇	1	2	yes	—
165	#500	C₂²×C₁₃₇	1	2	yes	—
166	#500	C₂²×C₁₃₇	1	1	yes	characteristic
167	#500	C₂²×C₁₃₇	1	4	yes	—
168	#497	—	2	4	no	—
169	#494	C₂²×C₁₃₇	1	4	yes	—
170	#493	C₂²×C₁₃₇	1	1	yes	characteristic
171	#492	C₂²×C₁₃₇	1	4	yes	—
172	#486	C₂²×C₁₃₇	1	2	yes	—
173	#486	C₂²×C₁₃₇	1	4	yes	—
174	#480	—	2	4	no	—
175	#474	—	2	4	no	—
176	#466	—	2	2	no	—
177	C₄×D₈	—	2	2	no	—
178	#431	C₂²×C₁₃₇	1	1	yes	characteristic
179	C₂×#107	—	2	2	no	—
180	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
182	#322	C₂²×C₁₃₇	1	2	yes	—
183	C₈²	C₂²×C₁₃₇	1	2	yes	—
184	C₂×#41	—	2	2	no	—
185	C₂²×Q₈	—	2	2	no	—
186	#137	—	4	16	no	—
187	C₂×Q₁₆	—	2	8	no	—
188	C₂×SD₁₆	—	2	8	no	—
189	C₂×D₈	—	2	2	no	—
190	C₂×M₁₆	D₄×C₁₃₇	1	2	yes	—
191	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	C₂³×C₁₃₇	1	4	yes	—
196	C₄⋊Q₈	C₂³×C₁₃₇	1	2	yes	—
197	#127	C₂³×C₁₃₇	1	2	yes	—
198	#124	—	2	2	no	—
199	#121	—	2	2	no	—
200	#121	—	2	2	no	—
201	C₄×D₄	—	2	2	no	—
202	C₄×D₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
206	#107	—	2	4	no	—
207	#107	—	2	4	no	—
208	#107	—	2	2	no	—
209	#107	—	2	8	no	—
210	C₄⋊C₈	D₄×C₁₃₇	1	2	yes	—
211	#102	—	2	4	no	—
212	#102	—	2	8	no	—
213	C₂²⋊C₈	D₄×C₁₃₇	1	2	yes	—
214	#97	D₄×C₁₃₇	1	2	yes	—
215	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
216	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
217	C₄×C₈	C₂³×C₁₃₇	1	2	yes	—
218	C₄×C₈	C₂³×C₁₃₇	1	2	yes	—
219	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
220	#41	—	4	8	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	4	2	no	—
224	C₂×Q₈	—	4	8	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
234	Q₁₆	—	4	16	no	—
235	SD₁₆	—	4	16	no	—
236	D₈	—	4	4	no	—
237	M₁₆	D₈×C₁₃₇	1	8	yes	complemented
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
240	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
241	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
242	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
252	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
253	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
254	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
255	C₂³	—	4	2	no	—
256	C₂³	#127×C₁₃₇	1	1	yes	characteristic
257	Q₈	—	4	8	no	—
258	Q₈	—	4	4	no	—
259	Q₈	—	8	4	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	4	no	—
262	D₄	—	4	4	no	—
263	D₄	—	8	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
271	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
272	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
273	C₂×C₄	C₂×D₈×C₁₃₇	1	4	yes	—
274	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	4	no	—
279	C₈	C₂×D₈×C₁₃₇	1	8	yes	—
280	C₈	#137×C₁₃₇	1	4	yes	complemented
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	#493×C₁₃₇	1	2	yes	—
287	C₂²	C₂×#127×C₁₃₇	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	4	no	—
292	C₄	#493×C₁₃₇	1	2	yes	—
293	C₄	C₂²×D₈×C₁₃₇	1	2	yes	—
294	C₄	C₂×#137×C₁₃₇	1	2	yes	—
295	C₄	#578×C₁₃₇	1	2	yes	—
296	C₂	—	2	2	no	—
297	C₂	—	16	2	no	—
298	C₂	C₂×#493×C₁₃₇	1	1	yes	characteristic
299	C₂	#3152×C₁₃₇	1	1	yes	characteristic
300	C₂	#3156×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
prime radical	#3169×C₁₃₇	C₂
LCS	G ▷ C₄² ▷ 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
1. #3226×C₁₃₇	C₂	semidirect
3. C₂×#500×C₁₃₇	C₂	semidirect
4. #3152×C₁₃₇	C₂	semidirect
5. #1722×C₁₃₇	C₂	semidirect
6. #1713×C₁₃₇	C₂	semidirect
7. #1706×C₁₃₇	C₂	semidirect
8. C₈×M₁₆×C₁₃₇	C₂	semidirect
86. M₁₆×C₁₃₇	D₈	semidirect
129. C₈×C₁₃₇	#137	semidirect
150. #19630	C₁₃₇	direct (N × H)
151. C₁₃₇	#19630	direct (N × H)
152. #3226	C₂×C₁₃₇	semidirect
154. C₂×#500	C₂×C₁₃₇	semidirect
155. #3152	C₂×C₁₃₇	semidirect
156. #1722	C₂×C₁₃₇	semidirect
157. #1713	C₂×C₁₃₇	semidirect
158. #1706	C₂×C₁₃₇	semidirect
159. C₈×M₁₆	C₂×C₁₃₇	semidirect
237. M₁₆	D₈×C₁₃₇	semidirect
280. C₈	#137×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3226×C₁₃₇	2
#3169×C₁₃₇	2
C₂×#500×C₁₃₇	2
#3152×C₁₃₇	2
#1722×C₁₃₇	2
#1713×C₁₃₇	2
#1706×C₁₃₇	2
C₈×M₁₆×C₁₃₇	2
#19630	137

Subgroups — C137×#19630 #46869016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#19630 #46869016