Subgroups — C₃₁×#3329 #16843262

Order 3968.

386 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#555	C₂	1	1	yes	maximal, characteristic
2	D₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#485	C₂	1	2	yes	maximal, complemented
5	C₃₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#468	C₂	1	2	yes	maximal, complemented
7	C₃₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#448	C₂	1	2	yes	maximal, complemented
9	C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#419	C₂	1	2	yes	maximal, complemented
12	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
13	C₂×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
16	C₃₁×C₄⋊Q₈	—	2	1	no	—
17	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₃₁×#124	—	2	2	no	—
20	C₃₁×C₂²⋊Q₈	—	2	2	no	—
21	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₃₁×#121	—	2	1	no	—
23	C₃₁×#121	C₂²	1	2	yes	—
24	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₃₁	—	2	1	no	—
26	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#117	—	2	1	no	—
29	C₃₁×#117	C₂²	1	2	yes	—
30	C₃₁×#107	C₂²	1	2	yes	—
31	C₃₁×#106	C₂²	1	1	yes	characteristic
32	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₃₁×#103	C₂²	1	1	yes	characteristic
34	C₃₁×#103	C₂²	1	2	yes	—
35	C₃₁×#103	C₂²	1	2	yes	—
36	C₃₁×#103	C₂²	1	2	yes	—
37	C₃₁×#103	C₂²	1	2	yes	—
38	C₃₁×#102	C₂²	1	1	yes	characteristic
39	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₃₁	—	2	2	no	—
42	C₂×Q₈×C₃₁	—	2	1	no	—
43	C₂×Q₈×C₃₁	—	2	4	no	—
44	C₂×Q₈×C₃₁	—	4	2	no	—
45	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
46	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₃₁	—	2	2	no	—
48	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	—	2	2	no	—
51	C₂²×C₄×C₃₁	—	2	1	no	—
52	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
53	SD₁₆×C₃₁	—	2	2	no	—
54	SD₁₆×C₃₁	—	4	2	no	—
55	M₁₆×C₃₁	—	4	2	no	—
56	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₃₁	C₂³	1	2	yes	—
59	C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
68	C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂²⋊C₄×C₃₁	—	2	2	no	—
70	C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₂²⋊C₄×C₃₁	—	2	1	no	—
72	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
73	C₄²×C₃₁	—	2	2	no	—
74	C₄²×C₃₁	—	2	1	no	—
75	C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂³×C₃₁	—	2	1	no	—
77	C₂³×C₃₁	C₂×D₄	1	2	yes	—
78	Q₈×C₃₁	—	2	1	no	—
79	Q₈×C₃₁	—	2	2	no	—
80	Q₈×C₃₁	—	2	2	no	—
81	Q₈×C₃₁	—	4	4	no	—
82	Q₈×C₃₁	—	8	1	no	—
83	Q₈×C₃₁	#41	1	2	yes	—
84	D₄×C₃₁	—	2	2	no	—
85	D₄×C₃₁	—	2	1	no	—
86	D₄×C₃₁	—	2	1	no	—
87	D₄×C₃₁	—	8	2	no	—
88	C₂×C₄×C₃₁	—	2	1	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	1	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	4	4	no	—
96	C₂×C₄×C₃₁	—	4	2	no	—
97	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
101	C₈×C₃₁	—	2	1	no	—
102	C₈×C₃₁	—	2	1	no	—
103	C₈×C₃₁	—	2	2	no	—
104	#3329	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₃₁	—	2	2	no	—
106	C₂²×C₃₁	—	2	2	no	—
107	C₂²×C₃₁	—	2	2	no	—
108	C₂²×C₃₁	—	4	1	no	—
109	C₂²×C₃₁	—	4	1	no	—
110	C₂²×C₃₁	—	4	1	no	—
111	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
112	C₄×C₃₁	—	2	2	no	—
113	C₄×C₃₁	—	2	2	no	—
114	C₄×C₃₁	—	2	1	no	—
115	C₄×C₃₁	—	4	2	no	—
116	C₄×C₃₁	—	4	1	no	—
117	C₄×C₃₁	—	4	1	no	—
118	C₄×C₃₁	—	4	2	no	—
119	C₄×C₃₁	#137	1	2	yes	—
120	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
121	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
122	#555	C₂×C₃₁	1	1	yes	characteristic
123	D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
124	#487	C₂×C₃₁	1	1	yes	characteristic, complemented
125	#485	C₂×C₃₁	1	2	yes	complemented
126	#477	C₂×C₃₁	1	1	yes	characteristic, complemented
127	#468	C₂×C₃₁	1	2	yes	complemented
128	#461	C₂×C₃₁	1	1	yes	characteristic, complemented
129	#448	C₂×C₃₁	1	2	yes	complemented
130	C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
131	#436	C₂×C₃₁	1	1	yes	characteristic, complemented
132	#419	C₂×C₃₁	1	2	yes	complemented
133	C₂×C₃₁	—	4	2	no	—
134	C₂×C₃₁	—	8	1	no	—
135	C₂×C₃₁	#546	1	1	yes	characteristic
136	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
137	C₂×C₃₁	#577	1	1	yes	characteristic
138	C₂²×Q₈	C₂²×C₃₁	1	2	yes	—
139	C₂×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
140	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
141	C₂×M₁₆	C₂²×C₃₁	1	2	yes	—
142	C₄⋊Q₈	—	2	1	no	—
143	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
144	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
145	#124	—	2	2	no	—
146	C₂²⋊Q₈	—	2	2	no	—
147	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
148	#121	—	2	1	no	—
149	#121	C₂²×C₃₁	1	2	yes	—
150	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
151	C₄×D₄	—	2	1	no	—
152	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
153	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
154	#117	—	2	1	no	—
155	#117	C₂²×C₃₁	1	2	yes	—
156	#107	C₂²×C₃₁	1	2	yes	—
157	#106	C₂²×C₃₁	1	1	yes	characteristic
158	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
159	#103	C₂²×C₃₁	1	1	yes	characteristic
160	#103	C₂²×C₃₁	1	2	yes	—
161	#103	C₂²×C₃₁	1	2	yes	—
162	#103	C₂²×C₃₁	1	2	yes	—
163	#103	C₂²×C₃₁	1	2	yes	—
164	#102	C₂²×C₃₁	1	1	yes	characteristic
165	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
166	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
167	C₃₁	#3329	1	1	yes	characteristic, direct product factor, Sylow
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	—	2	1	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
173	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	D₄×C₃₁	1	1	yes	characteristic
176	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
180	SD₁₆	—	2	2	no	—
181	SD₁₆	—	4	2	no	—
182	M₁₆	—	4	2	no	—
183	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
184	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
185	C₂×C₈	C₂³×C₃₁	1	2	yes	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
192	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
193	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
194	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	—	2	2	no	—
198	C₂²⋊C₄	—	2	1	no	—
199	C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	1	no	—
202	C₄²	C₂³×C₃₁	1	1	yes	characteristic
203	C₂³	—	2	1	no	—
204	C₂³	C₂×D₄×C₃₁	1	2	yes	—
205	Q₈	—	2	1	no	—
206	Q₈	—	2	2	no	—
207	Q₈	—	2	2	no	—
208	Q₈	—	4	4	no	—
209	Q₈	—	8	1	no	—
210	Q₈	#41×C₃₁	1	2	yes	—
211	D₄	—	2	2	no	—
212	D₄	—	2	1	no	—
213	D₄	—	2	1	no	—
214	D₄	—	8	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
225	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
226	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
227	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
228	C₈	—	2	1	no	—
229	C₈	—	2	1	no	—
230	C₈	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	4	1	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	1	no	—
237	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	2	1	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	1	no	—
245	C₄	C₃₁×#137	1	2	yes	—
246	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
247	C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
248	C₂	—	4	2	no	—
249	C₂	—	8	1	no	—
250	C₂	C₃₁×#546	1	1	yes	characteristic
251	C₂	C₂×C₃₁×#137	1	1	yes	characteristic
252	C₂	C₃₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#555	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. D₄×Q₈×C₃₁	C₂	semidirect
3. C₃₁×#487	C₂	semidirect
4. C₃₁×#485	C₂	semidirect
5. C₃₁×#477	C₂	semidirect
6. C₃₁×#468	C₂	semidirect
7. C₃₁×#461	C₂	semidirect
8. C₃₁×#448	C₂	semidirect
9. C₄×SD₁₆×C₃₁	C₂	semidirect
10. C₃₁×#436	C₂	semidirect
11. C₃₁×#419	C₂	semidirect
104. #3329	C₃₁	direct (N × H)
123. D₄×Q₈	C₂×C₃₁	semidirect
124. #487	C₂×C₃₁	semidirect
125. #485	C₂×C₃₁	semidirect
126. #477	C₂×C₃₁	semidirect
127. #468	C₂×C₃₁	semidirect
128. #461	C₂×C₃₁	semidirect
129. #448	C₂×C₃₁	semidirect
130. C₄×SD₁₆	C₂×C₃₁	semidirect
131. #436	C₂×C₃₁	semidirect
132. #419	C₂×C₃₁	semidirect
167. C₃₁	#3329	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#555	2
D₄×Q₈×C₃₁	2
C₃₁×#487	2
C₃₁×#485	2
C₃₁×#477	2
C₃₁×#468	2
C₃₁×#461	2
C₃₁×#448	2
C₄×SD₁₆×C₃₁	2
C₃₁×#436	2
C₃₁×#419	2
#3329	31

Subgroups — C31×#3329 #16843262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3329 #16843262