Subgroups — C₂×C₃₁×#1558 #21474491

Order 7936.

1182 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#432	C₂	1	2	yes	maximal
3	C₃₁×#1558	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#333	C₂	1	4	yes	maximal, complemented
5	C₂³×Q₈×C₃₁	C₄	1	2	yes	—
6	C₂×C₃₁×C₄⋊Q₈	—	2	2	no	—
7	C₂×C₃₁×C₄⋊Q₈	—	2	4	no	—
8	C₂×C₃₁×C₄⋊Q₈	C₄	1	4	yes	—
9	C₂×C₃₁×C₄⋊Q₈	C₄	1	2	yes	—
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₃₁	—	2	2	no	—
13	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#432	—	2	4	no	—
15	C₃₁×#432	C₂²	1	8	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
17	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
18	C₂×C₃₁×#97	—	2	2	no	—
19	C₃₁×#333	C₂²	1	16	yes	complemented
20	C₂²×Q₈×C₃₁	—	2	4	no	—
21	C₂²×Q₈×C₃₁	—	2	2	no	—
22	C₂²×Q₈×C₃₁	—	4	8	no	—
23	C₂²×Q₈×C₃₁	C₂×C₄	1	8	yes	—
24	C₂²×Q₈×C₃₁	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₃₁	—	2	2	no	—
26	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₃₁	D₄	1	2	yes	—
28	C₂²×C₈×C₃₁	—	2	4	no	—
29	C₃₁×C₄⋊Q₈	—	2	4	no	—
30	C₃₁×C₄⋊Q₈	—	2	8	no	—
31	C₃₁×C₄⋊Q₈	—	2	8	no	—
32	C₃₁×C₄⋊Q₈	—	2	4	no	—
33	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
34	C₃₁×C₄⋊Q₈	C₂×C₄	1	8	yes	—
35	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
36	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
40	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₃₁	D₄	1	2	yes	—
43	C₂×C₄²×C₃₁	D₄	1	4	yes	—
44	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
46	C₃₁×C₄⋊C₈	—	2	4	no	—
47	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
48	C₃₁×C₂²⋊C₈	—	2	8	no	—
49	C₃₁×#97	—	2	8	no	—
50	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Q₈×C₃₁	—	2	8	no	—
52	C₂×Q₈×C₃₁	—	2	4	no	—
53	C₂×Q₈×C₃₁	—	4	16	no	—
54	C₂×Q₈×C₃₁	—	4	8	no	—
55	C₂×Q₈×C₃₁	—	4	16	no	—
56	C₂×Q₈×C₃₁	—	4	8	no	—
57	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	2	8	no	—
60	C₂²×C₄×C₃₁	—	2	8	no	—
61	C₂²×C₄×C₃₁	—	2	4	no	—
62	C₂²×C₄×C₃₁	—	4	8	no	—
63	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
64	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₃₁	—	2	8	no	—
72	C₂×C₈×C₃₁	—	4	8	no	—
73	C₄⋊C₄×C₃₁	—	2	16	no	—
74	C₄⋊C₄×C₃₁	—	2	8	no	—
75	C₄⋊C₄×C₃₁	—	2	8	no	—
76	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
78	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
79	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₃₁	C₂×D₄	1	4	yes	—
81	C₄²×C₃₁	C₂×D₄	1	2	yes	—
82	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
83	C₂×#1558	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₃₁	—	2	4	no	—
85	C₂³×C₃₁	#100	1	4	yes	—
86	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
87	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	Q₈×C₃₁	—	4	8	no	—
89	Q₈×C₃₁	—	4	8	no	—
90	Q₈×C₃₁	—	4	16	no	—
91	C₂×C₄×C₃₁	—	2	4	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	4	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	4	no	—
96	C₂×C₄×C₃₁	—	2	8	no	—
97	C₂×C₄×C₃₁	—	2	8	no	—
98	C₂×C₄×C₃₁	—	2	8	no	—
99	C₂×C₄×C₃₁	—	4	16	no	—
100	C₂×C₄×C₃₁	—	4	8	no	—
101	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
102	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
105	C₂×C₄×C₃₁	#137	1	8	yes	—
106	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
108	C₈×C₃₁	—	4	8	no	—
109	C₂²×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
110	C₂×#432	C₂×C₃₁	1	2	yes	—
111	#1558	C₂×C₃₁	1	8	yes	direct product factor
112	C₂×#333	C₂×C₃₁	1	4	yes	complemented
113	C₂²×C₃₁	—	2	8	no	—
114	C₂²×C₃₁	—	2	4	no	—
115	C₂²×C₃₁	C₂×#100	1	4	yes	—
116	C₂²×C₃₁	C₂×#100	1	2	yes	—
117	C₂²×C₃₁	#419	1	4	yes	—
118	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₃₁	—	2	1	no	—
120	C₄×C₃₁	—	2	2	no	—
121	C₄×C₃₁	—	2	1	no	—
122	C₄×C₃₁	—	2	4	no	—
123	C₄×C₃₁	—	4	8	no	—
124	C₄×C₃₁	#419	1	4	yes	—
125	C₄×C₃₁	C₂×#137	1	4	yes	—
126	C₂³×Q₈	C₄×C₃₁	1	2	yes	—
127	C₂×C₄⋊Q₈	—	2	2	no	—
128	C₂×C₄⋊Q₈	—	2	4	no	—
129	C₂×C₄⋊Q₈	C₄×C₃₁	1	4	yes	—
130	C₂×C₄⋊Q₈	C₄×C₃₁	1	2	yes	—
131	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
132	C₂×C₄⋊Q₈	C₂²×C₃₁	1	4	yes	—
133	C₂²×C₄⋊C₄	—	2	2	no	—
134	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
135	#432	—	2	4	no	—
136	#432	C₂²×C₃₁	1	8	yes	—
137	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
138	C₂×C₂²⋊C₈	—	2	2	no	—
139	C₂×#97	—	2	2	no	—
140	#333	C₂²×C₃₁	1	16	yes	complemented
141	C₂×C₃₁	—	2	4	no	—
142	C₂×C₃₁	#1558	1	4	yes	direct product factor
143	C₂×C₃₁	C₂²×#100	1	1	yes	characteristic
144	C₂×C₃₁	C₂×#419	1	2	yes	—
145	C₂²×Q₈	—	2	4	no	—
146	C₂²×Q₈	—	2	2	no	—
147	C₂²×Q₈	—	4	8	no	—
148	C₂²×Q₈	C₂×C₄×C₃₁	1	4	yes	—
149	C₂²×Q₈	C₂×C₄×C₃₁	1	8	yes	—
150	C₂³×C₄	—	2	2	no	—
151	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
152	C₂³×C₄	D₄×C₃₁	1	2	yes	—
153	C₂²×C₈	—	2	4	no	—
154	C₄⋊Q₈	—	2	4	no	—
155	C₄⋊Q₈	—	2	8	no	—
156	C₄⋊Q₈	—	2	8	no	—
157	C₄⋊Q₈	—	2	4	no	—
158	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
159	C₄⋊Q₈	C₂×C₄×C₃₁	1	8	yes	—
160	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
161	C₂×C₄⋊C₄	—	2	4	no	—
162	C₂×C₄⋊C₄	—	2	8	no	—
163	C₂×C₄⋊C₄	—	2	8	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
166	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
167	C₂×C₄²	D₄×C₃₁	1	2	yes	—
168	C₂×C₄²	D₄×C₃₁	1	4	yes	—
169	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
170	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
171	C₄⋊C₈	—	2	4	no	—
172	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
173	C₂²⋊C₈	—	2	8	no	—
174	#97	—	2	8	no	—
175	C₃₁	C₂×#1558	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
177	C₂×Q₈	—	2	8	no	—
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	—	4	16	no	—
180	C₂×Q₈	—	4	8	no	—
181	C₂×Q₈	—	4	16	no	—
182	C₂×Q₈	—	4	8	no	—
183	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	4	8	no	—
189	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
190	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
191	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
192	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
193	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
194	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
196	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	—	4	8	no	—
199	C₄⋊C₄	—	2	16	no	—
200	C₄⋊C₄	—	2	8	no	—
201	C₄⋊C₄	—	2	8	no	—
202	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
203	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
204	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
205	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
206	C₄²	C₂×D₄×C₃₁	1	4	yes	—
207	C₄²	C₂×D₄×C₃₁	1	2	yes	—
208	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
209	C₂³	—	2	4	no	—
210	C₂³	C₃₁×#100	1	4	yes	—
211	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
212	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
213	Q₈	—	4	8	no	—
214	Q₈	—	4	8	no	—
215	Q₈	—	4	16	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	4	16	no	—
225	C₂×C₄	—	4	8	no	—
226	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
227	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
228	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
229	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
230	C₂×C₄	C₃₁×#137	1	8	yes	—
231	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
232	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
233	C₈	—	4	8	no	—
234	C₂²	—	2	8	no	—
235	C₂²	—	2	4	no	—
236	C₂²	C₂×C₃₁×#100	1	4	yes	—
237	C₂²	C₂×C₃₁×#100	1	2	yes	—
238	C₂²	C₃₁×#419	1	4	yes	—
239	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	4	no	—
244	C₄	—	4	8	no	—
245	C₄	C₃₁×#419	1	4	yes	—
246	C₄	C₂×C₃₁×#137	1	4	yes	—
247	C₂	—	2	4	no	—
248	C₂	C₃₁×#1558	1	4	yes	direct product factor
249	C₂	C₂²×C₃₁×#100	1	1	yes	characteristic
250	C₂	C₂×C₃₁×#419	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1558	C₂	direct (N × H)
4. C₂×C₃₁×#333	C₂	semidirect
19. C₃₁×#333	C₂²	semidirect
83. C₂×#1558	C₃₁	direct (N × H)
111. #1558	C₂×C₃₁	direct (N × H)
112. C₂×#333	C₂×C₃₁	semidirect
140. #333	C₂²×C₃₁	semidirect
142. C₂×C₃₁	#1558	direct (N × H)
175. C₃₁	C₂×#1558	direct (N × H)
248. C₂	C₃₁×#1558	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×C₄⋊Q₈	2
C₂×C₃₁×#432	2
C₃₁×#1558	2
C₂×C₃₁×#333	2
C₂×#1558	31

Subgroups — C2×C31×#1558 #21474491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1558 #21474491