Subgroups — C₁₃₇×#28695 #46878081

Order 35072.

714 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#501×C₁₃₇	C₂	1	2	yes	maximal
2	#2862×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#125×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2085×C₁₃₇	C₂	1	4	yes	maximal
5	#1993×C₁₃₇	C₂	1	2	yes	maximal
6	#1949×C₁₃₇	C₂	1	4	yes	maximal
7	#1767×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
9	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
10	#501×C₁₃₇	—	2	8	no	—
11	C₂×#107×C₁₃₇	C₂²	1	4	yes	—
12	C₂×#106×C₁₃₇	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
15	#391×C₁₃₇	—	2	2	no	—
16	#391×C₁₃₇	—	2	2	no	—
17	#389×C₁₃₇	C₂²	1	2	yes	—
18	#387×C₁₃₇	—	2	2	no	—
19	#382×C₁₃₇	C₂²	1	1	yes	characteristic
20	#382×C₁₃₇	C₂²	1	2	yes	—
21	#382×C₁₃₇	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
23	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
24	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
25	#340×C₁₃₇	C₂²	1	8	yes	—
26	#336×C₁₃₇	C₂²	1	2	yes	—
27	C₂²×Q₈×C₁₃₇	—	2	2	no	—
28	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
29	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
30	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
31	#125×C₁₃₇	—	2	2	no	—
32	#125×C₁₃₇	D₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
38	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
39	C₂×C₄²×C₁₃₇	—	2	2	no	—
40	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
41	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
42	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
45	#107×C₁₃₇	—	2	8	no	—
46	#106×C₁₃₇	—	2	8	no	—
47	C₄⋊C₈×C₁₃₇	Q₈	1	8	yes	complemented
48	#97×C₁₃₇	—	2	4	no	—
49	#95×C₁₃₇	—	2	2	no	—
50	#95×C₁₃₇	—	2	2	no	—
51	#95×C₁₃₇	—	2	2	no	—
52	#95×C₁₃₇	—	2	4	no	—
53	#95×C₁₃₇	—	2	2	no	—
54	#95×C₁₃₇	—	2	2	no	—
55	C₂×Q₈×C₁₃₇	—	4	4	no	—
56	C₂×Q₈×C₁₃₇	—	4	4	no	—
57	C₂×Q₈×C₁₃₇	—	4	4	no	—
58	C₂²×C₄×C₁₃₇	—	2	2	no	—
59	C₂²×C₄×C₁₃₇	—	2	2	no	—
60	C₂²×C₄×C₁₃₇	—	2	4	no	—
61	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₁₃₇	—	2	4	no	—
67	C₂×C₈×C₁₃₇	—	2	4	no	—
68	C₂×C₈×C₁₃₇	C₂×Q₈	1	8	yes	—
69	C₄⋊C₄×C₁₃₇	—	2	8	no	—
70	C₄⋊C₄×C₁₃₇	—	2	2	no	—
71	C₄⋊C₄×C₁₃₇	—	2	4	no	—
72	C₄⋊C₄×C₁₃₇	—	2	4	no	—
73	C₄⋊C₄×C₁₃₇	—	2	4	no	—
74	C₄⋊C₄×C₁₃₇	—	4	4	no	—
75	C₄⋊C₄×C₁₃₇	—	4	4	no	—
76	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
77	C₄²×C₁₃₇	—	2	2	no	—
78	C₄²×C₁₃₇	—	4	2	no	—
79	C₄²×C₁₃₇	—	4	2	no	—
80	C₄²×C₁₃₇	—	4	4	no	—
81	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
82	C₄²×C₁₃₇	C₂×Q₈	1	4	yes	—
83	C₄²×C₁₃₇	#41	1	2	yes	—
84	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
85	Q₈×C₁₃₇	—	8	8	no	—
86	C₂×C₄×C₁₃₇	—	2	2	no	—
87	C₂×C₄×C₁₃₇	—	2	2	no	—
88	C₂×C₄×C₁₃₇	—	2	2	no	—
89	C₂×C₄×C₁₃₇	—	2	2	no	—
90	C₂×C₄×C₁₃₇	—	2	4	no	—
91	C₂×C₄×C₁₃₇	—	2	2	no	—
92	C₂×C₄×C₁₃₇	—	2	2	no	—
93	C₂×C₄×C₁₃₇	—	2	2	no	—
94	C₂×C₄×C₁₃₇	—	4	2	no	—
95	C₂×C₄×C₁₃₇	—	4	2	no	—
96	C₂×C₄×C₁₃₇	—	4	4	no	—
97	C₂×C₄×C₁₃₇	—	4	2	no	—
98	C₂×C₄×C₁₃₇	—	4	2	no	—
99	C₂×C₄×C₁₃₇	—	4	2	no	—
100	C₂×C₄×C₁₃₇	—	4	4	no	—
101	C₂×C₄×C₁₃₇	—	4	4	no	—
102	C₂×C₄×C₁₃₇	#121	1	4	yes	—
103	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
104	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
105	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
107	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
109	C₈×C₁₃₇	—	2	8	no	—
110	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
111	C₂²×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
112	C₂²×C₁₃₇	#536	1	1	yes	characteristic
113	C₂²×C₁₃₇	#575	1	2	yes	—
114	C₂²×C₁₃₇	#575	1	2	yes	—
115	C₄×C₁₃₇	—	2	2	no	—
116	C₄×C₁₃₇	—	2	2	no	—
117	C₄×C₁₃₇	—	2	4	no	—
118	C₄×C₁₃₇	—	4	2	no	—
119	C₄×C₁₃₇	—	4	4	no	—
120	C₄×C₁₃₇	—	8	2	no	—
121	C₄×C₁₃₇	—	8	4	no	—
122	C₄×C₁₃₇	D₄×Q₈	1	2	yes	—
123	C₄×C₁₃₇	#557	1	2	yes	—
124	C₂×C₁₃₇	#2546	1	1	yes	characteristic
125	C₂×C₁₃₇	#3040	1	2	yes	—
126	C₂×C₁₃₇	#3162	1	1	yes	characteristic
127	C₂×C₁₃₇	#3166	1	1	yes	characteristic
128	C₂×C₁₃₇	#3240	1	2	yes	—
129	#28695	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₁₃₇	#28695	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×#501	C₂×C₁₃₇	1	2	yes	—
132	#2862	C₂×C₁₃₇	1	1	yes	characteristic
133	C₄×#125	C₂×C₁₃₇	1	1	yes	characteristic
134	#2085	C₂×C₁₃₇	1	4	yes	—
135	#1993	C₂×C₁₃₇	1	2	yes	—
136	#1949	C₂×C₁₃₇	1	4	yes	—
137	#1767	C₂×C₁₃₇	1	1	yes	characteristic
138	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
139	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
140	#501	—	2	8	no	—
141	C₂×#107	C₂²×C₁₃₇	1	4	yes	—
142	C₂×#106	C₂²×C₁₃₇	1	4	yes	—
143	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
144	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
145	#391	—	2	2	no	—
146	#391	—	2	2	no	—
147	#389	C₂²×C₁₃₇	1	2	yes	—
148	#387	—	2	2	no	—
149	#382	C₂²×C₁₃₇	1	1	yes	characteristic
150	#382	C₂²×C₁₃₇	1	2	yes	—
151	#382	C₂²×C₁₃₇	1	2	yes	—
152	C₄×C₄⋊C₄	—	2	2	no	—
153	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
154	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
155	#340	C₂²×C₁₃₇	1	8	yes	—
156	#336	C₂²×C₁₃₇	1	2	yes	—
157	C₂²×Q₈	—	2	2	no	—
158	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
159	C₄⋊Q₈	—	2	4	no	—
160	C₄⋊Q₈	—	2	4	no	—
161	#125	—	2	2	no	—
162	#125	D₄×C₁₃₇	1	4	yes	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
167	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
168	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
169	C₂×C₄²	—	2	2	no	—
170	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
171	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
172	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
173	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
175	#107	—	2	8	no	—
176	#106	—	2	8	no	—
177	C₄⋊C₈	Q₈×C₁₃₇	1	8	yes	complemented
178	#97	—	2	4	no	—
179	#95	—	2	2	no	—
180	#95	—	2	2	no	—
181	#95	—	2	2	no	—
182	#95	—	2	4	no	—
183	#95	—	2	2	no	—
184	#95	—	2	2	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂×Q₈	—	4	4	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
192	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
193	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
195	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	C₂×Q₈×C₁₃₇	1	8	yes	—
199	C₄⋊C₄	—	2	8	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
207	C₄²	—	2	2	no	—
208	C₄²	—	4	2	no	—
209	C₄²	—	4	2	no	—
210	C₄²	—	4	4	no	—
211	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
212	C₄²	C₂×Q₈×C₁₃₇	1	4	yes	—
213	C₄²	#41×C₁₃₇	1	2	yes	—
214	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
215	Q₈	—	8	8	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	#121×C₁₃₇	1	4	yes	—
233	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
234	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
235	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
236	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
237	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
238	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
239	C₈	—	2	8	no	—
240	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
241	C₂²	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
242	C₂²	#536×C₁₃₇	1	1	yes	characteristic
243	C₂²	#575×C₁₃₇	1	2	yes	—
244	C₂²	#575×C₁₃₇	1	2	yes	—
245	C₄	—	2	4	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	4	4	no	—
249	C₄	—	4	2	no	—
250	C₄	—	8	4	no	—
251	C₄	—	8	2	no	—
252	C₄	D₄×Q₈×C₁₃₇	1	2	yes	—
253	C₄	#557×C₁₃₇	1	2	yes	—
254	C₂	#2546×C₁₃₇	1	1	yes	characteristic
255	C₂	#3040×C₁₃₇	1	2	yes	—
256	C₂	#3162×C₁₃₇	1	1	yes	characteristic
257	C₂	#3166×C₁₃₇	1	1	yes	characteristic
258	C₂	#3240×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₄³×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
47. C₄⋊C₈×C₁₃₇	Q₈	semidirect
129. #28695	C₁₃₇	direct (N × H)
130. C₁₃₇	#28695	direct (N × H)
177. C₄⋊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×#501×C₁₃₇	2
#2862×C₁₃₇	2
C₄×#125×C₁₃₇	2
#2085×C₁₃₇	2
#1993×C₁₃₇	2
#1949×C₁₃₇	2
#1767×C₁₃₇	2
#28695	137

Subgroups — C137×#28695 #46878081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28695 #46878081