Subgroups — C₁₃₇×#33630 #46883016

Order 35072.

542 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#135×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2229×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	#2228×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#2159×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2150×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#2139×C₁₃₇	C₂	1	2	yes	maximal
8	C₂×#135×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×#125×C₁₃₇	—	2	1	no	—
10	C₄×#41×C₁₃₇	—	2	1	no	—
11	C₂×M₃₂×C₁₃₇	C₂²	1	2	yes	—
12	#499×C₁₃₇	—	2	2	no	—
13	#499×C₁₃₇	C₂²	1	4	yes	—
14	C₄×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
15	C₄×SD₁₆×C₁₃₇	—	2	2	no	—
16	C₄×D₈×C₁₃₇	C₂²	1	2	yes	—
17	#427×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
19	C₂×#106×C₁₃₇	—	2	1	no	—
20	#416×C₁₃₇	—	2	1	no	—
21	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
22	#367×C₁₃₇	C₂²	1	4	yes	—
23	#366×C₁₃₇	C₂²	1	4	yes	—
24	C₄⋊C₁₆×C₁₃₇	C₂²	1	4	yes	—
25	#358×C₁₃₇	C₂²	1	4	yes	—
26	#357×C₁₃₇	C₂²	1	4	yes	—
27	C₂×#41×C₁₃₇	—	2	1	no	—
28	#135×C₁₃₇	—	2	2	no	—
29	#135×C₁₃₇	Q₈	1	4	yes	—
30	C₂×Q₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
31	C₂×SD₁₆×C₁₃₇	—	2	1	no	—
32	C₂×D₈×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
35	#125×C₁₃₇	—	2	4	no	—
36	#125×C₁₃₇	—	4	1	no	—
37	#125×C₁₃₇	—	4	1	no	—
38	C₄×Q₈×C₁₃₇	—	2	2	no	—
39	C₄×D₄×C₁₃₇	—	2	2	no	—
40	C₄×D₄×C₁₃₇	—	4	2	no	—
41	#117×C₁₃₇	—	2	1	no	—
42	#117×C₁₃₇	—	4	1	no	—
43	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₇	—	4	2	no	—
45	C₂×C₄²×C₁₃₇	—	4	1	no	—
46	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
47	M₃₂×C₁₃₇	—	2	4	no	—
48	C₂×C₁₆×C₁₃₇	C₂³	1	4	yes	—
49	#107×C₁₃₇	—	2	2	no	—
50	#107×C₁₃₇	C₂³	1	2	yes	—
51	#107×C₁₃₇	C₂³	1	4	yes	—
52	#106×C₁₃₇	—	2	4	no	—
53	#106×C₁₃₇	—	2	1	no	—
54	#103×C₁₃₇	—	2	2	no	—
55	#102×C₁₃₇	—	2	2	no	—
56	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
57	C₄×C₈×C₁₃₇	C₂³	1	2	yes	—
58	#41×C₁₃₇	—	2	4	no	—
59	#41×C₁₃₇	—	4	2	no	—
60	C₂×Q₈×C₁₃₇	—	2	1	no	—
61	C₂×D₄×C₁₃₇	—	2	1	no	—
62	C₂×D₄×C₁₃₇	—	4	1	no	—
63	C₂²×C₄×C₁₃₇	—	4	1	no	—
64	C₂²×C₄×C₁₃₇	—	4	2	no	—
65	C₂²×C₄×C₁₃₇	—	4	2	no	—
66	C₂²×C₄×C₁₃₇	D₈	1	2	yes	—
67	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
68	Q₁₆×C₁₃₇	—	2	1	no	—
69	Q₁₆×C₁₃₇	C₂×Q₈	1	2	yes	—
70	SD₁₆×C₁₃₇	—	2	4	no	—
71	D₈×C₁₃₇	—	2	1	no	—
72	D₈×C₁₃₇	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₁₃₇	—	2	2	no	—
74	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
77	C₂×C₈×C₁₃₇	#41	1	2	yes	—
78	C₂×C₈×C₁₃₇	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₁₃₇	—	2	2	no	—
80	C₄⋊C₄×C₁₃₇	—	2	4	no	—
81	C₄⋊C₄×C₁₃₇	—	4	2	no	—
82	C₄⋊C₄×C₁₃₇	—	4	4	no	—
83	C₄⋊C₄×C₁₃₇	—	4	4	no	—
84	C₄⋊C₄×C₁₃₇	—	4	2	no	—
85	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
86	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
87	C₄²×C₁₃₇	—	4	1	no	—
88	C₄²×C₁₃₇	—	4	2	no	—
89	C₄²×C₁₃₇	D₈	1	1	yes	characteristic
90	C₄²×C₁₃₇	D₈	1	1	yes	characteristic
91	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
92	C₁₆×C₁₃₇	—	2	4	no	—
93	C₂³×C₁₃₇	—	4	1	no	—
94	C₂³×C₁₃₇	C₂×D₈	1	1	yes	characteristic
95	Q₈×C₁₃₇	—	2	2	no	—
96	Q₈×C₁₃₇	—	4	1	no	—
97	D₄×C₁₃₇	—	2	2	no	—
98	D₄×C₁₃₇	—	4	4	no	—
99	D₄×C₁₃₇	—	4	1	no	—
100	C₂×C₄×C₁₃₇	—	4	2	no	—
101	C₂×C₄×C₁₃₇	—	4	1	no	—
102	C₂×C₄×C₁₃₇	—	4	4	no	—
103	C₂×C₄×C₁₃₇	—	4	2	no	—
104	C₂×C₄×C₁₃₇	—	4	4	no	—
105	C₂×C₄×C₁₃₇	—	8	4	no	—
106	C₂×C₄×C₁₃₇	—	8	4	no	—
107	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃₇	#137	1	2	yes	—
112	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
113	C₈×C₁₃₇	—	2	2	no	—
114	C₈×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
115	C₈×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
116	C₈×C₁₃₇	C₂×#41	1	1	yes	characteristic
117	C₂²×C₁₃₇	—	2	2	no	—
118	C₂²×C₁₃₇	—	4	2	no	—
119	C₂²×C₁₃₇	—	8	2	no	—
120	C₂²×C₁₃₇	#474	1	2	yes	—
121	C₂²×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
122	C₄×C₁₃₇	—	2	2	no	—
123	C₄×C₁₃₇	—	2	2	no	—
124	C₄×C₁₃₇	—	4	2	no	—
125	C₄×C₁₃₇	—	8	2	no	—
126	C₄×C₁₃₇	—	8	4	no	—
127	C₄×C₁₃₇	#474	1	2	yes	—
128	C₄×C₁₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
129	C₄×C₁₃₇	C₂×#137	1	1	yes	characteristic
130	C₂×C₁₃₇	—	2	2	no	—
131	C₂×C₁₃₇	—	8	2	no	—
132	C₂×C₁₃₇	C₂×#474	1	1	yes	characteristic
133	C₂×C₁₃₇	#3414	1	2	yes	—
134	#33630	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₁₃₇	#33630	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×#499	C₂×C₁₃₇	1	1	yes	characteristic, complemented
137	C₄×#135	C₂×C₁₃₇	1	1	yes	characteristic
138	#2229	C₂×C₁₃₇	1	4	yes	complemented
139	#2228	C₂×C₁₃₇	1	4	yes	complemented
140	#2159	C₂×C₁₃₇	1	2	yes	complemented
141	#2150	C₂×C₁₃₇	1	1	yes	characteristic, complemented
142	#2139	C₂×C₁₃₇	1	2	yes	—
143	C₂×#135	C₂²×C₁₃₇	1	1	yes	characteristic
144	C₂×#125	—	2	1	no	—
145	C₄×#41	—	2	1	no	—
146	C₂×M₃₂	C₂²×C₁₃₇	1	2	yes	—
147	#499	—	2	2	no	—
148	#499	C₂²×C₁₃₇	1	4	yes	—
149	C₄×Q₁₆	C₂²×C₁₃₇	1	2	yes	—
150	C₄×SD₁₆	—	2	2	no	—
151	C₄×D₈	C₂²×C₁₃₇	1	2	yes	—
152	#427	C₂²×C₁₃₇	1	1	yes	characteristic
153	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
154	C₂×#106	—	2	1	no	—
155	#416	—	2	1	no	—
156	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
157	#367	C₂²×C₁₃₇	1	4	yes	—
158	#366	C₂²×C₁₃₇	1	4	yes	—
159	C₄⋊C₁₆	C₂²×C₁₃₇	1	4	yes	—
160	#358	C₂²×C₁₃₇	1	4	yes	—
161	#357	C₂²×C₁₃₇	1	4	yes	—
162	C₂×#41	—	2	1	no	—
163	#135	—	2	2	no	—
164	#135	Q₈×C₁₃₇	1	4	yes	—
165	C₂×Q₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
166	C₂×SD₁₆	—	2	1	no	—
167	C₂×D₈	C₂³×C₁₃₇	1	1	yes	characteristic
168	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
170	#125	—	2	4	no	—
171	#125	—	4	1	no	—
172	#125	—	4	1	no	—
173	C₄×Q₈	—	2	2	no	—
174	C₄×D₄	—	2	2	no	—
175	C₄×D₄	—	4	2	no	—
176	#117	—	2	1	no	—
177	#117	—	4	1	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	4	2	no	—
180	C₂×C₄²	—	4	1	no	—
181	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
182	M₃₂	—	2	4	no	—
183	C₂×C₁₆	C₂³×C₁₃₇	1	4	yes	—
184	#107	—	2	2	no	—
185	#107	C₂³×C₁₃₇	1	4	yes	—
186	#107	C₂³×C₁₃₇	1	2	yes	—
187	#106	—	2	4	no	—
188	#106	—	2	1	no	—
189	#103	—	2	2	no	—
190	#102	—	2	2	no	—
191	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
192	C₄×C₈	C₂³×C₁₃₇	1	2	yes	—
193	#41	—	2	4	no	—
194	#41	—	4	2	no	—
195	C₂×Q₈	—	2	1	no	—
196	C₂×D₄	—	2	1	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	D₈×C₁₃₇	1	2	yes	—
202	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
203	Q₁₆	—	2	1	no	—
204	Q₁₆	C₂×Q₈×C₁₃₇	1	2	yes	—
205	SD₁₆	—	2	4	no	—
206	D₈	—	2	1	no	—
207	D₈	C₂×Q₈×C₁₃₇	1	2	yes	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
210	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
211	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
212	C₂×C₈	#41×C₁₃₇	1	2	yes	—
213	C₂×C₈	C₂⁴×C₁₃₇	1	1	yes	characteristic
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	4	1	no	—
221	C₂²⋊C₄	—	4	2	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	2	no	—
224	C₄²	D₈×C₁₃₇	1	1	yes	characteristic
225	C₄²	D₈×C₁₃₇	1	1	yes	characteristic
226	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
227	C₁₆	—	2	4	no	—
228	C₂³	—	4	1	no	—
229	C₂³	C₂×D₈×C₁₃₇	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	Q₈	—	4	1	no	—
232	D₄	—	2	2	no	—
233	D₄	—	4	4	no	—
234	D₄	—	4	1	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	8	4	no	—
241	C₂×C₄	—	8	4	no	—
242	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
243	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
244	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
245	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
246	C₂×C₄	#137×C₁₃₇	1	2	yes	—
247	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
250	C₈	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
251	C₈	C₂×#41×C₁₃₇	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	8	2	no	—
255	C₂²	#474×C₁₃₇	1	2	yes	—
256	C₂²	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	4	2	no	—
260	C₄	—	8	2	no	—
261	C₄	—	8	4	no	—
262	C₄	#474×C₁₃₇	1	2	yes	—
263	C₄	C₂×C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
264	C₄	C₂×#137×C₁₃₇	1	1	yes	characteristic
265	C₂	—	2	2	no	—
266	C₂	—	8	2	no	—
267	C₂	C₂×#474×C₁₃₇	1	1	yes	characteristic
268	C₂	#3414×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₇
prime radical	C₂×#135×C₁₃₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#499×C₁₃₇	C₂	semidirect
3. #2229×C₁₃₇	C₂	semidirect
4. #2228×C₁₃₇	C₂	semidirect
5. #2159×C₁₃₇	C₂	semidirect
6. #2150×C₁₃₇	C₂	semidirect
134. #33630	C₁₃₇	direct (N × H)
135. C₁₃₇	#33630	direct (N × H)
136. C₂×#499	C₂×C₁₃₇	semidirect
138. #2229	C₂×C₁₃₇	semidirect
139. #2228	C₂×C₁₃₇	semidirect
140. #2159	C₂×C₁₃₇	semidirect
141. #2150	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#499×C₁₃₇	2
C₄×#135×C₁₃₇	2
#2229×C₁₃₇	2
#2228×C₁₃₇	2
#2159×C₁₃₇	2
#2150×C₁₃₇	2
#2139×C₁₃₇	2
#33630	137

Subgroups — C137×#33630 #46883016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#33630 #46883016