Subgroups — C₂×C₃₁×#2922 #21516878

Order 7936.

2222 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2922	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃₁×#433	C₂	1	4	yes	maximal, complemented
4	C₂×C₃₁×#431	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#424	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#408	C₂	1	2	yes	maximal
7	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
12	C₄×#41×C₃₁	C₄	1	8	yes	—
13	C₄×#41×C₃₁	C₂²	1	8	yes	—
14	C₂×C₄×Q₈×C₃₁	C₄	1	2	yes	—
15	C₂×C₄×D₄×C₃₁	C₄	1	4	yes	—
16	C₂×C₄×D₄×C₃₁	C₄	1	2	yes	—
17	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#117	C₂²	1	2	yes	—
19	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#433	C₂²	1	32	yes	complemented
22	C₃₁×#431	C₂²	1	32	yes	complemented
23	C₃₁×#424	C₂²	1	16	yes	complemented
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	8	yes	—
25	C₃₁×#408	C₂²	1	16	yes	—
26	C₂×C₃₁×C₂²⋊C₈	C₂²	1	8	yes	—
27	C₄×M₁₆×C₃₁	C₂²	1	8	yes	complemented
28	C₂×C₃₁×#97	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₃₁	C₂²	1	8	yes	complemented
33	C₂×#41×C₃₁	—	2	8	no	—
34	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
35	C₂×#41×C₃₁	C₂³	1	4	yes	—
36	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
38	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₃₁	C₂×C₄	1	4	yes	—
40	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
42	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₃₁	—	2	8	no	—
44	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
45	C₂²×C₈×C₃₁	—	2	8	no	—
46	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	8	yes	—
49	C₄×Q₈×C₃₁	C₂×C₄	1	16	yes	—
50	C₄×D₄×C₃₁	C₂×C₄	1	32	yes	—
51	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	—
52	C₃₁×#117	C₂×C₄	1	8	yes	—
53	C₃₁×#117	C₂×C₄	1	4	yes	—
54	C₃₁×#117	C₂³	1	4	yes	—
55	C₃₁×#117	C₂³	1	8	yes	—
56	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
57	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
58	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
59	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
61	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	—
63	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
66	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
68	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
69	C₃₁×C₄⋊C₈	C₂³	1	32	yes	—
70	C₃₁×C₂²⋊C₈	C₂³	1	32	yes	—
71	C₃₁×#97	C₂³	1	8	yes	—
72	C₄×C₈×C₃₁	C₂³	1	8	yes	—
73	C₄×C₈×C₃₁	C₂³	1	8	yes	—
74	C₄×C₈×C₃₁	C₂³	1	8	yes	—
75	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
76	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	#41×C₃₁	—	2	16	no	—
78	#41×C₃₁	—	2	16	no	—
79	C₂×Q₈×C₃₁	—	2	4	no	—
80	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
81	C₂×D₄×C₃₁	—	2	8	no	—
82	C₂×D₄×C₃₁	—	2	4	no	—
83	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
84	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
85	C₂²×C₄×C₃₁	—	2	8	no	—
86	C₂²×C₄×C₃₁	—	2	16	no	—
87	C₂²×C₄×C₃₁	—	2	8	no	—
88	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
90	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
92	C₂²×C₄×C₃₁	C₂²×C₄	1	16	yes	—
93	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
94	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₃₁	#41	1	8	yes	—
98	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
99	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
101	M₁₆×C₃₁	—	2	16	no	—
102	C₂×C₈×C₃₁	—	2	8	no	—
103	C₂×C₈×C₃₁	—	2	16	no	—
104	C₂×C₈×C₃₁	—	2	8	no	—
105	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
106	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
107	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
108	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
109	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
110	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
111	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
112	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
113	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
114	C₄²×C₃₁	C₂²×C₄	1	16	yes	—
115	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
116	C₄²×C₃₁	C₂⁴	1	2	yes	—
117	C₄²×C₃₁	C₂⁴	1	2	yes	—
118	C₄²×C₃₁	C₂⁴	1	4	yes	—
119	C₂×#2922	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₃₁	—	2	4	no	—
121	C₂³×C₃₁	—	2	4	no	—
122	C₂³×C₃₁	—	2	8	no	—
123	C₂³×C₃₁	#117	1	4	yes	—
124	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
125	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
126	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
127	Q₈×C₃₁	—	2	8	no	—
128	D₄×C₃₁	—	2	16	no	—
129	D₄×C₃₁	—	2	8	no	—
130	C₂×C₄×C₃₁	—	2	4	no	—
131	C₂×C₄×C₃₁	—	2	4	no	—
132	C₂×C₄×C₃₁	—	2	8	no	—
133	C₂×C₄×C₃₁	—	2	8	no	—
134	C₂×C₄×C₃₁	—	2	16	no	—
135	C₂×C₄×C₃₁	—	2	32	no	—
136	C₂×C₄×C₃₁	—	2	8	no	—
137	C₂×C₄×C₃₁	#117	1	4	yes	—
138	C₂×C₄×C₃₁	#117	1	8	yes	—
139	C₂×C₄×C₃₁	#131	1	8	yes	—
140	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
141	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
142	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
143	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
145	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
146	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
147	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
148	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
149	C₈×C₃₁	—	2	8	no	—
150	C₈×C₃₁	—	2	8	no	—
151	C₂×C₄×#41	C₂×C₃₁	1	1	yes	characteristic
152	#2922	C₂×C₃₁	1	16	yes	direct product factor
153	C₂×#433	C₂×C₃₁	1	4	yes	complemented
154	C₂×#431	C₂×C₃₁	1	4	yes	complemented
155	C₂×#424	C₂×C₃₁	1	2	yes	complemented
156	C₂×#408	C₂×C₃₁	1	2	yes	—
157	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
158	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
159	C₂²×C₃₁	—	2	8	no	—
160	C₂²×C₃₁	—	2	4	no	—
161	C₂²×C₃₁	—	2	4	no	—
162	C₂²×C₃₁	—	2	8	no	—
163	C₂²×C₃₁	—	4	8	no	—
164	C₂²×C₃₁	C₂×#117	1	4	yes	—
165	C₂²×C₃₁	C₂×#117	1	2	yes	—
166	C₂²×C₃₁	C₂×#131	1	4	yes	—
167	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
168	C₄×C₃₁	—	2	4	no	—
169	C₄×C₃₁	—	2	4	no	—
170	C₄×C₃₁	—	2	8	no	—
171	C₄×C₃₁	C₂×#117	1	2	yes	—
172	C₄×C₃₁	C₂×#117	1	4	yes	—
173	C₄×C₃₁	C₂×#131	1	8	yes	—
174	C₄×C₃₁	C₂²×#41	1	2	yes	—
175	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
176	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
177	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
178	C₄×#41	C₄×C₃₁	1	8	yes	—
179	C₄×#41	C₂²×C₃₁	1	8	yes	—
180	C₂×C₄×Q₈	C₄×C₃₁	1	2	yes	—
181	C₂×C₄×D₄	C₄×C₃₁	1	4	yes	—
182	C₂×C₄×D₄	C₄×C₃₁	1	2	yes	—
183	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
184	C₂×#117	C₂²×C₃₁	1	2	yes	—
185	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
186	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
187	#433	C₂²×C₃₁	1	32	yes	complemented
188	#431	C₂²×C₃₁	1	32	yes	complemented
189	#424	C₂²×C₃₁	1	16	yes	complemented
190	C₂×C₄⋊C₈	C₂²×C₃₁	1	8	yes	—
191	#408	C₂²×C₃₁	1	16	yes	—
192	C₂×C₂²⋊C₈	C₂²×C₃₁	1	8	yes	—
193	C₄×M₁₆	C₂²×C₃₁	1	8	yes	complemented
194	C₂×#97	C₂²×C₃₁	1	2	yes	—
195	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
196	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
197	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
198	C₂×C₄×C₈	C₂²×C₃₁	1	8	yes	complemented
199	C₂×C₃₁	—	2	4	no	—
200	C₂×C₃₁	—	4	4	no	—
201	C₂×C₃₁	#2922	1	4	yes	direct product factor
202	C₂×C₃₁	C₂²×#117	1	1	yes	characteristic
203	C₂×C₃₁	C₂²×#131	1	2	yes	—
204	C₂×#41	—	2	8	no	—
205	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
206	C₂×#41	C₂³×C₃₁	1	4	yes	—
207	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
208	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
209	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
210	C₂³×C₄	C₂×C₄×C₃₁	1	4	yes	—
211	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
212	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
213	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
214	C₂×M₁₆	—	2	8	no	—
215	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
216	C₂²×C₈	—	2	8	no	—
217	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
218	C₂²×C₈	C₂³×C₃₁	1	8	yes	—
219	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
220	C₄×Q₈	C₂×C₄×C₃₁	1	16	yes	—
221	C₄×D₄	C₂×C₄×C₃₁	1	32	yes	—
222	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	—
223	#117	C₂×C₄×C₃₁	1	8	yes	—
224	#117	C₂×C₄×C₃₁	1	4	yes	—
225	#117	C₂³×C₃₁	1	4	yes	—
226	#117	C₂³×C₃₁	1	8	yes	—
227	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
228	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
229	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
230	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
231	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
232	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
233	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	—
234	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
235	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
236	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
237	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
238	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
239	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
240	C₄⋊C₈	C₂³×C₃₁	1	32	yes	—
241	C₂²⋊C₈	C₂³×C₃₁	1	32	yes	—
242	#97	C₂³×C₃₁	1	8	yes	—
243	C₄×C₈	C₂³×C₃₁	1	8	yes	—
244	C₄×C₈	C₂³×C₃₁	1	8	yes	—
245	C₄×C₈	C₂³×C₃₁	1	8	yes	—
246	C₃₁	C₂×#2922	1	1	yes	characteristic, direct product factor, Sylow
247	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
248	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
249	#41	—	2	16	no	—
250	#41	—	2	16	no	—
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
253	C₂×D₄	—	2	8	no	—
254	C₂×D₄	—	2	4	no	—
255	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
256	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
257	C₂²×C₄	—	2	8	no	—
258	C₂²×C₄	—	2	16	no	—
259	C₂²×C₄	—	2	8	no	—
260	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
261	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
262	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
263	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
264	C₂²×C₄	C₂²×C₄×C₃₁	1	16	yes	—
265	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
266	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
267	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
268	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
269	C₂²×C₄	#41×C₃₁	1	8	yes	—
270	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
271	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
272	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
273	M₁₆	—	2	16	no	—
274	C₂×C₈	—	2	8	no	—
275	C₂×C₈	—	2	16	no	—
276	C₂×C₈	—	2	8	no	—
277	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
278	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
279	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
280	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
281	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
282	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
283	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
284	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
285	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
286	C₄²	C₂²×C₄×C₃₁	1	16	yes	—
287	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
288	C₄²	C₂⁴×C₃₁	1	2	yes	—
289	C₄²	C₂⁴×C₃₁	1	2	yes	—
290	C₄²	C₂⁴×C₃₁	1	4	yes	—
291	C₂³	—	2	4	no	—
292	C₂³	—	2	4	no	—
293	C₂³	—	2	8	no	—
294	C₂³	C₃₁×#117	1	4	yes	—
295	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
296	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
297	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
298	Q₈	—	2	8	no	—
299	D₄	—	2	16	no	—
300	D₄	—	2	8	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	8	no	—
304	C₂×C₄	—	2	8	no	—
305	C₂×C₄	—	2	16	no	—
306	C₂×C₄	—	2	32	no	—
307	C₂×C₄	—	2	8	no	—
308	C₂×C₄	C₃₁×#117	1	4	yes	—
309	C₂×C₄	C₃₁×#117	1	8	yes	—
310	C₂×C₄	C₃₁×#131	1	8	yes	—
311	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
312	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
313	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
314	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
315	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
316	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
317	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
318	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
319	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
320	C₈	—	2	8	no	—
321	C₈	—	2	8	no	—
322	C₂²	—	2	8	no	—
323	C₂²	—	2	4	no	—
324	C₂²	—	2	4	no	—
325	C₂²	—	2	8	no	—
326	C₂²	—	4	8	no	—
327	C₂²	C₂×C₃₁×#117	1	4	yes	—
328	C₂²	C₂×C₃₁×#117	1	2	yes	—
329	C₂²	C₂×C₃₁×#131	1	4	yes	—
330	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
331	C₄	—	2	4	no	—
332	C₄	—	2	4	no	—
333	C₄	—	2	8	no	—
334	C₄	C₂×C₃₁×#117	1	2	yes	—
335	C₄	C₂×C₃₁×#117	1	4	yes	—
336	C₄	C₂×C₃₁×#131	1	8	yes	—
337	C₄	C₂²×#41×C₃₁	1	2	yes	—
338	C₂	—	2	4	no	—
339	C₂	—	4	4	no	—
340	C₂	C₃₁×#2922	1	4	yes	direct product factor
341	C₂	C₂²×C₃₁×#117	1	1	yes	characteristic
342	C₂	C₂²×C₃₁×#131	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₃₁	C₂³
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2922	C₂	direct (N × H)
3. C₂×C₃₁×#433	C₂	semidirect
4. C₂×C₃₁×#431	C₂	semidirect
5. C₂×C₃₁×#424	C₂	semidirect
7. C₂×C₄×M₁₆×C₃₁	C₂	semidirect
8. C₂²×C₄×C₈×C₃₁	C₂	semidirect
21. C₃₁×#433	C₂²	semidirect
22. C₃₁×#431	C₂²	semidirect
23. C₃₁×#424	C₂²	semidirect
27. C₄×M₁₆×C₃₁	C₂²	semidirect
32. C₂×C₄×C₈×C₃₁	C₂²	semidirect
119. C₂×#2922	C₃₁	direct (N × H)
152. #2922	C₂×C₃₁	direct (N × H)
153. C₂×#433	C₂×C₃₁	semidirect
154. C₂×#431	C₂×C₃₁	semidirect
155. C₂×#424	C₂×C₃₁	semidirect
157. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
158. C₂²×C₄×C₈	C₂×C₃₁	semidirect
187. #433	C₂²×C₃₁	semidirect
188. #431	C₂²×C₃₁	semidirect
189. #424	C₂²×C₃₁	semidirect
193. C₄×M₁₆	C₂²×C₃₁	semidirect
198. C₂×C₄×C₈	C₂²×C₃₁	semidirect
201. C₂×C₃₁	#2922	direct (N × H)
246. C₃₁	C₂×#2922	direct (N × H)
340. C₂	C₃₁×#2922	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×#41×C₃₁	2
C₃₁×#2922	2
C₂×C₃₁×#433	2
C₂×C₃₁×#431	2
C₂×C₃₁×#424	2
C₂×C₃₁×#408	2
C₂×C₄×M₁₆×C₃₁	2
C₂²×C₄×C₈×C₃₁	2
C₂×#2922	31

Subgroups — C2×C31×#2922 #21516878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2922 #21516878