Subgroups — C₃₁×C₄⋊#338 #21467597

Order 7936.

762 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#423	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#338	C₂	1	4	yes	maximal
4	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
5	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
6	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#423	—	2	2	no	—
8	C₃₁×#423	—	2	2	no	—
9	C₃₁×#423	—	2	4	no	—
10	C₂×C₃₁×C₄⋊C₈	C₄	1	8	yes	—
11	C₄×M₁₆×C₃₁	—	2	2	no	—
12	C₄×M₁₆×C₃₁	—	2	2	no	—
13	C₂×C₃₁×#97	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
15	C₃₁×#338	—	2	8	no	—
16	C₂³×C₄×C₃₁	Q₈	1	2	yes	—
17	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₃₁	—	2	4	no	—
19	C₂×M₁₆×C₃₁	—	2	4	no	—
20	C₂×M₁₆×C₃₁	—	2	4	no	—
21	C₂×M₁₆×C₃₁	—	2	4	no	—
22	C₂×M₁₆×C₃₁	—	2	8	no	—
23	C₂×M₁₆×C₃₁	—	2	8	no	—
24	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
25	C₂×M₁₆×C₃₁	Q₈	1	4	yes	—
26	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₃₁	D₄	1	2	yes	—
32	C₂×C₄²×C₃₁	D₄	1	2	yes	—
33	C₂×C₄²×C₃₁	D₄	1	2	yes	—
34	C₃₁×C₄⋊C₈	—	2	8	no	—
35	C₃₁×C₄⋊C₈	—	2	8	no	—
36	C₃₁×#97	—	2	4	no	—
37	C₄×C₈×C₃₁	—	2	2	no	—
38	C₄×C₈×C₃₁	—	2	2	no	—
39	C₂⁴×C₃₁	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	—	4	4	no	—
41	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₃₁	#41	1	4	yes	—
52	M₁₆×C₃₁	—	2	8	no	—
53	M₁₆×C₃₁	—	2	8	no	—
54	M₁₆×C₃₁	—	4	8	no	—
55	M₁₆×C₃₁	—	4	4	no	—
56	M₁₆×C₃₁	—	4	4	no	—
57	C₂×C₈×C₃₁	—	2	4	no	—
58	C₂×C₈×C₃₁	—	2	4	no	—
59	C₂×C₈×C₃₁	—	2	8	no	—
60	C₂×C₈×C₃₁	—	2	16	no	—
61	C₂×C₈×C₃₁	C₄⋊C₄	1	8	yes	—
62	C₄²×C₃₁	C₄²	1	2	yes	—
63	C₄²×C₃₁	C₄²	1	2	yes	—
64	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
68	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
69	C₄⋊#338	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₃₁	—	4	2	no	—
71	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
73	C₂³×C₃₁	#125	1	2	yes	—
74	C₂³×C₃₁	C₄⋊Q₈	1	2	yes	—
75	C₂×C₄×C₃₁	—	2	4	no	—
76	C₂×C₄×C₃₁	—	2	8	no	—
77	C₂×C₄×C₃₁	—	4	2	no	—
78	C₂×C₄×C₃₁	—	4	4	no	—
79	C₂×C₄×C₃₁	#95	1	2	yes	—
80	C₂×C₄×C₃₁	#95	1	4	yes	—
81	C₂×C₄×C₃₁	#95	1	2	yes	—
82	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
85	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₃₁	#117	1	2	yes	—
87	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
88	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
89	C₂×C₄×C₃₁	#121	1	4	yes	—
90	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
91	C₂×C₄×C₃₁	#124	1	2	yes	—
92	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
93	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
94	C₈×C₃₁	—	2	8	no	—
95	C₈×C₃₁	—	4	8	no	—
96	C₂×#423	C₂×C₃₁	1	2	yes	—
97	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
98	C₂×#338	C₂×C₃₁	1	4	yes	—
99	C₂²×C₃₁	—	2	2	no	—
100	C₂²×C₃₁	—	2	4	no	—
101	C₂²×C₃₁	—	4	2	no	—
102	C₂²×C₃₁	—	4	2	no	—
103	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
104	C₂²×C₃₁	#382	1	1	yes	characteristic
105	C₂²×C₃₁	#384	1	1	yes	characteristic
106	C₂²×C₃₁	#389	1	4	yes	—
107	C₄×C₃₁	—	2	2	no	—
108	C₄×C₃₁	—	2	4	no	—
109	C₄×C₃₁	#338	1	4	yes	complemented
110	C₄×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
111	C₄×C₃₁	#380	1	2	yes	—
112	C₄×C₃₁	#390	1	2	yes	—
113	C₄×C₃₁	#391	1	2	yes	—
114	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
115	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
116	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
117	#423	—	2	2	no	—
118	#423	—	2	2	no	—
119	#423	—	2	4	no	—
120	C₂×C₄⋊C₈	C₄×C₃₁	1	8	yes	—
121	C₄×M₁₆	—	2	2	no	—
122	C₄×M₁₆	—	2	2	no	—
123	C₂×#97	C₄×C₃₁	1	2	yes	—
124	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
125	#338	—	2	8	no	—
126	C₂×C₃₁	—	2	1	no	—
127	C₂×C₃₁	—	2	1	no	—
128	C₂×C₃₁	—	2	4	no	—
129	C₂×C₃₁	C₄⋊#95	1	1	yes	characteristic
130	C₂×C₃₁	C₂×#338	1	2	yes	—
131	C₂³×C₄	Q₈×C₃₁	1	2	yes	—
132	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
133	C₂×M₁₆	—	2	4	no	—
134	C₂×M₁₆	—	2	4	no	—
135	C₂×M₁₆	—	2	8	no	—
136	C₂×M₁₆	—	2	4	no	—
137	C₂×M₁₆	—	2	4	no	—
138	C₂×M₁₆	—	2	8	no	—
139	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
140	C₂×M₁₆	Q₈×C₃₁	1	4	yes	—
141	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
142	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
143	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
144	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
145	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
146	C₂×C₄²	D₄×C₃₁	1	2	yes	—
147	C₂×C₄²	D₄×C₃₁	1	2	yes	—
148	C₂×C₄²	D₄×C₃₁	1	2	yes	—
149	C₄⋊C₈	—	2	8	no	—
150	C₄⋊C₈	—	2	8	no	—
151	#97	—	2	4	no	—
152	C₄×C₈	—	2	2	no	—
153	C₄×C₈	—	2	2	no	—
154	C₃₁	C₄⋊#338	1	1	yes	characteristic, direct product factor, Sylow
155	C₂⁴	C₂×Q₈×C₃₁	1	1	yes	characteristic
156	C₂²×C₄	—	4	4	no	—
157	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
158	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
159	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
160	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
161	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
162	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
163	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
164	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
166	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
167	C₂²×C₄	#41×C₃₁	1	4	yes	—
168	M₁₆	—	2	8	no	—
169	M₁₆	—	2	8	no	—
170	M₁₆	—	4	8	no	—
171	M₁₆	—	4	4	no	—
172	M₁₆	—	4	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	—	2	16	no	—
177	C₂×C₈	C₄⋊C₄×C₃₁	1	8	yes	—
178	C₄²	C₄²×C₃₁	1	2	yes	—
179	C₄²	C₄²×C₃₁	1	2	yes	—
180	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
181	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
182	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
183	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
184	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
185	C₂³	—	4	2	no	—
186	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
187	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
188	C₂³	C₃₁×#125	1	2	yes	—
189	C₂³	C₃₁×C₄⋊Q₈	1	2	yes	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	C₃₁×#95	1	2	yes	—
195	C₂×C₄	C₃₁×#95	1	4	yes	—
196	C₂×C₄	C₃₁×#95	1	2	yes	—
197	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
198	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
199	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
200	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
201	C₂×C₄	C₃₁×#117	1	2	yes	—
202	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
203	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
204	C₂×C₄	C₃₁×#121	1	4	yes	—
205	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
206	C₂×C₄	C₃₁×#124	1	2	yes	—
207	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
208	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
209	C₈	—	2	8	no	—
210	C₈	—	4	8	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
216	C₂²	C₃₁×#382	1	1	yes	characteristic
217	C₂²	C₃₁×#384	1	1	yes	characteristic
218	C₂²	C₃₁×#389	1	4	yes	—
219	C₄	—	2	2	no	—
220	C₄	—	2	4	no	—
221	C₄	C₃₁×#338	1	4	yes	complemented
222	C₄	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
223	C₄	C₃₁×#380	1	2	yes	—
224	C₄	C₃₁×#390	1	2	yes	—
225	C₄	C₃₁×#391	1	2	yes	—
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	—	2	4	no	—
229	C₂	C₃₁×C₄⋊#95	1	1	yes	characteristic
230	C₂	C₂×C₃₁×#338	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#95
prime radical	C₂⁴×C₃₁	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
69. C₄⋊#338	C₃₁	direct (N × H)
109. C₄×C₃₁	#338	semidirect
154. C₃₁	C₄⋊#338	direct (N × H)
221. C₄	C₃₁×#338	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#423	2
C₂×C₄×M₁₆×C₃₁	2
C₂×C₃₁×#338	2
C₄⋊#338	31

Subgroups — C31×C4⋊#338 #21467597

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×C₄⋊#338 #21467597