Subgroups — C₁₃₇×#28675 #46878061

Order 35072.

790 conjugacy classes of subgroups.

The subgroup lattice has 264 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	#2546×C₁₃₇	C₂	1	2	yes	maximal
4	#2086×C₁₃₇	C₂	1	4	yes	maximal
5	#1988×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	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	#500×C₁₃₇	—	2	4	no	—
12	C₈×Q₈×C₁₃₇	—	2	4	no	—
13	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
15	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	#398×C₁₃₇	—	2	4	no	—
21	#391×C₁₃₇	—	2	2	no	—
22	#389×C₁₃₇	C₂²	1	8	yes	—
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	—	4	8	no	—
34	C₄×Q₈×C₁₃₇	—	2	2	no	—
35	C₄×Q₈×C₁₃₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₃₇	—	2	8	no	—
37	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
42	C₂×C₄²×C₁₃₇	—	2	2	no	—
43	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
45	#107×C₁₃₇	—	2	4	no	—
46	#107×C₁₃₇	—	2	4	no	—
47	#107×C₁₃₇	Q₈	1	8	yes	complemented
48	C₄⋊C₈×C₁₃₇	—	2	4	no	—
49	C₄⋊C₈×C₁₃₇	—	2	2	no	—
50	C₄×C₈×C₁₃₇	—	2	4	no	—
51	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
52	#95×C₁₃₇	—	2	2	no	—
53	#95×C₁₃₇	—	2	8	no	—
54	C₂×Q₈×C₁₃₇	—	2	2	no	—
55	C₂×Q₈×C₁₃₇	—	2	2	no	—
56	C₂×Q₈×C₁₃₇	—	4	4	no	—
57	C₂×Q₈×C₁₃₇	—	4	4	no	—
58	C₂×Q₈×C₁₃₇	—	4	4	no	—
59	C₂×Q₈×C₁₃₇	D₈	1	4	yes	—
60	C₂²×C₄×C₁₃₇	—	2	4	no	—
61	C₂²×C₄×C₁₃₇	—	2	4	no	—
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	2	yes	—
65	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₁₃₇	—	2	2	no	—
68	C₂×C₈×C₁₃₇	—	2	2	no	—
69	C₂×C₈×C₁₃₇	—	2	4	no	—
70	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
73	C₄⋊C₄×C₁₃₇	—	2	4	no	—
74	C₄⋊C₄×C₁₃₇	—	2	2	no	—
75	C₄⋊C₄×C₁₃₇	—	2	4	no	—
76	C₄⋊C₄×C₁₃₇	—	2	4	no	—
77	C₄⋊C₄×C₁₃₇	—	2	4	no	—
78	C₄⋊C₄×C₁₃₇	—	4	4	no	—
79	C₄⋊C₄×C₁₃₇	—	4	16	no	—
80	C₄⋊C₄×C₁₃₇	C₂×Q₈	1	8	yes	—
81	C₄²×C₁₃₇	—	2	2	no	—
82	C₄²×C₁₃₇	—	2	2	no	—
83	C₄²×C₁₃₇	—	4	4	no	—
84	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
85	C₄²×C₁₃₇	#41	1	2	yes	—
86	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
87	Q₈×C₁₃₇	—	2	8	no	—
88	Q₈×C₁₃₇	—	8	8	no	—
89	C₂×C₄×C₁₃₇	—	2	2	no	—
90	C₂×C₄×C₁₃₇	—	2	2	no	—
91	C₂×C₄×C₁₃₇	—	2	2	no	—
92	C₂×C₄×C₁₃₇	—	2	2	no	—
93	C₂×C₄×C₁₃₇	—	2	8	no	—
94	C₂×C₄×C₁₃₇	—	2	4	no	—
95	C₂×C₄×C₁₃₇	—	4	4	no	—
96	C₂×C₄×C₁₃₇	—	4	4	no	—
97	C₂×C₄×C₁₃₇	—	4	4	no	—
98	C₂×C₄×C₁₃₇	—	4	4	no	—
99	C₂×C₄×C₁₃₇	—	4	4	no	—
100	C₂×C₄×C₁₃₇	#121	1	2	yes	—
101	C₂×C₄×C₁₃₇	#121	1	2	yes	—
102	C₂×C₄×C₁₃₇	C₂×D₈	1	4	yes	—
103	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₁₃₇	#135	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	2	no	—
110	C₈×C₁₃₇	—	4	2	no	—
111	C₈×C₁₃₇	C₄⋊Q₈	1	4	yes	complemented
112	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
113	C₂²×C₁₃₇	D₄×Q₈	1	2	yes	—
114	C₂²×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
115	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
116	C₂²×C₁₃₇	#578	1	2	yes	—
117	C₄×C₁₃₇	—	2	4	no	—
118	C₄×C₁₃₇	—	2	2	no	—
119	C₄×C₁₃₇	—	4	8	no	—
120	C₄×C₁₃₇	—	4	2	no	—
121	C₄×C₁₃₇	—	8	4	no	—
122	C₄×C₁₃₇	#466	1	4	yes	—
123	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
124	C₄×C₁₃₇	#536	1	1	yes	characteristic
125	C₄×C₁₃₇	#557	1	2	yes	—
126	C₂×C₁₃₇	#2546	1	1	yes	characteristic
127	C₂×C₁₃₇	C₂×#466	1	1	yes	characteristic
128	C₂×C₁₃₇	#3059	1	1	yes	characteristic
129	C₂×C₁₃₇	Q₈×D₈	1	2	yes	—
130	C₂×C₁₃₇	#3383	1	2	yes	—
131	#28675	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₁₃₇	#28675	1	1	yes	characteristic, direct product factor, Sylow
133	C₂×#500	C₂×C₁₃₇	1	1	yes	characteristic
134	C₂×C₈×Q₈	C₂×C₁₃₇	1	1	yes	characteristic
135	#2546	C₂×C₁₃₇	1	2	yes	—
136	#2086	C₂×C₁₃₇	1	4	yes	—
137	#1988	C₂×C₁₃₇	1	2	yes	—
138	#1942	C₂×C₁₃₇	1	4	yes	—
139	C₄×#107	C₂×C₁₃₇	1	1	yes	characteristic
140	C₂×C₄⋊Q₈	—	2	2	no	—
141	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
142	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
143	#500	—	2	4	no	—
144	C₈×Q₈	—	2	4	no	—
145	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
146	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
147	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
148	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
149	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
151	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
152	#398	—	2	4	no	—
153	#391	—	2	2	no	—
154	#389	C₂²×C₁₃₇	1	8	yes	—
155	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
156	#340	C₂²×C₁₃₇	1	2	yes	—
157	#340	C₂²×C₁₃₇	1	8	yes	—
158	C₂²×Q₈	—	2	2	no	—
159	C₂²×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
161	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
163	C₄⋊Q₈	—	2	4	no	—
164	C₄⋊Q₈	—	2	4	no	—
165	C₄⋊Q₈	—	4	8	no	—
166	C₄×Q₈	—	2	2	no	—
167	C₄×Q₈	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	8	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
171	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
172	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
173	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
176	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
177	#107	—	2	4	no	—
178	#107	—	2	4	no	—
179	#107	Q₈×C₁₃₇	1	8	yes	complemented
180	C₄⋊C₈	—	2	4	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄×C₈	—	2	4	no	—
183	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
184	#95	—	2	2	no	—
185	#95	—	2	8	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	4	4	no	—
189	C₂×Q₈	—	4	4	no	—
190	C₂×Q₈	—	4	4	no	—
191	C₂×Q₈	D₈×C₁₃₇	1	4	yes	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
198	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
204	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	—	4	16	no	—
212	C₄⋊C₄	C₂×Q₈×C₁₃₇	1	8	yes	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	4	4	no	—
216	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
217	C₄²	#41×C₁₃₇	1	2	yes	—
218	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
219	Q₈	—	2	8	no	—
220	Q₈	—	8	8	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	#121×C₁₃₇	1	2	yes	—
233	C₂×C₄	#121×C₁₃₇	1	2	yes	—
234	C₂×C₄	C₂×D₈×C₁₃₇	1	4	yes	—
235	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
236	C₂×C₄	#135×C₁₃₇	1	2	yes	—
237	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
238	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
239	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
240	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
241	C₈	—	2	2	no	—
242	C₈	—	4	2	no	—
243	C₈	C₄⋊Q₈×C₁₃₇	1	4	yes	complemented
244	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
245	C₂²	D₄×Q₈×C₁₃₇	1	2	yes	—
246	C₂²	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
247	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
248	C₂²	#578×C₁₃₇	1	2	yes	—
249	C₄	—	2	4	no	—
250	C₄	—	2	2	no	—
251	C₄	—	4	2	no	—
252	C₄	—	4	8	no	—
253	C₄	—	8	4	no	—
254	C₄	#466×C₁₃₇	1	4	yes	—
255	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
256	C₄	#536×C₁₃₇	1	1	yes	characteristic
257	C₄	#557×C₁₃₇	1	2	yes	—
258	C₂	#2546×C₁₃₇	1	1	yes	characteristic
259	C₂	C₂×#466×C₁₃₇	1	1	yes	characteristic
260	C₂	#3059×C₁₃₇	1	1	yes	characteristic
261	C₂	Q₈×D₈×C₁₃₇	1	2	yes	—
262	C₂	#3383×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
47. #107×C₁₃₇	Q₈	semidirect
111. C₈×C₁₃₇	C₄⋊Q₈	semidirect
131. #28675	C₁₃₇	direct (N × H)
132. C₁₃₇	#28675	direct (N × H)
179. #107	Q₈×C₁₃₇	semidirect
243. C₈	C₄⋊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
#2546×C₁₃₇	2
#2086×C₁₃₇	2
#1988×C₁₃₇	2
#1942×C₁₃₇	2
C₄×#107×C₁₃₇	2
#28675	137

Subgroups — C137×#28675 #46878061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28675 #46878061