Subgroups — C₃₁×#3098 #16843031

Order 3968.

398 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#537	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#485	C₂	1	2	yes	maximal, complemented
4	C₃₁×#484	C₂	1	2	yes	maximal, complemented
5	C₃₁×#482	C₂	1	2	yes	maximal, complemented
6	C₃₁×#480	C₂	1	2	yes	maximal, complemented
7	C₃₁×#428	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#107	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#417	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#416	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#408	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×C₂²⋊Q₈	—	2	1	no	—
17	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
18	C₃₁×#121	—	2	1	no	—
19	C₃₁×#121	—	2	2	no	—
20	C₃₁×#121	C₂²	1	2	yes	—
21	C₄×Q₈×C₃₁	—	2	1	no	—
22	C₄×D₄×C₃₁	—	2	2	no	—
23	C₄×D₄×C₃₁	—	2	1	no	—
24	C₃₁×#117	C₂²	1	1	yes	characteristic
25	C₃₁×#117	C₂²	1	2	yes	—
26	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#107	C₂²	1	2	yes	—
30	C₃₁×#107	C₂²	1	2	yes	—
31	C₃₁×#107	C₂²	1	2	yes	—
32	C₃₁×#106	C₂²	1	2	yes	—
33	C₃₁×#103	C₂²	1	1	yes	characteristic
34	C₃₁×#103	C₂²	1	1	yes	characteristic
35	C₃₁×#103	C₂²	1	2	yes	—
36	C₃₁×#102	C₂²	1	1	yes	characteristic
37	C₃₁×#102	C₂²	1	1	yes	characteristic
38	C₃₁×#102	C₂²	1	2	yes	—
39	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
41	#41×C₃₁	—	2	2	no	—
42	#41×C₃₁	—	2	2	no	—
43	#41×C₃₁	—	2	2	no	—
44	#41×C₃₁	—	2	2	no	—
45	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
46	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₃₁	—	2	1	no	—
48	C₂×D₄×C₃₁	—	2	2	no	—
49	C₂×D₄×C₃₁	D₄	1	2	yes	—
50	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
51	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	—	2	2	no	—
53	C₂²×C₄×C₃₁	—	2	1	no	—
54	C₂²×C₄×C₃₁	—	2	1	no	—
55	C₂²×C₄×C₃₁	—	2	1	no	—
56	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	M₁₆×C₃₁	—	2	2	no	—
59	C₂×C₈×C₃₁	—	2	2	no	—
60	C₂×C₈×C₃₁	C₂³	1	2	yes	—
61	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₄⋊C₄×C₃₁	—	2	2	no	—
67	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
70	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
71	C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂²⋊C₄×C₃₁	—	2	1	no	—
73	C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂²⋊C₄×C₃₁	—	2	2	no	—
75	C₄²×C₃₁	—	2	2	no	—
76	C₄²×C₃₁	—	2	1	no	—
77	C₂³×C₃₁	—	2	1	no	—
78	C₂³×C₃₁	C₂×D₄	1	2	yes	—
79	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
80	Q₈×C₃₁	—	2	2	no	—
81	Q₈×C₃₁	—	2	1	no	—
82	Q₈×C₃₁	—	2	1	no	—
83	D₄×C₃₁	—	2	4	no	—
84	D₄×C₃₁	—	2	2	no	—
85	D₄×C₃₁	—	2	1	no	—
86	D₄×C₃₁	—	2	1	no	—
87	D₄×C₃₁	—	4	2	no	—
88	D₄×C₃₁	—	8	2	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	2	no	—
93	C₂×C₄×C₃₁	—	2	1	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	1	no	—
96	C₂×C₄×C₃₁	—	4	2	no	—
97	C₂×C₄×C₃₁	—	8	1	no	—
98	C₂×C₄×C₃₁	—	8	2	no	—
99	C₂×C₄×C₃₁	—	8	1	no	—
100	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	#41	1	2	yes	—
103	C₂×C₄×C₃₁	#41	1	2	yes	—
104	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
105	C₈×C₃₁	—	2	1	no	—
106	C₈×C₃₁	—	2	1	no	—
107	C₈×C₃₁	—	2	2	no	—
108	#3098	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂²×C₃₁	—	2	1	no	—
110	C₂²×C₃₁	—	2	1	no	—
111	C₂²×C₃₁	—	2	2	no	—
112	C₂²×C₃₁	—	2	2	no	—
113	C₂²×C₃₁	—	2	2	no	—
114	C₂²×C₃₁	—	4	1	no	—
115	C₂²×C₃₁	—	4	1	no	—
116	C₂²×C₃₁	—	4	1	no	—
117	C₂²×C₃₁	#123	1	2	yes	—
118	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
119	C₄×C₃₁	—	2	2	no	—
120	C₄×C₃₁	—	4	1	no	—
121	C₄×C₃₁	—	4	2	no	—
122	C₄×C₃₁	—	4	1	no	—
123	C₄×C₃₁	—	4	2	no	—
124	C₄×C₃₁	—	4	1	no	—
125	C₄×C₃₁	#123	1	2	yes	—
126	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
127	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
128	#537	C₂×C₃₁	1	1	yes	characteristic
129	#520	C₂×C₃₁	1	1	yes	characteristic, complemented
130	#485	C₂×C₃₁	1	2	yes	complemented
131	#484	C₂×C₃₁	1	2	yes	complemented
132	#482	C₂×C₃₁	1	2	yes	complemented
133	#480	C₂×C₃₁	1	2	yes	complemented
134	#428	C₂×C₃₁	1	1	yes	characteristic, complemented
135	C₂×#107	C₂×C₃₁	1	1	yes	characteristic, complemented
136	#417	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#416	C₂×C₃₁	1	1	yes	characteristic, complemented
138	#408	C₂×C₃₁	1	1	yes	characteristic, complemented
139	C₂×C₃₁	—	2	2	no	—
140	C₂×C₃₁	—	4	2	no	—
141	C₂×C₃₁	—	8	1	no	—
142	C₂×C₃₁	C₂×#123	1	1	yes	characteristic
143	C₂×C₃₁	#576	1	1	yes	characteristic
144	C₂×C₃₁	#578	1	1	yes	characteristic
145	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
146	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
147	C₂×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
148	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
149	C₂²⋊Q₈	—	2	1	no	—
150	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
151	#121	—	2	1	no	—
152	#121	—	2	2	no	—
153	#121	C₂²×C₃₁	1	2	yes	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×D₄	—	2	2	no	—
156	C₄×D₄	—	2	1	no	—
157	#117	C₂²×C₃₁	1	1	yes	characteristic
158	#117	C₂²×C₃₁	1	2	yes	—
159	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
160	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
161	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
162	#107	C₂²×C₃₁	1	2	yes	—
163	#107	C₂²×C₃₁	1	2	yes	—
164	#107	C₂²×C₃₁	1	2	yes	—
165	#106	C₂²×C₃₁	1	2	yes	—
166	#103	C₂²×C₃₁	1	1	yes	characteristic
167	#103	C₂²×C₃₁	1	1	yes	characteristic
168	#103	C₂²×C₃₁	1	2	yes	—
169	#102	C₂²×C₃₁	1	1	yes	characteristic
170	#102	C₂²×C₃₁	1	1	yes	characteristic
171	#102	C₂²×C₃₁	1	2	yes	—
172	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
173	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
174	C₃₁	#3098	1	1	yes	characteristic, direct product factor, Sylow
175	#41	—	2	2	no	—
176	#41	—	2	2	no	—
177	#41	—	2	2	no	—
178	#41	—	2	2	no	—
179	C₂×Q₈	D₄×C₃₁	1	1	yes	characteristic
180	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
181	C₂×D₄	—	2	1	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	D₄×C₃₁	1	2	yes	—
184	C₂×D₄	D₄×C₃₁	1	1	yes	characteristic
185	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
191	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
192	M₁₆	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	C₂³×C₃₁	1	2	yes	—
195	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
196	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
202	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
203	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
204	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	C₂×D₄×C₃₁	1	2	yes	—
213	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
214	Q₈	—	2	2	no	—
215	Q₈	—	2	1	no	—
216	Q₈	—	2	1	no	—
217	D₄	—	2	4	no	—
218	D₄	—	2	2	no	—
219	D₄	—	2	1	no	—
220	D₄	—	2	1	no	—
221	D₄	—	4	2	no	—
222	D₄	—	8	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	8	1	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	8	1	no	—
234	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
235	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
236	C₂×C₄	#41×C₃₁	1	2	yes	—
237	C₂×C₄	#41×C₃₁	1	2	yes	—
238	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	2	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	1	no	—
250	C₂²	C₃₁×#123	1	2	yes	—
251	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	1	no	—
258	C₄	C₃₁×#123	1	2	yes	—
259	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
260	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	4	2	no	—
263	C₂	—	8	1	no	—
264	C₂	C₂×C₃₁×#123	1	1	yes	characteristic
265	C₂	C₃₁×#576	1	1	yes	characteristic
266	C₂	C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#537	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#520	C₂	semidirect
3. C₃₁×#485	C₂	semidirect
4. C₃₁×#484	C₂	semidirect
5. C₃₁×#482	C₂	semidirect
6. C₃₁×#480	C₂	semidirect
7. C₃₁×#428	C₂	semidirect
8. C₂×C₃₁×#107	C₂	semidirect
9. C₃₁×#417	C₂	semidirect
10. C₃₁×#416	C₂	semidirect
11. C₃₁×#408	C₂	semidirect
108. #3098	C₃₁	direct (N × H)
129. #520	C₂×C₃₁	semidirect
130. #485	C₂×C₃₁	semidirect
131. #484	C₂×C₃₁	semidirect
132. #482	C₂×C₃₁	semidirect
133. #480	C₂×C₃₁	semidirect
134. #428	C₂×C₃₁	semidirect
135. C₂×#107	C₂×C₃₁	semidirect
136. #417	C₂×C₃₁	semidirect
137. #416	C₂×C₃₁	semidirect
138. #408	C₂×C₃₁	semidirect
174. C₃₁	#3098	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×#537	2
C₃₁×#520	2
C₃₁×#485	2
C₃₁×#484	2
C₃₁×#482	2
C₃₁×#480	2
C₃₁×#428	2
C₂×C₃₁×#107	2
C₃₁×#417	2
C₃₁×#416	2
C₃₁×#408	2
#3098	31

Subgroups — C31×#3098 #16843031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3098 #16843031