Subgroups — C₃₁×#29144 #21485903

Order 7936.

722 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×#2859	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2026	C₂	1	4	yes	maximal
4	C₃₁×#1956	C₂	1	2	yes	maximal
5	C₃₁×#1834	C₂	1	2	yes	maximal
6	C₃₁×#1805	C₂	1	4	yes	maximal
7	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#125	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₃₁×#103	C₂²	1	8	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#391	C₂²	1	2	yes	—
15	C₃₁×#389	—	2	2	no	—
16	C₃₁×#389	—	2	2	no	—
17	C₃₁×#389	—	2	2	no	—
18	C₃₁×#384	C₂²	1	1	yes	characteristic
19	C₃₁×#384	C₂²	1	2	yes	—
20	C₃₁×#382	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
22	C₄³×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#340	C₂²	1	8	yes	—
24	C₃₁×#336	C₂²	1	2	yes	—
25	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
26	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
27	C₃₁×C₄⋊Q₈	—	2	2	no	—
28	C₃₁×C₄⋊Q₈	Q₈	1	4	yes	—
29	C₃₁×#125	—	4	4	no	—
30	C₄×Q₈×C₃₁	—	2	8	no	—
31	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
38	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
39	C₂×C₄²×C₃₁	—	2	2	no	—
40	C₂×C₄²×C₃₁	D₄	1	2	yes	—
41	C₂×C₄²×C₃₁	D₄	1	2	yes	—
42	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₃₁×C₄⋊C₈	—	2	4	no	—
46	C₃₁×#103	—	4	8	no	—
47	C₄×C₈×C₃₁	—	2	4	no	—
48	C₃₁×#95	—	2	4	no	—
49	C₃₁×#95	—	2	2	no	—
50	C₃₁×#95	—	2	2	no	—
51	C₂×Q₈×C₃₁	—	2	2	no	—
52	C₂×Q₈×C₃₁	—	2	4	no	—
53	C₂×Q₈×C₃₁	—	2	4	no	—
54	C₂×Q₈×C₃₁	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₃₁	—	2	2	no	—
56	C₂²×C₄×C₃₁	—	2	2	no	—
57	C₂²×C₄×C₃₁	—	2	4	no	—
58	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
63	C₂×C₈×C₃₁	—	2	4	no	—
64	C₂×C₈×C₃₁	—	2	4	no	—
65	C₂×C₈×C₃₁	—	2	4	no	—
66	C₄⋊C₄×C₃₁	—	2	8	no	—
67	C₄⋊C₄×C₃₁	—	2	4	no	—
68	C₄⋊C₄×C₃₁	—	2	8	no	—
69	C₄⋊C₄×C₃₁	—	4	8	no	—
70	C₄⋊C₄×C₃₁	—	4	4	no	—
71	C₄⋊C₄×C₃₁	—	4	4	no	—
72	C₄⋊C₄×C₃₁	—	4	4	no	—
73	C₄⋊C₄×C₃₁	—	4	4	no	—
74	C₄²×C₃₁	—	2	4	no	—
75	C₄²×C₃₁	—	2	4	no	—
76	C₄²×C₃₁	—	2	2	no	—
77	C₄²×C₃₁	—	4	4	no	—
78	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
79	C₄²×C₃₁	#41	1	2	yes	—
80	C₄²×C₃₁	#41	1	4	yes	—
81	#29144	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
83	Q₈×C₃₁	—	2	8	no	—
84	Q₈×C₃₁	—	4	4	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	4	2	no	—
92	C₂×C₄×C₃₁	—	4	2	no	—
93	C₂×C₄×C₃₁	—	4	2	no	—
94	C₂×C₄×C₃₁	—	4	4	no	—
95	C₂×C₄×C₃₁	—	4	4	no	—
96	C₂×C₄×C₃₁	—	4	4	no	—
97	C₂×C₄×C₃₁	—	4	4	no	—
98	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
99	C₂×C₄×C₃₁	#124	1	2	yes	—
100	C₂×C₄×C₃₁	#124	1	2	yes	—
101	C₂×C₄×C₃₁	#135	1	4	yes	—
102	C₂×C₄×C₃₁	#137	1	4	yes	—
103	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
105	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
106	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
107	C₈×C₃₁	—	4	4	no	—
108	#2859	C₂×C₃₁	1	1	yes	characteristic
109	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
110	#2026	C₂×C₃₁	1	4	yes	—
111	#1956	C₂×C₃₁	1	2	yes	—
112	#1834	C₂×C₃₁	1	2	yes	—
113	#1805	C₂×C₃₁	1	4	yes	—
114	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
115	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
116	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
117	C₂²×C₃₁	#535	1	1	yes	characteristic
118	C₂²×C₃₁	C₂×#135	1	2	yes	—
119	C₂²×C₃₁	C₂×#137	1	2	yes	—
120	C₄×C₃₁	—	2	2	no	—
121	C₄×C₃₁	—	2	2	no	—
122	C₄×C₃₁	—	4	4	no	—
123	C₄×C₃₁	—	4	2	no	—
124	C₄×C₃₁	—	8	2	no	—
125	C₄×C₃₁	—	8	4	no	—
126	C₄×C₃₁	#479	1	8	yes	—
127	C₄×C₃₁	#554	1	2	yes	—
128	C₄×C₃₁	#555	1	2	yes	—
129	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
130	C₂×#125	C₂²×C₃₁	1	2	yes	—
131	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
132	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
133	C₂×#103	C₂²×C₃₁	1	8	yes	—
134	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
135	#391	C₂²×C₃₁	1	2	yes	—
136	#389	—	2	2	no	—
137	#389	—	2	2	no	—
138	#389	—	2	2	no	—
139	#384	C₂²×C₃₁	1	1	yes	characteristic
140	#384	C₂²×C₃₁	1	2	yes	—
141	#382	C₂²×C₃₁	1	2	yes	—
142	C₄×C₄⋊C₄	—	2	2	no	—
143	C₄³	C₂²×C₃₁	1	1	yes	characteristic
144	#340	C₂²×C₃₁	1	8	yes	—
145	#336	C₂²×C₃₁	1	2	yes	—
146	C₂×C₃₁	#2563	1	1	yes	characteristic
147	C₂×C₃₁	C₂×#479	1	2	yes	—
148	C₂×C₃₁	#3131	1	1	yes	characteristic
149	C₂×C₃₁	#3139	1	1	yes	characteristic
150	C₂×C₃₁	#3210	1	2	yes	—
151	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
152	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	Q₈×C₃₁	1	4	yes	—
155	#125	—	4	4	no	—
156	C₄×Q₈	—	2	8	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	4	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
164	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	D₄×C₃₁	1	2	yes	—
167	C₂×C₄²	D₄×C₃₁	1	2	yes	—
168	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
170	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
171	C₄⋊C₈	—	2	4	no	—
172	#103	—	4	8	no	—
173	C₄×C₈	—	2	4	no	—
174	#95	—	2	4	no	—
175	#95	—	2	2	no	—
176	#95	—	2	2	no	—
177	C₃₁	#29144	1	1	yes	characteristic, direct product factor, Sylow
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	C₂×Q₈×C₃₁	1	4	yes	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
186	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
189	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	4	no	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	8	no	—
196	C₄⋊C₄	—	4	8	no	—
197	C₄⋊C₄	—	4	4	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₄²	—	2	4	no	—
202	C₄²	—	2	4	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	4	4	no	—
205	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
206	C₄²	#41×C₃₁	1	2	yes	—
207	C₄²	#41×C₃₁	1	4	yes	—
208	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
209	Q₈	—	2	8	no	—
210	Q₈	—	4	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
225	C₂×C₄	C₃₁×#124	1	2	yes	—
226	C₂×C₄	C₃₁×#124	1	2	yes	—
227	C₂×C₄	C₃₁×#135	1	4	yes	—
228	C₂×C₄	C₃₁×#137	1	4	yes	—
229	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
230	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
231	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
232	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
233	C₈	—	4	4	no	—
234	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
235	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
236	C₂²	C₃₁×#535	1	1	yes	characteristic
237	C₂²	C₂×C₃₁×#135	1	2	yes	—
238	C₂²	C₂×C₃₁×#137	1	2	yes	—
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	4	4	no	—
242	C₄	—	4	2	no	—
243	C₄	—	8	2	no	—
244	C₄	—	8	4	no	—
245	C₄	C₃₁×#479	1	8	yes	—
246	C₄	C₃₁×#554	1	2	yes	—
247	C₄	C₃₁×#555	1	2	yes	—
248	C₂	C₃₁×#2563	1	1	yes	characteristic
249	C₂	C₂×C₃₁×#479	1	2	yes	—
250	C₂	C₃₁×#3131	1	1	yes	characteristic
251	C₂	C₃₁×#3139	1	1	yes	characteristic
252	C₂	C₃₁×#3210	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
81. #29144	C₃₁	direct (N × H)
177. C₃₁	#29144	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×#2859	2
C₄×C₃₁×C₄⋊Q₈	2
C₃₁×#2026	2
C₃₁×#1956	2
C₃₁×#1834	2
C₃₁×#1805	2
C₄×C₃₁×C₄⋊C₈	2
#29144	31

Subgroups — C31×#29144 #21485903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29144 #21485903