Subgroups — C₄×C₃₁×#344 #21467469

Order 7936.

906 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#344	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂×C₃₁×#117	C₂²	1	2	yes	—
8	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₄×M₁₆×C₃₁	—	2	2	no	—
10	C₄×M₁₆×C₃₁	—	2	1	no	—
11	C₄×M₁₆×C₃₁	—	2	1	no	—
12	C₄×M₁₆×C₃₁	—	2	1	no	—
13	C₄×M₁₆×C₃₁	—	2	1	no	—
14	C₄×M₁₆×C₃₁	C₄	1	4	yes	complemented
15	C₂×C₃₁×#97	—	2	1	no	—
16	C₂×C₃₁×#97	C₄	1	2	yes	complemented
17	C₂×C₄×C₈×C₃₁	—	2	1	no	—
18	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	complemented
19	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
20	C₄×C₂²⋊C₄×C₃₁	C₄	1	4	yes	—
21	C₃₁×#376	—	2	1	no	—
22	C₄³×C₃₁	—	2	1	no	—
23	C₃₁×#344	C₄	1	16	yes	direct product factor
24	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₃₁	—	2	2	no	—
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂×M₁₆×C₃₁	—	2	4	no	—
30	C₂×M₁₆×C₃₁	—	2	4	no	—
31	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	—
33	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₃₁	—	2	2	no	—
35	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
36	C₃₁×#117	—	2	4	no	—
37	C₃₁×#117	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₃₁	—	2	2	no	—
41	C₂×C₄²×C₃₁	—	2	4	no	—
42	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	D₄	1	2	yes	—
49	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
51	C₃₁×#97	—	2	4	no	—
52	C₃₁×#97	—	2	2	no	—
53	C₃₁×#97	—	2	2	no	—
54	C₄×C₈×C₃₁	—	2	4	no	—
55	C₄×C₈×C₃₁	—	2	2	no	—
56	C₄×C₈×C₃₁	—	2	2	no	—
57	C₃₁×#95	—	2	4	no	—
58	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	—	4	4	no	—
61	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
62	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
63	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	#41	1	4	yes	—
75	M₁₆×C₃₁	—	2	4	no	—
76	M₁₆×C₃₁	—	2	8	no	—
77	M₁₆×C₃₁	—	2	8	no	—
78	M₁₆×C₃₁	—	2	8	no	—
79	M₁₆×C₃₁	C₄²	1	8	yes	complemented
80	C₂×C₈×C₃₁	—	2	4	no	—
81	C₂×C₈×C₃₁	—	2	4	no	—
82	C₂×C₈×C₃₁	—	2	4	no	—
83	C₂×C₈×C₃₁	—	2	8	no	—
84	C₂×C₈×C₃₁	C₄²	1	8	yes	complemented
85	C₄⋊C₄×C₃₁	—	2	8	no	—
86	C₂²⋊C₄×C₃₁	—	2	4	no	—
87	C₂²⋊C₄×C₃₁	C₄²	1	8	yes	—
88	C₄²×C₃₁	—	2	8	no	—
89	C₄²×C₃₁	—	4	8	no	—
90	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
95	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
97	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
98	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
99	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
100	C₄×#344	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₃₁	—	4	2	no	—
102	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
103	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
104	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₃₁	#117	1	2	yes	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	2	4	no	—
109	C₂×C₄×C₃₁	—	2	4	no	—
110	C₂×C₄×C₃₁	—	2	8	no	—
111	C₂×C₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	4	2	no	—
114	C₂×C₄×C₃₁	—	4	8	no	—
115	C₂×C₄×C₃₁	#95	1	2	yes	—
116	C₂×C₄×C₃₁	#95	1	2	yes	—
117	C₂×C₄×C₃₁	#95	1	2	yes	—
118	C₂×C₄×C₃₁	#95	1	2	yes	—
119	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	#117	1	2	yes	—
126	C₂×C₄×C₃₁	#117	1	2	yes	—
127	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
128	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
129	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
130	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
131	C₈×C₃₁	—	2	8	no	—
132	C₈×C₃₁	—	2	8	no	—
133	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
134	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
135	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
136	C₂×#344	C₂×C₃₁	1	4	yes	—
137	C₂²×C₃₁	—	2	2	no	—
138	C₂²×C₃₁	—	2	2	no	—
139	C₂²×C₃₁	—	2	2	no	—
140	C₂²×C₃₁	—	4	2	no	—
141	C₂²×C₃₁	—	4	2	no	—
142	C₂²×C₃₁	C₄³	1	1	yes	characteristic
143	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
144	C₂²×C₃₁	#376	1	1	yes	characteristic
145	C₂²×C₃₁	C₄×C₂²⋊C₄	1	2	yes	—
146	C₂²×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
147	C₄×C₃₁	—	2	2	no	—
148	C₄×C₃₁	—	2	2	no	—
149	C₄×C₃₁	—	2	2	no	—
150	C₄×C₃₁	—	4	8	no	—
151	C₄×C₃₁	#344	1	4	yes	direct product factor
152	C₄×C₃₁	C₄×C₂²⋊C₄	1	2	yes	—
153	C₄×C₃₁	C₄×C₂²⋊C₄	1	2	yes	—
154	C₄×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
155	C₄×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
156	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
157	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
158	C₂×#117	C₂²×C₃₁	1	2	yes	—
159	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
160	C₄×M₁₆	—	2	2	no	—
161	C₄×M₁₆	—	2	1	no	—
162	C₄×M₁₆	—	2	1	no	—
163	C₄×M₁₆	—	2	1	no	—
164	C₄×M₁₆	—	2	1	no	—
165	C₄×M₁₆	C₄×C₃₁	1	4	yes	complemented
166	C₂×#97	—	2	1	no	—
167	C₂×#97	C₄×C₃₁	1	2	yes	complemented
168	C₂×C₄×C₈	—	2	1	no	—
169	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	complemented
170	C₄×C₄⋊C₄	—	2	2	no	—
171	C₄×C₂²⋊C₄	C₄×C₃₁	1	4	yes	—
172	#376	—	2	1	no	—
173	C₄³	—	2	1	no	—
174	#344	C₄×C₃₁	1	16	yes	direct product factor
175	C₂×C₃₁	—	2	2	no	—
176	C₂×C₃₁	—	2	1	no	—
177	C₂×C₃₁	—	2	1	no	—
178	C₂×C₃₁	—	2	2	no	—
179	C₂×C₃₁	C₄×#95	1	1	yes	characteristic
180	C₂×C₃₁	C₂×#344	1	2	yes	—
181	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
182	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	4	no	—
187	C₂×M₁₆	—	2	4	no	—
188	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	—
189	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
190	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
193	#117	—	2	4	no	—
194	#117	C₂×C₄×C₃₁	1	8	yes	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	2	4	no	—
199	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
200	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
201	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
202	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
203	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
204	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
205	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
206	C₂×C₄²	D₄×C₃₁	1	2	yes	—
207	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
208	#97	—	2	4	no	—
209	#97	—	2	2	no	—
210	#97	—	2	2	no	—
211	C₄×C₈	—	2	4	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	—	2	2	no	—
214	#95	—	2	4	no	—
215	C₃₁	C₄×#344	1	1	yes	characteristic, direct product factor, Sylow
216	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	4	4	no	—
219	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
220	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
221	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
222	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
223	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
224	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
225	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
228	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	#41×C₃₁	1	4	yes	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	8	no	—
235	M₁₆	—	2	8	no	—
236	M₁₆	—	2	8	no	—
237	M₁₆	C₄²×C₃₁	1	8	yes	complemented
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	8	no	—
242	C₂×C₈	C₄²×C₃₁	1	8	yes	complemented
243	C₄⋊C₄	—	2	8	no	—
244	C₂²⋊C₄	—	2	4	no	—
245	C₂²⋊C₄	C₄²×C₃₁	1	8	yes	—
246	C₄²	—	2	8	no	—
247	C₄²	—	4	8	no	—
248	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
249	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
250	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
251	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
252	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
253	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
254	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
255	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
256	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
257	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
258	C₂³	—	4	2	no	—
259	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
260	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
261	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
262	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
263	C₂³	C₃₁×#117	1	2	yes	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	8	no	—
272	C₂×C₄	C₃₁×#95	1	2	yes	—
273	C₂×C₄	C₃₁×#95	1	2	yes	—
274	C₂×C₄	C₃₁×#95	1	2	yes	—
275	C₂×C₄	C₃₁×#95	1	2	yes	—
276	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
277	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
278	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
281	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
282	C₂×C₄	C₃₁×#117	1	2	yes	—
283	C₂×C₄	C₃₁×#117	1	2	yes	—
284	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
285	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
286	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
287	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
288	C₈	—	2	8	no	—
289	C₈	—	2	8	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₄³×C₃₁	1	1	yes	characteristic
296	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
297	C₂²	C₃₁×#376	1	1	yes	characteristic
298	C₂²	C₄×C₂²⋊C₄×C₃₁	1	2	yes	—
299	C₂²	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	8	no	—
304	C₄	C₃₁×#344	1	4	yes	direct product factor
305	C₄	C₄×C₂²⋊C₄×C₃₁	1	2	yes	—
306	C₄	C₄×C₂²⋊C₄×C₃₁	1	2	yes	—
307	C₄	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
308	C₄	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
309	C₂	—	2	2	no	—
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	—	2	2	no	—
313	C₂	C₄×C₃₁×#95	1	1	yes	characteristic
314	C₂	C₂×C₃₁×#344	1	2	yes	—

	subgroup	G/H
center	C₄²×C₃₁	C₂²⋊C₄
commutator	C₂²	C₄³×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#95
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₄×M₁₆×C₃₁	C₄	semidirect
16. C₂×C₃₁×#97	C₄	semidirect
18. C₂×C₄×C₈×C₃₁	C₄	semidirect
23. C₃₁×#344	C₄	direct (N × H)
79. M₁₆×C₃₁	C₄²	semidirect
84. C₂×C₈×C₃₁	C₄²	semidirect
100. C₄×#344	C₃₁	direct (N × H)
151. C₄×C₃₁	#344	direct (N × H)
165. C₄×M₁₆	C₄×C₃₁	semidirect
167. C₂×#97	C₄×C₃₁	semidirect
169. C₂×C₄×C₈	C₄×C₃₁	semidirect
174. #344	C₄×C₃₁	direct (N × H)
215. C₃₁	C₄×#344	direct (N × H)
237. M₁₆	C₄²×C₃₁	semidirect
242. C₂×C₈	C₄²×C₃₁	semidirect
304. C₄	C₃₁×#344	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×M₁₆×C₃₁	2
C₄×C₃₁×#117	2
C₂×C₃₁×#344	2
C₄×#344	31

Subgroups — C4×C31×#344 #21467469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₃₁×#344 #21467469