Subgroups — C₂×C₄×C₃₁×#106 #21477206

Order 7936.

1358 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#106	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal
4	C₄×C₃₁×#106	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#340	C₂	1	2	yes	maximal
6	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
9	C₂²×C₄²×C₃₁	—	2	2	no	—
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#106	C₄	1	16	yes	direct product factor
12	C₂×C₃₁×#106	C₂²	1	8	yes	—
13	C₂×C₄×C₈×C₃₁	C₄	1	8	yes	complemented
14	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
15	C₄×C₄⋊C₄×C₃₁	—	2	4	no	—
16	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	—
17	C₂×C₃₁×#95	—	2	2	no	—
18	C₃₁×#340	C₂²	1	8	yes	—
19	C₂³×C₄×C₃₁	—	2	4	no	—
20	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₃₁	—	2	4	no	—
24	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
28	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
29	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
31	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	16	yes	—
32	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
33	C₂×C₄²×C₃₁	—	2	8	no	—
34	C₂×C₄²×C₃₁	—	4	8	no	—
35	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₃₁	D₄	1	2	yes	—
37	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
38	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
39	C₃₁×#106	C₂×C₄	1	32	yes	direct product factor
40	C₄×C₈×C₃₁	C₂×C₄	1	16	yes	complemented
41	C₃₁×#95	—	2	8	no	—
42	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	—	2	4	no	—
44	C₂²×C₄×C₃₁	—	2	16	no	—
45	C₂²×C₄×C₃₁	—	2	8	no	—
46	C₂²×C₄×C₃₁	—	4	8	no	—
47	C₂²×C₄×C₃₁	—	4	8	no	—
48	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
49	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₃₁	SD₁₆	1	8	yes	—
51	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₃₁	#41	1	4	yes	—
58	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
59	C₂×C₈×C₃₁	—	2	8	no	—
60	C₂×C₈×C₃₁	—	2	4	no	—
61	C₂×C₈×C₃₁	C₄²	1	16	yes	complemented
62	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
65	C₄⋊C₄×C₃₁	—	2	16	no	—
66	C₄⋊C₄×C₃₁	—	2	8	no	—
67	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
68	C₄²×C₃₁	—	4	16	no	—
69	C₄²×C₃₁	C₄⋊C₄	1	8	yes	—
70	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
71	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
72	C₂×C₄×#106	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
74	C₂³×C₃₁	C₄×D₄	1	4	yes	—
75	C₂³×C₃₁	C₂×SD₁₆	1	4	yes	—
76	C₂³×C₃₁	#135	1	4	yes	—
77	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₃₁	—	2	4	no	—
79	C₂×C₄×C₃₁	—	2	8	no	—
80	C₂×C₄×C₃₁	—	2	16	no	—
81	C₂×C₄×C₃₁	—	4	8	no	—
82	C₂×C₄×C₃₁	—	4	8	no	—
83	C₂×C₄×C₃₁	—	4	32	no	—
84	C₂×C₄×C₃₁	#106	1	16	yes	direct product factor
85	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
86	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
87	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₃₁	#117	1	4	yes	—
90	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₃₁	C₄×Q₈	1	8	yes	—
92	C₂×C₄×C₃₁	C₂×SD₁₆	1	8	yes	—
93	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
95	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
97	C₈×C₃₁	—	2	4	no	—
98	C₈×C₃₁	C₂×C₄²	1	8	yes	complemented
99	C₂²×#106	C₂×C₃₁	1	2	yes	—
100	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
101	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	2	yes	—
102	C₄×#106	C₂×C₃₁	1	8	yes	direct product factor
103	C₂×#340	C₂×C₃₁	1	2	yes	—
104	C₂²×C₃₁	C₄×C₄⋊C₄	1	4	yes	—
105	C₂²×C₃₁	C₂×#106	1	4	yes	—
106	C₂²×C₃₁	#427	1	4	yes	—
107	C₂²×C₃₁	C₄×SD₁₆	1	16	yes	—
108	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
110	C₂²×C₃₁	C₂²×SD₁₆	1	2	yes	—
111	C₂²×C₃₁	C₂×#135	1	2	yes	—
112	C₄×C₃₁	—	2	4	no	—
113	C₄×C₃₁	—	4	16	no	—
114	C₄×C₃₁	C₄×C₄⋊C₄	1	4	yes	—
115	C₄×C₃₁	C₂×#106	1	8	yes	direct product factor
116	C₄×C₃₁	C₂²×C₄²	1	1	yes	characteristic
117	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
118	C₄×C₃₁	C₂×C₄×Q₈	1	2	yes	—
119	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
120	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
121	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
122	C₂²×C₄²	—	2	2	no	—
123	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
124	C₂×#106	C₄×C₃₁	1	16	yes	direct product factor
125	C₂×#106	C₂²×C₃₁	1	8	yes	—
126	C₂×C₄×C₈	C₄×C₃₁	1	8	yes	complemented
127	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
128	C₄×C₄⋊C₄	—	2	4	no	—
129	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	—
130	C₂×#95	—	2	2	no	—
131	#340	C₂²×C₃₁	1	8	yes	—
132	C₂×C₃₁	C₄×#106	1	8	yes	direct product factor
133	C₂×C₃₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₃₁	C₂²×#106	1	1	yes	characteristic
135	C₂×C₃₁	C₂×#427	1	1	yes	characteristic
136	C₂×C₃₁	C₂×C₄×SD₁₆	1	4	yes	—
137	C₂³×C₄	—	2	4	no	—
138	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
139	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
140	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
143	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
144	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
145	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
146	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
147	C₂×C₄⋊C₄	—	2	4	no	—
148	C₂×C₄⋊C₄	—	2	8	no	—
149	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	16	yes	—
150	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
151	C₂×C₄²	—	2	8	no	—
152	C₂×C₄²	—	4	8	no	—
153	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
154	C₂×C₄²	D₄×C₃₁	1	2	yes	—
155	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
156	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
157	#106	C₂×C₄×C₃₁	1	32	yes	direct product factor
158	C₄×C₈	C₂×C₄×C₃₁	1	16	yes	complemented
159	#95	—	2	8	no	—
160	C₃₁	C₂×C₄×#106	1	1	yes	characteristic, direct product factor, Sylow
161	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	16	no	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	—	4	8	no	—
166	C₂²×C₄	—	4	8	no	—
167	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
168	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
169	C₂²×C₄	SD₁₆×C₃₁	1	8	yes	—
170	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
171	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
172	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
173	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
174	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
175	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
176	C₂²×C₄	#41×C₃₁	1	4	yes	—
177	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
178	C₂×C₈	—	2	8	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	C₄²×C₃₁	1	16	yes	complemented
181	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
182	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
183	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
184	C₄⋊C₄	—	2	16	no	—
185	C₄⋊C₄	—	2	8	no	—
186	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
187	C₄²	—	4	16	no	—
188	C₄²	C₄⋊C₄×C₃₁	1	8	yes	—
189	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
190	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
191	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
192	C₂³	C₄×D₄×C₃₁	1	4	yes	—
193	C₂³	C₂×SD₁₆×C₃₁	1	4	yes	—
194	C₂³	C₃₁×#135	1	4	yes	—
195	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
196	C₂×C₄	—	2	8	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	16	no	—
199	C₂×C₄	—	4	32	no	—
200	C₂×C₄	—	4	8	no	—
201	C₂×C₄	—	4	8	no	—
202	C₂×C₄	C₃₁×#106	1	16	yes	direct product factor
203	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
204	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
205	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
206	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
207	C₂×C₄	C₃₁×#117	1	4	yes	—
208	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
209	C₂×C₄	C₄×Q₈×C₃₁	1	8	yes	—
210	C₂×C₄	C₂×SD₁₆×C₃₁	1	8	yes	—
211	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
212	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
213	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
215	C₈	—	2	4	no	—
216	C₈	C₂×C₄²×C₃₁	1	8	yes	complemented
217	C₂²	C₄×C₄⋊C₄×C₃₁	1	4	yes	—
218	C₂²	C₂×C₃₁×#106	1	4	yes	—
219	C₂²	C₃₁×#427	1	4	yes	—
220	C₂²	C₄×SD₁₆×C₃₁	1	16	yes	—
221	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
222	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
223	C₂²	C₂²×SD₁₆×C₃₁	1	2	yes	—
224	C₂²	C₂×C₃₁×#135	1	2	yes	—
225	C₄	—	2	4	no	—
226	C₄	—	4	16	no	—
227	C₄	C₄×C₄⋊C₄×C₃₁	1	4	yes	—
228	C₄	C₂×C₃₁×#106	1	8	yes	direct product factor
229	C₄	C₂²×C₄²×C₃₁	1	1	yes	characteristic
230	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
231	C₄	C₂×C₄×Q₈×C₃₁	1	2	yes	—
232	C₂	C₄×C₃₁×#106	1	8	yes	direct product factor
233	C₂	C₂×C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
234	C₂	C₂²×C₃₁×#106	1	1	yes	characteristic
235	C₂	C₂×C₃₁×#427	1	1	yes	characteristic
236	C₂	C₂×C₄×SD₁₆×C₃₁	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₃₁	D₄
commutator	C₄	C₂²×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle, prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄×C₃₁×#106	C₂	direct (N × H)
11. C₂×C₃₁×#106	C₄	direct (N × H)
13. C₂×C₄×C₈×C₃₁	C₄	semidirect
39. C₃₁×#106	C₂×C₄	direct (N × H)
40. C₄×C₈×C₃₁	C₂×C₄	semidirect
61. C₂×C₈×C₃₁	C₄²	semidirect
72. C₂×C₄×#106	C₃₁	direct (N × H)
84. C₂×C₄×C₃₁	#106	direct (N × H)
98. C₈×C₃₁	C₂×C₄²	semidirect
102. C₄×#106	C₂×C₃₁	direct (N × H)
115. C₄×C₃₁	C₂×#106	direct (N × H)
124. C₂×#106	C₄×C₃₁	direct (N × H)
126. C₂×C₄×C₈	C₄×C₃₁	semidirect
132. C₂×C₃₁	C₄×#106	direct (N × H)
157. #106	C₂×C₄×C₃₁	direct (N × H)
158. C₄×C₈	C₂×C₄×C₃₁	semidirect
160. C₃₁	C₂×C₄×#106	direct (N × H)
180. C₂×C₈	C₄²×C₃₁	semidirect
202. C₂×C₄	C₃₁×#106	direct (N × H)
216. C₈	C₂×C₄²×C₃₁	semidirect
228. C₄	C₂×C₃₁×#106	direct (N × H)
232. C₂	C₄×C₃₁×#106	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#106	2
C₂²×C₄×C₈×C₃₁	2
C₂×C₄×C₄⋊C₄×C₃₁	2
C₄×C₃₁×#106	2
C₂×C₃₁×#340	2
C₂×C₄×#106	31

Subgroups — C2×C4×C31×#106 #21477206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₃₁×#106 #21477206