Subgroups — D₄×M₃₂×C₁₃₁ #45719321

Order 33536.

734 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₁₃₁	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2169×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#2168×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	D₄×C₁₆×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#2150×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#2112×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	C₄×M₃₂×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₈×C₁₃₁	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₁₃₁	C₄	1	1	yes	characteristic
11	C₂×M₃₂×C₁₃₁	—	2	8	no	—
12	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	—
14	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₁₃₁	C₂²	1	4	yes	complemented
16	C₂²×C₁₆×C₁₃₁	C₂²	1	4	yes	complemented
17	C₈×D₄×C₁₃₁	C₄	1	2	yes	—
18	C₈×D₄×C₁₃₁	C₄	1	1	yes	characteristic
19	C₈×D₄×C₁₃₁	C₄	1	1	yes	characteristic
20	C₈×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₈×D₄×C₁₃₁	C₂²	1	2	yes	—
22	C₈×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₁₃₁	C₄	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₁₃₁	C₄	1	2	yes	—
25	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
26	C₄⋊C₁₆×C₁₃₁	C₂²	1	2	yes	complemented
27	C₄⋊C₁₆×C₁₃₁	C₂²	1	2	yes	complemented
28	C₂²⋊C₁₆×C₁₃₁	C₂²	1	4	yes	complemented
29	C₂²⋊C₁₆×C₁₃₁	C₂²	1	4	yes	complemented
30	#346×C₁₃₁	C₂²	1	2	yes	complemented
31	C₄×C₁₆×C₁₃₁	C₂²	1	2	yes	complemented
32	C₂²×D₄×C₁₃₁	C₈	1	1	yes	characteristic
33	C₂³×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₁₃₁	—	2	4	no	—
35	C₂²×C₈×C₁₃₁	—	2	4	no	—
36	C₂²×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
43	C₄×D₄×C₁₃₁	C₈	1	2	yes	—
44	C₄×D₄×C₁₃₁	C₈	1	1	yes	characteristic
45	C₄×D₄×C₁₃₁	C₈	1	1	yes	characteristic
46	C₄×D₄×C₁₃₁	C₂×C₄	1	2	yes	—
47	C₄×D₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
48	C₄×D₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₁₃₁	C₈	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₁₃₁	C₈	1	2	yes	—
51	C₂×C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
52	M₃₂×C₁₃₁	—	2	4	no	—
53	M₃₂×C₁₃₁	—	2	2	no	—
54	M₃₂×C₁₃₁	—	2	8	no	—
55	M₃₂×C₁₃₁	D₄	1	4	yes	direct product factor
56	C₂×C₁₆×C₁₃₁	—	2	8	no	—
57	C₂×C₁₆×C₁₃₁	C₂³	1	4	yes	—
58	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
59	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
60	C₄⋊C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
62	C₄⋊C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂²⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
64	C₂²⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
65	C₂²⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
66	C₂²⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
71	C₂⁴×C₁₃₁	C₂×C₈	1	2	yes	—
72	C₂×D₄×C₁₃₁	—	2	4	no	—
73	C₂×D₄×C₁₃₁	C₂×C₈	1	2	yes	—
74	C₂×D₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
75	C₂×D₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₁	—	2	4	no	—
77	C₂²×C₄×C₁₃₁	—	2	4	no	—
78	C₂²×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₁	C₂×C₈	1	2	yes	—
80	C₂²×C₄×C₁₃₁	C₂×C₈	1	2	yes	—
81	C₂²×C₄×C₁₃₁	C₂×C₈	1	2	yes	—
82	C₂²×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃₁	—	2	4	no	—
87	C₂×C₈×C₁₃₁	—	2	2	no	—
88	C₂×C₈×C₁₃₁	—	2	4	no	—
89	C₂×C₈×C₁₃₁	—	2	4	no	—
90	C₂×C₈×C₁₃₁	—	2	2	no	—
91	C₂×C₈×C₁₃₁	—	2	2	no	—
92	C₂×C₈×C₁₃₁	—	2	2	no	—
93	C₂×C₈×C₁₃₁	—	2	2	no	—
94	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₁₃₁	#41	1	2	yes	—
101	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₁	C₂×C₈	1	2	yes	—
103	C₄⋊C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
104	C₄⋊C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₃₁	C₂×C₈	1	2	yes	—
106	C₂²⋊C₄×C₁₃₁	C₂×C₈	1	2	yes	—
107	C₂²⋊C₄×C₁₃₁	C₂×C₈	1	2	yes	—
108	C₂²⋊C₄×C₁₃₁	C₂×C₈	1	2	yes	—
109	C₄²×C₁₃₁	C₂×C₈	1	1	yes	characteristic
110	C₄²×C₁₃₁	C₂×C₈	1	1	yes	characteristic
111	C₄²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₁₆×C₁₃₁	—	2	4	no	—
114	C₁₆×C₁₃₁	—	2	2	no	—
115	C₁₆×C₁₃₁	C₂×D₄	1	4	yes	complemented
116	C₂³×C₁₃₁	—	2	4	no	—
117	C₂³×C₁₃₁	—	2	4	no	—
118	C₂³×C₁₃₁	—	4	2	no	—
119	C₂³×C₁₃₁	C₂²×C₈	1	2	yes	—
120	C₂³×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
121	C₂³×C₁₃₁	C₂²×C₈	1	2	yes	—
122	D₄×C₁₃₁	—	2	4	no	—
123	D₄×C₁₃₁	—	2	2	no	—
124	D₄×C₁₃₁	M₃₂	1	4	yes	direct product factor
125	C₂×C₄×C₁₃₁	—	2	4	no	—
126	C₂×C₄×C₁₃₁	—	2	2	no	—
127	C₂×C₄×C₁₃₁	—	2	2	no	—
128	C₂×C₄×C₁₃₁	—	2	2	no	—
129	C₂×C₄×C₁₃₁	—	2	2	no	—
130	C₂×C₄×C₁₃₁	—	2	4	no	—
131	C₂×C₄×C₁₃₁	—	2	4	no	—
132	C₂×C₄×C₁₃₁	—	2	2	no	—
133	C₂×C₄×C₁₃₁	—	4	1	no	—
134	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃₁	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₁₃₁	C₂²×C₈	1	2	yes	—
140	C₂×C₄×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃₁	#131	1	2	yes	—
143	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
144	C₈×C₁₃₁	—	2	2	no	—
145	C₈×C₁₃₁	—	2	2	no	—
146	C₈×C₁₃₁	—	2	1	no	—
147	C₈×C₁₃₁	—	2	1	no	—
148	C₈×C₁₃₁	C₄×D₄	1	2	yes	—
149	C₈×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
150	C₈×C₁₃₁	C₂×#41	1	1	yes	characteristic
151	C₂²×C₁₃₁	—	2	1	no	—
152	C₂²×C₁₃₁	—	2	1	no	—
153	C₂²×C₁₃₁	—	2	2	no	—
154	C₂²×C₁₃₁	—	2	2	no	—
155	C₂²×C₁₃₁	—	2	2	no	—
156	C₂²×C₁₃₁	—	2	2	no	—
157	C₂²×C₁₃₁	—	2	2	no	—
158	C₂²×C₁₃₁	—	4	8	no	—
159	C₂²×C₁₃₁	C₈×D₄	1	2	yes	—
160	C₂²×C₁₃₁	C₂×M₃₂	1	4	yes	complemented
161	C₂²×C₁₃₁	C₂³×C₈	1	1	yes	characteristic
162	C₄×C₁₃₁	—	2	1	no	—
163	C₄×C₁₃₁	—	2	1	no	—
164	C₄×C₁₃₁	—	2	1	no	—
165	C₄×C₁₃₁	—	2	2	no	—
166	C₄×C₁₃₁	—	2	2	no	—
167	C₄×C₁₃₁	C₈×D₄	1	2	yes	—
168	C₄×C₁₃₁	C₂×M₃₂	1	2	yes	complemented
169	C₄×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
170	C₄×C₁₃₁	C₂×#131	1	1	yes	characteristic
171	C₂×C₁₃₁	—	2	2	no	—
172	C₂×C₁₃₁	—	2	4	no	—
173	C₂×C₁₃₁	—	4	2	no	—
174	C₂×C₁₃₁	C₂×C₈×D₄	1	1	yes	characteristic
175	C₂×C₁₃₁	C₂²×M₃₂	1	1	yes	characteristic
176	C₂×C₁₃₁	#3407	1	1	yes	characteristic
177	D₄×M₃₂	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₁₃₁	D₄×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
179	C₂²×M₃₂	C₂×C₁₃₁	1	2	yes	complemented
180	C₂×C₈×D₄	C₂×C₁₃₁	1	1	yes	characteristic
181	#2169	C₂×C₁₃₁	1	2	yes	complemented
182	#2168	C₂×C₁₃₁	1	4	yes	complemented
183	D₄×C₁₆	C₂×C₁₃₁	1	2	yes	complemented
184	#2150	C₂×C₁₃₁	1	1	yes	characteristic, complemented
185	#2112	C₂×C₁₃₁	1	2	yes	complemented
186	C₄×M₃₂	C₂×C₁₃₁	1	1	yes	characteristic, complemented
187	C₂³×C₈	C₂²×C₁₃₁	1	2	yes	—
188	C₂×C₄×D₄	C₄×C₁₃₁	1	1	yes	characteristic
189	C₂×M₃₂	—	2	8	no	—
190	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
191	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	—
192	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
193	C₂×M₃₂	C₂²×C₁₃₁	1	4	yes	complemented
194	C₂²×C₁₆	C₂²×C₁₃₁	1	4	yes	complemented
195	C₈×D₄	C₄×C₁₃₁	1	2	yes	—
196	C₈×D₄	C₄×C₁₃₁	1	1	yes	characteristic
197	C₈×D₄	C₄×C₁₃₁	1	1	yes	characteristic
198	C₈×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
199	C₈×D₄	C₂²×C₁₃₁	1	2	yes	—
200	C₈×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
201	C₂×C₄⋊C₈	C₄×C₁₃₁	1	1	yes	characteristic
202	C₂×C₂²⋊C₈	C₄×C₁₃₁	1	2	yes	—
203	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
204	C₄⋊C₁₆	C₂²×C₁₃₁	1	2	yes	complemented
205	C₄⋊C₁₆	C₂²×C₁₃₁	1	2	yes	complemented
206	C₂²⋊C₁₆	C₂²×C₁₃₁	1	4	yes	complemented
207	C₂²⋊C₁₆	C₂²×C₁₃₁	1	4	yes	complemented
208	#346	C₂²×C₁₃₁	1	2	yes	complemented
209	C₄×C₁₆	C₂²×C₁₃₁	1	2	yes	complemented
210	C₂²×D₄	C₈×C₁₃₁	1	1	yes	characteristic
211	C₂³×C₄	C₂×C₄×C₁₃₁	1	2	yes	—
212	C₂²×C₈	—	2	4	no	—
213	C₂²×C₈	—	2	4	no	—
214	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
215	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
216	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
217	C₂²×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
218	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
219	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
220	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
221	C₄×D₄	C₈×C₁₃₁	1	2	yes	—
222	C₄×D₄	C₈×C₁₃₁	1	1	yes	characteristic
223	C₄×D₄	C₈×C₁₃₁	1	1	yes	characteristic
224	C₄×D₄	C₂×C₄×C₁₃₁	1	2	yes	—
225	C₄×D₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
226	C₄×D₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
227	C₂×C₄⋊C₄	C₈×C₁₃₁	1	1	yes	characteristic
228	C₂×C₂²⋊C₄	C₈×C₁₃₁	1	2	yes	—
229	C₂×C₄²	C₂×C₄×C₁₃₁	1	1	yes	characteristic
230	M₃₂	—	2	4	no	—
231	M₃₂	—	2	2	no	—
232	M₃₂	—	2	8	no	—
233	M₃₂	D₄×C₁₃₁	1	4	yes	direct product factor
234	C₂×C₁₆	—	2	8	no	—
235	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
236	C₂×C₁₆	C₂³×C₁₃₁	1	4	yes	—
237	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
238	C₄⋊C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
239	C₄⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
240	C₄⋊C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
241	C₂²⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
242	C₂²⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
243	C₂²⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
244	C₂²⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
245	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
246	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
249	C₂⁴	C₂×C₈×C₁₃₁	1	2	yes	—
250	C₂×D₄	—	2	4	no	—
251	C₂×D₄	C₂×C₈×C₁₃₁	1	2	yes	—
252	C₂×D₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
253	C₂×D₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₈×C₁₃₁	1	2	yes	—
258	C₂²×C₄	C₂×C₈×C₁₃₁	1	2	yes	—
259	C₂²×C₄	C₂×C₈×C₁₃₁	1	2	yes	—
260	C₂²×C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
261	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
262	C₂²×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
263	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
273	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
274	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
275	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
276	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
277	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
278	C₂×C₈	#41×C₁₃₁	1	2	yes	—
279	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
280	C₄⋊C₄	C₂×C₈×C₁₃₁	1	2	yes	—
281	C₄⋊C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
282	C₄⋊C₄	C₂×C₈×C₁₃₁	1	1	yes	characteristic
283	C₂²⋊C₄	C₂×C₈×C₁₃₁	1	2	yes	—
284	C₂²⋊C₄	C₂×C₈×C₁₃₁	1	2	yes	—
285	C₂²⋊C₄	C₂×C₈×C₁₃₁	1	2	yes	—
286	C₂²⋊C₄	C₂×C₈×C₁₃₁	1	2	yes	—
287	C₄²	C₂×C₈×C₁₃₁	1	1	yes	characteristic
288	C₄²	C₂×C₈×C₁₃₁	1	1	yes	characteristic
289	C₄²	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
290	C₄²	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
291	C₁₆	—	2	4	no	—
292	C₁₆	—	2	2	no	—
293	C₁₆	C₂×D₄×C₁₃₁	1	4	yes	complemented
294	C₂³	—	2	4	no	—
295	C₂³	—	2	4	no	—
296	C₂³	—	4	2	no	—
297	C₂³	C₂²×C₈×C₁₃₁	1	2	yes	—
298	C₂³	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
299	C₂³	C₂²×C₈×C₁₃₁	1	2	yes	—
300	D₄	—	2	4	no	—
301	D₄	—	2	2	no	—
302	D₄	M₃₂×C₁₃₁	1	4	yes	direct product factor
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
313	C₂×C₄	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
316	C₂×C₄	C₂²×C₈×C₁₃₁	1	2	yes	—
317	C₂×C₄	C₂²×C₈×C₁₃₁	1	2	yes	—
318	C₂×C₄	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
319	C₂×C₄	C₂²×C₈×C₁₃₁	1	1	yes	characteristic
320	C₂×C₄	#131×C₁₃₁	1	2	yes	—
321	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	C₄×D₄×C₁₃₁	1	2	yes	—
327	C₈	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
328	C₈	C₂×#41×C₁₃₁	1	1	yes	characteristic
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	4	8	no	—
337	C₂²	C₈×D₄×C₁₃₁	1	2	yes	—
338	C₂²	C₂×M₃₂×C₁₃₁	1	4	yes	complemented
339	C₂²	C₂³×C₈×C₁₃₁	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	2	2	no	—
345	C₄	C₈×D₄×C₁₃₁	1	2	yes	—
346	C₄	C₂×M₃₂×C₁₃₁	1	2	yes	complemented
347	C₄	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
348	C₄	C₂×#131×C₁₃₁	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	2	4	no	—
351	C₂	—	4	2	no	—
352	C₂	C₂×C₈×D₄×C₁₃₁	1	1	yes	characteristic
353	C₂	C₂²×M₃₂×C₁₃₁	1	1	yes	characteristic
354	C₂	#3407×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₃₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₁₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
socle	C₂²×C₁₃₁	C₂³×C₈
prime radical	C₂²×D₄×C₁₃₁	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×M₃₂×C₁₃₁	C₂	semidirect
3. #2169×C₁₃₁	C₂	semidirect
4. #2168×C₁₃₁	C₂	semidirect
5. D₄×C₁₆×C₁₃₁	C₂	semidirect
6. #2150×C₁₃₁	C₂	semidirect
7. #2112×C₁₃₁	C₂	semidirect
8. C₄×M₃₂×C₁₃₁	C₂	semidirect
15. C₂×M₃₂×C₁₃₁	C₂²	semidirect
16. C₂²×C₁₆×C₁₃₁	C₂²	semidirect
26. C₄⋊C₁₆×C₁₃₁	C₂²	semidirect
27. C₄⋊C₁₆×C₁₃₁	C₂²	semidirect
28. C₂²⋊C₁₆×C₁₃₁	C₂²	semidirect
29. C₂²⋊C₁₆×C₁₃₁	C₂²	semidirect
30. #346×C₁₃₁	C₂²	semidirect
31. C₄×C₁₆×C₁₃₁	C₂²	semidirect
55. M₃₂×C₁₃₁	D₄	direct (N × H)
115. C₁₆×C₁₃₁	C₂×D₄	semidirect
124. D₄×C₁₃₁	M₃₂	direct (N × H)
160. C₂²×C₁₃₁	C₂×M₃₂	semidirect
168. C₄×C₁₃₁	C₂×M₃₂	semidirect
177. D₄×M₃₂	C₁₃₁	direct (N × H)
178. C₁₃₁	D₄×M₃₂	direct (N × H)
179. C₂²×M₃₂	C₂×C₁₃₁	semidirect
181. #2169	C₂×C₁₃₁	semidirect
182. #2168	C₂×C₁₃₁	semidirect
183. D₄×C₁₆	C₂×C₁₃₁	semidirect
184. #2150	C₂×C₁₃₁	semidirect
185. #2112	C₂×C₁₃₁	semidirect
186. C₄×M₃₂	C₂×C₁₃₁	semidirect
193. C₂×M₃₂	C₂²×C₁₃₁	semidirect
194. C₂²×C₁₆	C₂²×C₁₃₁	semidirect
204. C₄⋊C₁₆	C₂²×C₁₃₁	semidirect
205. C₄⋊C₁₆	C₂²×C₁₃₁	semidirect
206. C₂²⋊C₁₆	C₂²×C₁₃₁	semidirect
207. C₂²⋊C₁₆	C₂²×C₁₃₁	semidirect
208. #346	C₂²×C₁₃₁	semidirect
209. C₄×C₁₆	C₂²×C₁₃₁	semidirect
233. M₃₂	D₄×C₁₃₁	direct (N × H)
293. C₁₆	C₂×D₄×C₁₃₁	semidirect
302. D₄	M₃₂×C₁₃₁	direct (N × H)
338. C₂²	C₂×M₃₂×C₁₃₁	semidirect
346. C₄	C₂×M₃₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₁₃₁	2
C₂×C₈×D₄×C₁₃₁	2
#2169×C₁₃₁	2
#2168×C₁₃₁	2
D₄×C₁₆×C₁₃₁	2
#2150×C₁₃₁	2
#2112×C₁₃₁	2
C₄×M₃₂×C₁₃₁	2
D₄×M₃₂	131

Subgroups — D4×M32×C131 #45719321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×M₃₂×C₁₃₁ #45719321