Subgroups — C₄×#494×C₁₃₇ #46871074

Order 35072.

1086 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₁₃₇	C₂	1	4	yes	maximal
3	C₄×C₄⋊Q₈×C₁₃₇	C₂	1	2	yes	maximal
4	#1969×C₁₃₇	C₂	1	4	yes	maximal
5	#1956×C₁₃₇	C₂	1	2	yes	maximal
6	#1845×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₄²×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₁₃₇	C₂²	1	4	yes	—
9	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₁₃₇	C₂²	1	4	yes	—
11	#494×C₁₃₇	C₄	1	8	yes	direct product factor
12	C₄×Q₁₆×C₁₃₇	—	2	16	no	—
13	C₂×#107×C₁₃₇	C₂²	1	4	yes	—
14	C₂×#103×C₁₃₇	C₂²	1	8	yes	—
15	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
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	#391×C₁₃₇	C₂²	1	4	yes	—
20	#389×C₁₃₇	—	2	4	no	—
21	#384×C₁₃₇	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₁₃₇	—	2	4	no	—
23	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂²×Q₈×C₁₃₇	C₂³	1	4	yes	—
25	C₂×Q₁₆×C₁₃₇	—	2	8	no	—
26	C₂×Q₁₆×C₁₃₇	—	2	8	no	—
27	C₂×Q₁₆×C₁₃₇	C₂×C₄	1	16	yes	—
28	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
30	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
31	C₄⋊Q₈×C₁₃₇	C₂×C₄	1	8	yes	—
32	C₄×Q₈×C₁₃₇	—	2	16	no	—
33	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
37	C₂×C₄²×C₁₃₇	—	2	4	no	—
38	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
39	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
40	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
43	#107×C₁₃₇	—	2	8	no	—
44	#103×C₁₃₇	—	2	16	no	—
45	C₄×C₈×C₁₃₇	—	2	4	no	—
46	C₄×C₈×C₁₃₇	—	2	2	no	—
47	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
48	C₄×C₈×C₁₃₇	D₄	1	8	yes	—
49	#95×C₁₃₇	—	2	8	no	—
50	C₂×Q₈×C₁₃₇	—	2	4	no	—
51	C₂×Q₈×C₁₃₇	—	2	8	no	—
52	C₂×Q₈×C₁₃₇	—	2	8	no	—
53	C₂×Q₈×C₁₃₇	C₂²×C₄	1	8	yes	—
54	C₂²×C₄×C₁₃₇	—	2	4	no	—
55	C₂²×C₄×C₁₃₇	—	2	4	no	—
56	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
61	Q₁₆×C₁₃₇	—	2	32	no	—
62	C₂×C₈×C₁₃₇	—	2	2	no	—
63	C₂×C₈×C₁₃₇	—	2	2	no	—
64	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
66	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
67	C₂×C₈×C₁₃₇	#41	1	4	yes	—
68	C₄⋊C₄×C₁₃₇	—	2	16	no	—
69	C₄⋊C₄×C₁₃₇	—	2	8	no	—
70	C₄⋊C₄×C₁₃₇	—	4	8	no	—
71	C₄⋊C₄×C₁₃₇	—	4	8	no	—
72	C₄²×C₁₃₇	—	2	4	no	—
73	C₄²×C₁₃₇	—	2	2	no	—
74	C₄²×C₁₃₇	—	4	8	no	—
75	C₄²×C₁₃₇	Q₁₆	1	8	yes	complemented
76	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₁₃₇	C₂×D₄	1	4	yes	—
78	C₄²×C₁₃₇	#41	1	2	yes	—
79	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
80	Q₈×C₁₃₇	—	2	16	no	—
81	Q₈×C₁₃₇	—	4	8	no	—
82	C₂×C₄×C₁₃₇	—	2	2	no	—
83	C₂×C₄×C₁₃₇	—	2	2	no	—
84	C₂×C₄×C₁₃₇	—	2	2	no	—
85	C₂×C₄×C₁₃₇	—	2	8	no	—
86	C₂×C₄×C₁₃₇	—	4	4	no	—
87	C₂×C₄×C₁₃₇	—	4	4	no	—
88	C₂×C₄×C₁₃₇	—	4	4	no	—
89	C₂×C₄×C₁₃₇	—	4	8	no	—
90	C₂×C₄×C₁₃₇	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₁₃₇	#127	1	2	yes	—
92	C₂×C₄×C₁₃₇	#127	1	2	yes	—
93	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	4	yes	—
94	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	4	yes	—
95	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	4	yes	—
96	C₂×C₄×C₁₃₇	#135	1	4	yes	—
97	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
99	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
100	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
101	C₈×C₁₃₇	—	2	4	no	—
102	C₈×C₁₃₇	C₄×D₄	1	8	yes	—
103	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
104	C₂²×C₁₃₇	C₂×#127	1	1	yes	characteristic
105	C₂²×C₁₃₇	#532	1	1	yes	characteristic
106	C₂²×C₁₃₇	C₂²×Q₁₆	1	2	yes	—
107	C₂²×C₁₃₇	C₂×#135	1	2	yes	—
108	C₄×C₁₃₇	—	2	2	no	—
109	C₄×C₁₃₇	—	2	4	no	—
110	C₄×C₁₃₇	—	4	8	no	—
111	C₄×C₁₃₇	—	8	4	no	—
112	C₄×C₁₃₇	C₄×Q₁₆	1	8	yes	complemented
113	C₄×C₁₃₇	#494	1	4	yes	direct product factor
114	C₄×C₁₃₇	C₂×C₄×D₄	1	2	yes	—
115	C₄×C₁₃₇	#532	1	2	yes	—
116	C₂×C₁₃₇	C₄×#127	1	1	yes	characteristic
117	C₂×C₁₃₇	C₂×C₄×Q₁₆	1	2	yes	—
118	C₂×C₁₃₇	C₂×#494	1	1	yes	characteristic
119	C₂×C₁₃₇	#3147	1	1	yes	characteristic
120	C₂×C₁₃₇	#3170	1	2	yes	—
121	C₄×#494	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₁₃₇	C₄×#494	1	1	yes	characteristic, direct product factor, Sylow
123	C₂×#494	C₂×C₁₃₇	1	1	yes	characteristic
124	C₂×C₄×Q₁₆	C₂×C₁₃₇	1	4	yes	—
125	C₄×C₄⋊Q₈	C₂×C₁₃₇	1	2	yes	—
126	#1969	C₂×C₁₃₇	1	4	yes	—
127	#1956	C₂×C₁₃₇	1	2	yes	—
128	#1845	C₂×C₁₃₇	1	1	yes	characteristic
129	C₄²×C₈	C₂×C₁₃₇	1	1	yes	characteristic
130	C₂²×Q₁₆	C₂²×C₁₃₇	1	4	yes	—
131	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
132	C₂×C₄×Q₈	C₂²×C₁₃₇	1	4	yes	—
133	#494	C₄×C₁₃₇	1	8	yes	direct product factor
134	C₄×Q₁₆	—	2	16	no	—
135	C₂×#107	C₂²×C₁₃₇	1	4	yes	—
136	C₂×#103	C₂²×C₁₃₇	1	8	yes	—
137	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
139	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
141	#391	C₂²×C₁₃₇	1	4	yes	—
142	#389	—	2	4	no	—
143	#384	C₂²×C₁₃₇	1	2	yes	—
144	C₄×C₄⋊C₄	—	2	4	no	—
145	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
146	C₂²×Q₈	C₂³×C₁₃₇	1	4	yes	—
147	C₂×Q₁₆	—	2	8	no	—
148	C₂×Q₁₆	—	2	8	no	—
149	C₂×Q₁₆	C₂×C₄×C₁₃₇	1	16	yes	—
150	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
151	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
152	C₄⋊Q₈	—	2	4	no	—
153	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	8	yes	—
154	C₄×Q₈	—	2	16	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄⋊C₄	—	2	4	no	—
158	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
159	C₂×C₄²	—	2	4	no	—
160	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
161	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
162	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
163	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
164	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
165	#107	—	2	8	no	—
166	#103	—	2	16	no	—
167	C₄×C₈	—	2	4	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
170	C₄×C₈	D₄×C₁₃₇	1	8	yes	—
171	#95	—	2	8	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	2	8	no	—
174	C₂×Q₈	—	2	8	no	—
175	C₂×Q₈	C₂²×C₄×C₁₃₇	1	8	yes	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
180	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
181	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
182	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
183	Q₁₆	—	2	32	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
187	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
188	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
189	C₂×C₈	#41×C₁₃₇	1	4	yes	—
190	C₄⋊C₄	—	2	16	no	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	4	8	no	—
193	C₄⋊C₄	—	4	8	no	—
194	C₄²	—	2	4	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	4	8	no	—
197	C₄²	Q₁₆×C₁₃₇	1	8	yes	complemented
198	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
199	C₄²	C₂×D₄×C₁₃₇	1	4	yes	—
200	C₄²	#41×C₁₃₇	1	2	yes	—
201	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
202	Q₈	—	2	16	no	—
203	Q₈	—	4	8	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	8	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	8	no	—
212	C₂×C₄	C₄×D₄×C₁₃₇	1	4	yes	—
213	C₂×C₄	#127×C₁₃₇	1	2	yes	—
214	C₂×C₄	#127×C₁₃₇	1	2	yes	—
215	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	4	yes	—
216	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	4	yes	—
217	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	4	yes	—
218	C₂×C₄	#135×C₁₃₇	1	4	yes	—
219	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
220	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
221	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
222	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
223	C₈	—	2	4	no	—
224	C₈	C₄×D₄×C₁₃₇	1	8	yes	—
225	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
226	C₂²	C₂×#127×C₁₃₇	1	1	yes	characteristic
227	C₂²	#532×C₁₃₇	1	1	yes	characteristic
228	C₂²	C₂²×Q₁₆×C₁₃₇	1	2	yes	—
229	C₂²	C₂×#135×C₁₃₇	1	2	yes	—
230	C₄	—	2	2	no	—
231	C₄	—	2	4	no	—
232	C₄	—	4	8	no	—
233	C₄	—	8	4	no	—
234	C₄	C₄×Q₁₆×C₁₃₇	1	8	yes	complemented
235	C₄	#494×C₁₃₇	1	4	yes	direct product factor
236	C₄	C₂×C₄×D₄×C₁₃₇	1	2	yes	—
237	C₄	#532×C₁₃₇	1	2	yes	—
238	C₂	C₄×#127×C₁₃₇	1	1	yes	characteristic
239	C₂	C₂×C₄×Q₁₆×C₁₃₇	1	2	yes	—
240	C₂	C₂×#494×C₁₃₇	1	1	yes	characteristic
241	C₂	#3147×C₁₃₇	1	1	yes	characteristic
242	C₂	#3170×C₁₃₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. #494×C₁₃₇	C₄	direct (N × H)
75. C₄²×C₁₃₇	Q₁₆	semidirect
112. C₄×C₁₃₇	C₄×Q₁₆	semidirect
113. C₄×C₁₃₇	#494	direct (N × H)
121. C₄×#494	C₁₃₇	direct (N × H)
122. C₁₃₇	C₄×#494	direct (N × H)
133. #494	C₄×C₁₃₇	direct (N × H)
197. C₄²	Q₁₆×C₁₃₇	semidirect
234. C₄	C₄×Q₁₆×C₁₃₇	semidirect
235. C₄	#494×C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#494×C₁₃₇	2
C₂×C₄×Q₁₆×C₁₃₇	2
C₄×C₄⋊Q₈×C₁₃₇	2
#1969×C₁₃₇	2
#1956×C₁₃₇	2
#1845×C₁₃₇	2
C₄²×C₈×C₁₃₇	2
C₄×#494	137

Subgroups — C4×#494×C137 #46871074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#494×C₁₃₇ #46871074