Subgroups — C₁₃₇×#28676 #46878062

Order 35072.

750 conjugacy classes of subgroups.

The subgroup lattice has 258 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×C₈×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2567×C₁₃₇	C₂	1	2	yes	maximal
4	#2087×C₁₃₇	C₂	1	4	yes	maximal
5	#1987×C₁₃₇	C₂	1	2	yes	maximal
6	#1942×C₁₃₇	C₂	1	4	yes	maximal
7	C₄×#107×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	C₈×Q₈×C₁₃₇	—	2	4	no	—
12	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
13	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#107×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×C₄×C₈×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	#107×C₁₃₇	Q₈	1	8	yes	complemented
47	C₄⋊C₈×C₁₃₇	—	2	4	no	—
48	C₄⋊C₈×C₁₃₇	—	2	2	no	—
49	C₄×C₈×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	2	no	—
55	C₂×Q₈×C₁₃₇	—	4	4	no	—
56	C₂×Q₈×C₁₃₇	—	8	4	no	—
57	C₂×Q₈×C₁₃₇	Q₁₆	1	4	yes	—
58	C₂²×C₄×C₁₃₇	—	2	4	no	—
59	C₂²×C₄×C₁₃₇	—	2	4	no	—
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	2	yes	—
63	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
65	C₂×C₈×C₁₃₇	—	2	2	no	—
66	C₂×C₈×C₁₃₇	—	2	2	no	—
67	C₂×C₈×C₁₃₇	—	2	4	no	—
68	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
71	C₄⋊C₄×C₁₃₇	—	2	4	no	—
72	C₄⋊C₄×C₁₃₇	—	2	2	no	—
73	C₄⋊C₄×C₁₃₇	—	2	4	no	—
74	C₄⋊C₄×C₁₃₇	—	2	4	no	—
75	C₄⋊C₄×C₁₃₇	—	2	4	no	—
76	C₄⋊C₄×C₁₃₇	—	4	4	no	—
77	C₄⋊C₄×C₁₃₇	—	8	8	no	—
78	C₄⋊C₄×C₁₃₇	C₂×Q₈	1	8	yes	—
79	C₄²×C₁₃₇	—	2	2	no	—
80	C₄²×C₁₃₇	—	2	2	no	—
81	C₄²×C₁₃₇	—	4	4	no	—
82	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
83	C₄²×C₁₃₇	#41	1	2	yes	—
84	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
85	Q₈×C₁₃₇	—	2	8	no	—
86	Q₈×C₁₃₇	—	8	8	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	2	no	—
91	C₂×C₄×C₁₃₇	—	2	8	no	—
92	C₂×C₄×C₁₃₇	—	2	4	no	—
93	C₂×C₄×C₁₃₇	—	4	4	no	—
94	C₂×C₄×C₁₃₇	—	4	4	no	—
95	C₂×C₄×C₁₃₇	—	4	4	no	—
96	C₂×C₄×C₁₃₇	—	4	8	no	—
97	C₂×C₄×C₁₃₇	#121	1	2	yes	—
98	C₂×C₄×C₁₃₇	#121	1	2	yes	—
99	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	4	yes	—
100	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
101	C₂×C₄×C₁₃₇	#135	1	2	yes	—
102	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
104	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
106	C₈×C₁₃₇	—	2	2	no	—
107	C₈×C₁₃₇	—	4	2	no	—
108	C₈×C₁₃₇	C₄⋊Q₈	1	4	yes	—
109	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
110	C₂²×C₁₃₇	D₄×Q₈	1	2	yes	—
111	C₂²×C₁₃₇	C₂²×Q₁₆	1	1	yes	characteristic
112	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
113	C₂²×C₁₃₇	#576	1	2	yes	—
114	C₄×C₁₃₇	—	2	4	no	—
115	C₄×C₁₃₇	—	2	2	no	—
116	C₄×C₁₃₇	—	4	8	no	—
117	C₄×C₁₃₇	—	4	2	no	—
118	C₄×C₁₃₇	—	8	4	no	—
119	C₄×C₁₃₇	#467	1	4	yes	—
120	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₄×C₁₃₇	#536	1	1	yes	characteristic
122	C₄×C₁₃₇	#557	1	2	yes	—
123	C₂×C₁₃₇	#2546	1	1	yes	characteristic
124	C₂×C₁₃₇	C₂×#467	1	1	yes	characteristic
125	C₂×C₁₃₇	#3058	1	1	yes	characteristic
126	C₂×C₁₃₇	#3380	1	2	yes	—
127	C₂×C₁₃₇	Q₈×Q₁₆	1	2	yes	—
128	#28676	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₁₃₇	#28676	1	1	yes	characteristic, direct product factor, Sylow
130	C₂×#500	C₂×C₁₃₇	1	1	yes	characteristic
131	C₂×C₈×Q₈	C₂×C₁₃₇	1	1	yes	characteristic
132	#2567	C₂×C₁₃₇	1	2	yes	—
133	#2087	C₂×C₁₃₇	1	4	yes	—
134	#1987	C₂×C₁₃₇	1	2	yes	—
135	#1942	C₂×C₁₃₇	1	4	yes	—
136	C₄×#107	C₂×C₁₃₇	1	1	yes	characteristic
137	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
138	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
139	#500	—	2	4	no	—
140	C₈×Q₈	—	2	4	no	—
141	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
142	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
143	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
144	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
145	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
147	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
148	#395	—	2	4	no	—
149	#391	—	2	2	no	—
150	#389	C₂²×C₁₃₇	1	8	yes	—
151	#384	—	2	2	no	—
152	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
153	#340	C₂²×C₁₃₇	1	2	yes	—
154	#340	C₂²×C₁₃₇	1	8	yes	—
155	C₂²×Q₈	—	2	2	no	—
156	C₂²×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
157	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
158	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
159	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
160	C₄⋊Q₈	—	2	4	no	—
161	C₄⋊Q₈	—	2	4	no	—
162	C₄×Q₈	—	2	2	no	—
163	C₄×Q₈	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	8	no	—
166	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
167	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
168	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
169	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
170	C₂×C₄²	—	2	2	no	—
171	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
172	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
173	#107	—	2	4	no	—
174	#107	—	2	4	no	—
175	#107	Q₈×C₁₃₇	1	8	yes	complemented
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄×C₈	—	2	4	no	—
179	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
180	#95	—	2	2	no	—
181	#95	—	2	8	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	4	4	no	—
185	C₂×Q₈	—	8	4	no	—
186	C₂×Q₈	Q₁₆×C₁₃₇	1	4	yes	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
192	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
193	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
198	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
199	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	8	8	no	—
207	C₄⋊C₄	C₂×Q₈×C₁₃₇	1	8	yes	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	4	4	no	—
211	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
212	C₄²	#41×C₁₃₇	1	2	yes	—
213	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
214	Q₈	—	2	8	no	—
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	8	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	8	no	—
226	C₂×C₄	#121×C₁₃₇	1	2	yes	—
227	C₂×C₄	#121×C₁₃₇	1	2	yes	—
228	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	4	yes	—
229	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	2	yes	—
230	C₂×C₄	#135×C₁₃₇	1	2	yes	—
231	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
232	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
233	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
234	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
235	C₈	—	2	2	no	—
236	C₈	—	4	2	no	—
237	C₈	C₄⋊Q₈×C₁₃₇	1	4	yes	—
238	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
239	C₂²	D₄×Q₈×C₁₃₇	1	2	yes	—
240	C₂²	C₂²×Q₁₆×C₁₃₇	1	1	yes	characteristic
241	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
242	C₂²	#576×C₁₃₇	1	2	yes	—
243	C₄	—	2	4	no	—
244	C₄	—	2	2	no	—
245	C₄	—	4	2	no	—
246	C₄	—	4	8	no	—
247	C₄	—	8	4	no	—
248	C₄	#467×C₁₃₇	1	4	yes	—
249	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
250	C₄	#536×C₁₃₇	1	1	yes	characteristic
251	C₄	#557×C₁₃₇	1	2	yes	—
252	C₂	#2546×C₁₃₇	1	1	yes	characteristic
253	C₂	C₂×#467×C₁₃₇	1	1	yes	characteristic
254	C₂	#3058×C₁₃₇	1	1	yes	characteristic
255	C₂	#3380×C₁₃₇	1	2	yes	—
256	C₂	Q₈×Q₁₆×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 ◁ C₂×C₄×Q₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
46. #107×C₁₃₇	Q₈	semidirect
128. #28676	C₁₃₇	direct (N × H)
129. C₁₃₇	#28676	direct (N × H)
175. #107	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×C₈×Q₈×C₁₃₇	2
#2567×C₁₃₇	2
#2087×C₁₃₇	2
#1987×C₁₃₇	2
#1942×C₁₃₇	2
C₄×#107×C₁₃₇	2
#28676	137

Subgroups — C137×#28676 #46878062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28676 #46878062