Subgroups — C₁₃₇×#28682 #46878068

Order 35072.

730 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#446×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2567×C₁₃₇	C₂	1	2	yes	maximal
4	#2086×C₁₃₇	C₂	1	4	yes	maximal
5	#1987×C₁₃₇	C₂	1	2	yes	maximal
6	#1944×C₁₃₇	C₂	1	4	yes	maximal
7	C₄×#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	#500×C₁₃₇	—	2	4	no	—
11	#446×C₁₃₇	—	2	4	no	—
12	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
13	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	#395×C₁₃₇	—	2	4	no	—
20	#391×C₁₃₇	—	2	2	no	—
21	#389×C₁₃₇	C₂²	1	8	yes	—
22	#384×C₁₃₇	—	2	2	no	—
23	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
24	#340×C₁₃₇	C₂²	1	2	yes	—
25	#340×C₁₃₇	C₂²	1	8	yes	—
26	C₂²×Q₈×C₁₃₇	—	2	2	no	—
27	C₂²×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
32	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
33	C₄×Q₈×C₁₃₇	—	2	2	no	—
34	C₄×Q₈×C₁₃₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₃₇	—	2	8	no	—
37	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
41	C₂×C₄²×C₁₃₇	—	2	2	no	—
42	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
44	#107×C₁₃₇	—	2	4	no	—
45	#107×C₁₃₇	—	2	4	no	—
46	#106×C₁₃₇	Q₈	1	8	yes	complemented
47	C₄⋊C₈×C₁₃₇	—	2	4	no	—
48	C₄⋊C₈×C₁₃₇	—	2	2	no	—
49	#97×C₁₃₇	—	2	4	no	—
50	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
51	#95×C₁₃₇	—	2	2	no	—
52	#95×C₁₃₇	—	2	8	no	—
53	C₂×Q₈×C₁₃₇	—	2	2	no	—
54	C₂×Q₈×C₁₃₇	—	2	4	no	—
55	C₂×Q₈×C₁₃₇	—	4	4	no	—
56	C₂×Q₈×C₁₃₇	—	8	4	no	—
57	C₂²×C₄×C₁₃₇	—	2	4	no	—
58	C₂²×C₄×C₁₃₇	—	2	4	no	—
59	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₁₃₇	—	2	2	no	—
65	C₂×C₈×C₁₃₇	—	2	2	no	—
66	C₂×C₈×C₁₃₇	—	2	4	no	—
67	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
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	—	2	4	no	—
75	C₄⋊C₄×C₁₃₇	—	4	4	no	—
76	C₄⋊C₄×C₁₃₇	—	8	8	no	—
77	C₄⋊C₄×C₁₃₇	C₂×Q₈	1	8	yes	—
78	C₄²×C₁₃₇	—	2	2	no	—
79	C₄²×C₁₃₇	—	2	2	no	—
80	C₄²×C₁₃₇	—	4	4	no	—
81	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
82	C₄²×C₁₃₇	#41	1	2	yes	—
83	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
84	Q₈×C₁₃₇	—	4	4	no	—
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	8	no	—
90	C₂×C₄×C₁₃₇	—	2	4	no	—
91	C₂×C₄×C₁₃₇	—	2	4	no	—
92	C₂×C₄×C₁₃₇	—	4	4	no	—
93	C₂×C₄×C₁₃₇	—	4	4	no	—
94	C₂×C₄×C₁₃₇	—	4	4	no	—
95	C₂×C₄×C₁₃₇	—	4	8	no	—
96	C₂×C₄×C₁₃₇	#121	1	2	yes	—
97	C₂×C₄×C₁₃₇	#121	1	2	yes	—
98	C₂×C₄×C₁₃₇	#137	1	4	yes	—
99	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
101	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
102	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
103	C₈×C₁₃₇	—	2	2	no	—
104	C₈×C₁₃₇	—	4	2	no	—
105	C₈×C₁₃₇	C₄⋊Q₈	1	4	yes	—
106	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
107	C₂²×C₁₃₇	D₄×Q₈	1	2	yes	—
108	C₂²×C₁₃₇	C₂×#137	1	2	yes	—
109	C₂²×C₁₃₇	#576	1	2	yes	—
110	C₄×C₁₃₇	—	2	2	no	—
111	C₄×C₁₃₇	—	4	8	no	—
112	C₄×C₁₃₇	—	4	2	no	—
113	C₄×C₁₃₇	—	4	2	no	—
114	C₄×C₁₃₇	—	8	4	no	—
115	C₄×C₁₃₇	#470	1	4	yes	—
116	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
117	C₄×C₁₃₇	#536	1	1	yes	characteristic
118	C₄×C₁₃₇	#557	1	2	yes	—
119	C₂×C₁₃₇	#2546	1	1	yes	characteristic
120	C₂×C₁₃₇	C₂×#470	1	1	yes	characteristic
121	C₂×C₁₃₇	#3058	1	1	yes	characteristic
122	C₂×C₁₃₇	#3385	1	4	yes	—
123	#28682	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₁₃₇	#28682	1	1	yes	characteristic, direct product factor, Sylow
125	C₂×#500	C₂×C₁₃₇	1	1	yes	characteristic
126	C₂×#446	C₂×C₁₃₇	1	1	yes	characteristic
127	#2567	C₂×C₁₃₇	1	2	yes	—
128	#2086	C₂×C₁₃₇	1	4	yes	—
129	#1987	C₂×C₁₃₇	1	2	yes	—
130	#1944	C₂×C₁₃₇	1	4	yes	—
131	C₄×#106	C₂×C₁₃₇	1	1	yes	characteristic
132	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
133	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
134	#500	—	2	4	no	—
135	#446	—	2	4	no	—
136	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
137	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
138	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
139	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
140	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
141	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
142	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
143	#395	—	2	4	no	—
144	#391	—	2	2	no	—
145	#389	C₂²×C₁₃₇	1	8	yes	—
146	#384	—	2	2	no	—
147	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
148	#340	C₂²×C₁₃₇	1	2	yes	—
149	#340	C₂²×C₁₃₇	1	8	yes	—
150	C₂²×Q₈	—	2	2	no	—
151	C₂²×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
152	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
153	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
154	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
155	C₄⋊Q₈	—	2	4	no	—
156	C₄⋊Q₈	—	2	4	no	—
157	C₄×Q₈	—	2	2	no	—
158	C₄×Q₈	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	8	no	—
161	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
162	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
163	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
164	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
167	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
168	#107	—	2	4	no	—
169	#107	—	2	4	no	—
170	#106	Q₈×C₁₃₇	1	8	yes	complemented
171	C₄⋊C₈	—	2	4	no	—
172	C₄⋊C₈	—	2	2	no	—
173	#97	—	2	4	no	—
174	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
175	#95	—	2	2	no	—
176	#95	—	2	8	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	—	4	4	no	—
180	C₂×Q₈	—	8	4	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
184	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
185	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
187	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
192	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
193	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄⋊C₄	—	8	8	no	—
201	C₄⋊C₄	C₂×Q₈×C₁₃₇	1	8	yes	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	4	4	no	—
205	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
206	C₄²	#41×C₁₃₇	1	2	yes	—
207	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
208	Q₈	—	4	4	no	—
209	Q₈	—	8	8	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	8	no	—
220	C₂×C₄	#121×C₁₃₇	1	2	yes	—
221	C₂×C₄	#121×C₁₃₇	1	2	yes	—
222	C₂×C₄	#137×C₁₃₇	1	4	yes	—
223	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
224	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
225	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
226	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
227	C₈	—	2	2	no	—
228	C₈	—	4	2	no	—
229	C₈	C₄⋊Q₈×C₁₃₇	1	4	yes	—
230	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
231	C₂²	D₄×Q₈×C₁₃₇	1	2	yes	—
232	C₂²	C₂×#137×C₁₃₇	1	2	yes	—
233	C₂²	#576×C₁₃₇	1	2	yes	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	8	no	—
238	C₄	—	8	4	no	—
239	C₄	#470×C₁₃₇	1	4	yes	—
240	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
241	C₄	#536×C₁₃₇	1	1	yes	characteristic
242	C₄	#557×C₁₃₇	1	2	yes	—
243	C₂	#2546×C₁₃₇	1	1	yes	characteristic
244	C₂	C₂×#470×C₁₃₇	1	1	yes	characteristic
245	C₂	#3058×C₁₃₇	1	1	yes	characteristic
246	C₂	#3385×C₁₃₇	1	4	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 ◁ C₂×C₄×Q₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
46. #106×C₁₃₇	Q₈	semidirect
123. #28682	C₁₃₇	direct (N × H)
124. C₁₃₇	#28682	direct (N × H)
170. #106	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×#500×C₁₃₇	2
C₂×#446×C₁₃₇	2
#2567×C₁₃₇	2
#2086×C₁₃₇	2
#1987×C₁₃₇	2
#1944×C₁₃₇	2
C₄×#106×C₁₃₇	2
#28682	137

Subgroups — C137×#28682 #46878068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28682 #46878068