Subgroups — C₃₁×#33360 #21490119

Order 7936.

1188 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3499	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2884	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2129	C₂	1	6	yes	maximal, complemented
4	C₃₁×#2126	C₂	1	6	yes	maximal, complemented
5	C₂×C₃₁×2₋¹⁺⁴	C₄	1	1	yes	characteristic, complemented
6	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
7	D₄×Q₈×C₃₁	—	2	2	no	—
8	C₃₁×#547	—	2	6	no	—
9	C₃₁×#536	C₄	1	3	yes	complemented
10	C₃₁×#536	C₂²	1	3	yes	—
11	C₃₁×#532	C₂²	1	3	yes	—
12	C₂×C₃₁×C₄⋊Q₈	C₄	1	3	yes	complemented
13	C₃₁×#519	—	2	2	no	—
14	C₄×#41×C₃₁	C₄	1	1	yes	characteristic, complemented
15	C₃₁×#410	—	2	6	no	—
16	C₃₁×#410	C₂²	1	12	yes	—
17	C₃₁×#354	C₂²	1	12	yes	—
18	C₃₁×#353	C₂²	1	4	yes	—
19	C₃₁×2₋¹⁺⁴	—	4	4	no	—
20	C₃₁×2₋¹⁺⁴	—	4	4	no	—
21	C₂×#41×C₃₁	—	2	3	no	—
22	C₂×#41×C₃₁	C₂×C₄	1	3	yes	—
23	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂×#41×C₃₁	D₄	1	6	yes	—
25	C₂×#41×C₃₁	C₂³	1	3	yes	—
26	C₂²×Q₈×C₃₁	—	2	1	no	—
27	C₂²×Q₈×C₃₁	C₂×C₄	1	3	yes	—
28	C₂²×Q₈×C₃₁	C₂×C₄	1	3	yes	—
29	C₂²×Q₈×C₃₁	D₄	1	2	yes	—
30	C₃₁×C₄⋊Q₈	—	2	6	no	—
31	C₃₁×C₄⋊Q₈	C₂×C₄	1	3	yes	—
32	C₃₁×C₄⋊Q₈	C₂×C₄	1	6	yes	—
33	C₃₁×#127	C₂³	1	1	yes	characteristic
34	C₃₁×#124	C₂×C₄	1	3	yes	—
35	C₃₁×#124	C₂³	1	3	yes	—
36	C₃₁×#123	—	2	6	no	—
37	C₃₁×#123	—	2	6	no	—
38	C₃₁×C₂²⋊Q₈	—	2	6	no	—
39	C₃₁×C₂²⋊Q₈	—	2	6	no	—
40	C₃₁×#121	—	2	6	no	—
41	C₄×Q₈×C₃₁	—	2	1	no	—
42	C₄×Q₈×C₃₁	—	4	8	no	—
43	C₄×D₄×C₃₁	—	2	3	no	—
44	C₄×D₄×C₃₁	—	2	6	no	—
45	C₃₁×#117	—	2	12	no	—
46	C₃₁×#117	C₂×C₄	1	3	yes	—
47	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
48	C₂×C₄²×C₃₁	C₂×C₄	1	3	yes	—
49	C₃₁×#99	—	2	12	no	—
50	C₃₁×#99	C₂³	1	8	yes	—
51	#41×C₃₁	—	2	4	no	—
52	#41×C₃₁	—	4	6	no	—
53	#41×C₃₁	—	4	12	no	—
54	#41×C₃₁	—	4	6	no	—
55	#41×C₃₁	—	4	12	no	—
56	C₂×Q₈×C₃₁	—	2	8	no	—
57	C₂×Q₈×C₃₁	—	2	3	no	—
58	C₂×Q₈×C₃₁	—	4	6	no	—
59	C₂×Q₈×C₃₁	—	4	6	no	—
60	C₂×Q₈×C₃₁	—	4	4	no	—
61	C₂×Q₈×C₃₁	C₂²⋊C₄	1	6	yes	—
62	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×Q₈×C₃₁	C₂²×C₄	1	3	yes	—
64	C₂×Q₈×C₃₁	C₂²×C₄	1	3	yes	—
65	C₂×D₄×C₃₁	—	2	3	no	—
66	C₂×D₄×C₃₁	—	2	2	no	—
67	C₂×D₄×C₃₁	C₂²⋊C₄	1	3	yes	—
68	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×D₄×C₃₁	C₂×D₄	1	6	yes	—
70	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	—	2	3	no	—
72	C₂²×C₄×C₃₁	—	2	3	no	—
73	C₂²×C₄×C₃₁	—	2	3	no	—
74	C₂²×C₄×C₃₁	C₂²⋊C₄	1	6	yes	—
75	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
76	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
77	C₄⋊C₄×C₃₁	—	2	6	no	—
78	C₄⋊C₄×C₃₁	—	2	12	no	—
79	C₄⋊C₄×C₃₁	—	2	3	no	—
80	C₄⋊C₄×C₃₁	—	4	12	no	—
81	C₂²⋊C₄×C₃₁	—	2	3	no	—
82	C₂²⋊C₄×C₃₁	—	2	6	no	—
83	C₂²⋊C₄×C₃₁	—	2	6	no	—
84	C₂²⋊C₄×C₃₁	—	2	8	no	—
85	C₄²×C₃₁	—	2	3	no	—
86	C₄²×C₃₁	—	4	12	no	—
87	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₃₁	C₂²×C₄	1	3	yes	—
89	#33360	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	2	1	no	—
91	C₂³×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
92	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
93	Q₈×C₃₁	—	2	12	no	—
94	Q₈×C₃₁	—	4	3	no	—
95	Q₈×C₃₁	—	4	3	no	—
96	Q₈×C₃₁	—	4	6	no	—
97	Q₈×C₃₁	#99	1	4	yes	—
98	D₄×C₃₁	—	2	6	no	—
99	D₄×C₃₁	—	4	6	no	—
100	D₄×C₃₁	—	4	1	no	—
101	D₄×C₃₁	—	4	1	no	—
102	D₄×C₃₁	—	4	4	no	—
103	D₄×C₃₁	—	4	6	no	—
104	C₂×C₄×C₃₁	—	2	1	no	—
105	C₂×C₄×C₃₁	—	2	12	no	—
106	C₂×C₄×C₃₁	—	2	6	no	—
107	C₂×C₄×C₃₁	—	2	3	no	—
108	C₂×C₄×C₃₁	—	2	3	no	—
109	C₂×C₄×C₃₁	—	4	6	no	—
110	C₂×C₄×C₃₁	—	4	6	no	—
111	C₂×C₄×C₃₁	—	4	6	no	—
112	C₂×C₄×C₃₁	—	4	6	no	—
113	C₂×C₄×C₃₁	—	4	8	no	—
114	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
115	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
116	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
117	#3499	C₂×C₃₁	1	1	yes	characteristic
118	#2884	C₂×C₃₁	1	2	yes	complemented
119	#2129	C₂×C₃₁	1	6	yes	complemented
120	#2126	C₂×C₃₁	1	6	yes	complemented
121	C₂²×C₃₁	—	2	2	no	—
122	C₂²×C₃₁	—	2	3	no	—
123	C₂²×C₃₁	—	4	3	no	—
124	C₂²×C₃₁	—	4	2	no	—
125	C₂²×C₃₁	—	4	2	no	—
126	C₂²×C₃₁	—	4	1	no	—
127	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
128	C₄×C₃₁	—	2	2	no	—
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	2	6	no	—
131	C₄×C₃₁	—	4	3	no	—
132	C₄×C₃₁	—	4	3	no	—
133	C₄×C₃₁	—	8	8	no	—
134	C₄×C₃₁	C₂×#99	1	6	yes	—
135	C₂×2₋¹⁺⁴	C₄×C₃₁	1	1	yes	characteristic, complemented
136	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	1	yes	characteristic
137	D₄×Q₈	—	2	2	no	—
138	#547	—	2	6	no	—
139	#536	C₄×C₃₁	1	3	yes	complemented
140	#536	C₂²×C₃₁	1	3	yes	—
141	#532	C₂²×C₃₁	1	3	yes	—
142	C₂×C₄⋊Q₈	C₄×C₃₁	1	3	yes	complemented
143	#519	—	2	2	no	—
144	C₄×#41	C₄×C₃₁	1	1	yes	characteristic, complemented
145	#410	—	2	6	no	—
146	#410	C₂²×C₃₁	1	12	yes	—
147	#354	C₂²×C₃₁	1	12	yes	—
148	#353	C₂²×C₃₁	1	4	yes	—
149	C₂×C₃₁	—	2	1	no	—
150	C₂×C₃₁	—	4	2	no	—
151	C₂×C₃₁	—	4	3	no	—
152	C₂×C₃₁	—	8	1	no	—
153	C₂×C₃₁	C₂²×#99	1	1	yes	characteristic
154	2₋¹⁺⁴	—	4	4	no	—
155	2₋¹⁺⁴	—	4	4	no	—
156	C₂×#41	—	2	3	no	—
157	C₂×#41	C₂×C₄×C₃₁	1	3	yes	—
158	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
159	C₂×#41	D₄×C₃₁	1	6	yes	—
160	C₂×#41	C₂³×C₃₁	1	3	yes	—
161	C₂²×Q₈	—	2	1	no	—
162	C₂²×Q₈	C₂×C₄×C₃₁	1	3	yes	—
163	C₂²×Q₈	C₂×C₄×C₃₁	1	3	yes	—
164	C₂²×Q₈	D₄×C₃₁	1	2	yes	—
165	C₄⋊Q₈	—	2	6	no	—
166	C₄⋊Q₈	C₂×C₄×C₃₁	1	3	yes	—
167	C₄⋊Q₈	C₂×C₄×C₃₁	1	6	yes	—
168	#127	C₂³×C₃₁	1	1	yes	characteristic
169	#124	C₂×C₄×C₃₁	1	3	yes	—
170	#124	C₂³×C₃₁	1	3	yes	—
171	#123	—	2	6	no	—
172	#123	—	2	6	no	—
173	C₂²⋊Q₈	—	2	6	no	—
174	C₂²⋊Q₈	—	2	6	no	—
175	#121	—	2	6	no	—
176	C₄×Q₈	—	2	1	no	—
177	C₄×Q₈	—	4	8	no	—
178	C₄×D₄	—	2	3	no	—
179	C₄×D₄	—	2	6	no	—
180	#117	—	2	12	no	—
181	#117	C₂×C₄×C₃₁	1	3	yes	—
182	C₂×C₄⋊C₄	—	2	6	no	—
183	C₂×C₄²	C₂×C₄×C₃₁	1	3	yes	—
184	#99	—	2	12	no	—
185	#99	C₂³×C₃₁	1	8	yes	—
186	C₃₁	#33360	1	1	yes	characteristic, direct product factor, Sylow
187	#41	—	2	4	no	—
188	#41	—	4	6	no	—
189	#41	—	4	12	no	—
190	#41	—	4	6	no	—
191	#41	—	4	12	no	—
192	C₂×Q₈	—	2	8	no	—
193	C₂×Q₈	—	2	3	no	—
194	C₂×Q₈	—	4	6	no	—
195	C₂×Q₈	—	4	6	no	—
196	C₂×Q₈	—	4	4	no	—
197	C₂×Q₈	C₂²⋊C₄×C₃₁	1	6	yes	—
198	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
199	C₂×Q₈	C₂²×C₄×C₃₁	1	3	yes	—
200	C₂×Q₈	C₂²×C₄×C₃₁	1	3	yes	—
201	C₂×D₄	—	2	3	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	C₂²⋊C₄×C₃₁	1	3	yes	—
204	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
205	C₂×D₄	C₂×D₄×C₃₁	1	6	yes	—
206	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
207	C₂²×C₄	—	2	3	no	—
208	C₂²×C₄	—	2	3	no	—
209	C₂²×C₄	—	2	3	no	—
210	C₂²×C₄	C₂²⋊C₄×C₃₁	1	6	yes	—
211	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
212	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
213	C₄⋊C₄	—	2	6	no	—
214	C₄⋊C₄	—	2	12	no	—
215	C₄⋊C₄	—	2	3	no	—
216	C₄⋊C₄	—	4	12	no	—
217	C₂²⋊C₄	—	2	3	no	—
218	C₂²⋊C₄	—	2	6	no	—
219	C₂²⋊C₄	—	2	6	no	—
220	C₂²⋊C₄	—	2	8	no	—
221	C₄²	—	2	3	no	—
222	C₄²	—	4	12	no	—
223	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
224	C₄²	C₂²×C₄×C₃₁	1	3	yes	—
225	C₂³	—	2	1	no	—
226	C₂³	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
227	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
228	Q₈	—	2	12	no	—
229	Q₈	—	4	3	no	—
230	Q₈	—	4	3	no	—
231	Q₈	—	4	6	no	—
232	Q₈	C₃₁×#99	1	4	yes	—
233	D₄	—	2	6	no	—
234	D₄	—	4	6	no	—
235	D₄	—	4	1	no	—
236	D₄	—	4	1	no	—
237	D₄	—	4	4	no	—
238	D₄	—	4	6	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	12	no	—
241	C₂×C₄	—	2	6	no	—
242	C₂×C₄	—	2	3	no	—
243	C₂×C₄	—	2	3	no	—
244	C₂×C₄	—	4	6	no	—
245	C₂×C₄	—	4	6	no	—
246	C₂×C₄	—	4	6	no	—
247	C₂×C₄	—	4	6	no	—
248	C₂×C₄	—	4	8	no	—
249	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
250	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
251	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	2	3	no	—
254	C₂²	—	4	3	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	6	no	—
262	C₄	—	4	3	no	—
263	C₄	—	4	3	no	—
264	C₄	—	8	8	no	—
265	C₄	C₂×C₃₁×#99	1	6	yes	—
266	C₂	—	2	1	no	—
267	C₂	—	4	2	no	—
268	C₂	—	4	3	no	—
269	C₂	—	8	1	no	—
270	C₂	C₂²×C₃₁×#99	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	C₂²×#99
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×D₄	C₂⁴×C₃₁
prime radical	C₃₁×#3499	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂×Q₈×C₃₁ ◁ C₂×C₃₁×2₋¹⁺⁴ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2884	C₂	semidirect
3. C₃₁×#2129	C₂	semidirect
4. C₃₁×#2126	C₂	semidirect
5. C₂×C₃₁×2₋¹⁺⁴	C₄	semidirect
9. C₃₁×#536	C₄	semidirect
12. C₂×C₃₁×C₄⋊Q₈	C₄	semidirect
14. C₄×#41×C₃₁	C₄	semidirect
89. #33360	C₃₁	direct (N × H)
118. #2884	C₂×C₃₁	semidirect
119. #2129	C₂×C₃₁	semidirect
120. #2126	C₂×C₃₁	semidirect
135. C₂×2₋¹⁺⁴	C₄×C₃₁	semidirect
139. #536	C₄×C₃₁	semidirect
142. C₂×C₄⋊Q₈	C₄×C₃₁	semidirect
144. C₄×#41	C₄×C₃₁	semidirect
186. C₃₁	#33360	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×#3499	2
C₃₁×#2884	2
C₃₁×#2129	2
C₃₁×#2126	2
#33360	31

Subgroups — C31×#33360 #21490119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33360 #21490119