Subgroups — C₁₃₁×#22669 #45708553

Order 33536.

862 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×M₁₆×C₁₃₁	C₂	1	2	yes	maximal
2	C₂×#436×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#430×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#1950×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1855×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#1848×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#1811×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	C₄×#108×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×M₁₆×C₁₃₁	C₂²	1	4	yes	—
10	C₂²×M₁₆×C₁₃₁	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
12	#436×C₁₃₁	—	2	2	no	—
13	#436×C₁₃₁	C₄	1	4	yes	—
14	#436×C₁₃₁	C₄	1	4	yes	—
15	#435×C₁₃₁	—	2	4	no	—
16	C₈×D₄×C₁₃₁	—	2	2	no	—
17	#430×C₁₃₁	—	2	8	no	—
18	C₂×#108×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×#108×C₁₃₁	C₂²	1	2	yes	—
20	C₂×#108×C₁₃₁	C₂²	1	4	yes	—
21	C₂×#108×C₁₃₁	C₂²	1	2	yes	—
22	C₂×#108×C₁₃₁	C₂²	1	1	yes	characteristic
23	#423×C₁₃₁	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
25	#407×C₁₃₁	C₂²	1	4	yes	—
26	C₂×C₂²⋊C₈×C₁₃₁	C₂²	1	2	yes	—
27	C₄×M₁₆×C₁₃₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
29	#341×C₁₃₁	C₂²	1	2	yes	—
30	#341×C₁₃₁	C₂²	1	4	yes	—
31	C₂²×D₄×C₁₃₁	Q₈	1	1	yes	characteristic
32	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
33	C₂×M₁₆×C₁₃₁	—	2	4	no	—
34	C₂×M₁₆×C₁₃₁	—	2	4	no	—
35	C₂×M₁₆×C₁₃₁	—	2	4	no	—
36	C₂×M₁₆×C₁₃₁	—	2	8	no	—
37	C₂×M₁₆×C₁₃₁	—	4	4	no	—
38	C₂×M₁₆×C₁₃₁	C₂×C₄	1	8	yes	—
39	C₂×M₁₆×C₁₃₁	C₂³	1	2	yes	—
40	C₂×M₁₆×C₁₃₁	C₂³	1	4	yes	—
41	C₂×M₁₆×C₁₃₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃₁	—	2	4	no	—
43	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
46	C₄×D₄×C₁₃₁	C₂×C₄	1	2	yes	—
47	C₄×D₄×C₁₃₁	C₂×C₄	1	2	yes	—
48	C₄×D₄×C₁₃₁	D₄	1	2	yes	—
49	C₄×D₄×C₁₃₁	D₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₁₃₁	Q₈	1	1	yes	characteristic
51	C₂×C₂²⋊C₄×C₁₃₁	Q₈	1	2	yes	—
52	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
53	#108×C₁₃₁	—	2	8	no	—
54	#108×C₁₃₁	—	2	4	no	—
55	#108×C₁₃₁	—	2	2	no	—
56	#108×C₁₃₁	—	4	2	no	—
57	#108×C₁₃₁	D₄	1	4	yes	complemented
58	C₄⋊C₈×C₁₃₁	—	2	2	no	—
59	C₄⋊C₈×C₁₃₁	—	2	2	no	—
60	C₄⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
61	C₄⋊C₈×C₁₃₁	C₂×C₄	1	2	yes	—
62	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
63	C₂²⋊C₈×C₁₃₁	—	2	4	no	—
64	C₂²⋊C₈×C₁₃₁	C₂×C₄	1	8	yes	—
65	#97×C₁₃₁	—	2	2	no	—
66	C₄×C₈×C₁₃₁	—	2	2	no	—
67	C₄×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
69	C₂⁴×C₁₃₁	C₂×Q₈	1	2	yes	—
70	C₂×D₄×C₁₃₁	—	4	2	no	—
71	C₂×D₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
72	C₂×D₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₁₃₁	—	2	2	no	—
74	C₂²×C₄×C₁₃₁	—	2	2	no	—
75	C₂²×C₄×C₁₃₁	—	2	2	no	—
76	C₂²×C₄×C₁₃₁	—	2	2	no	—
77	C₂²×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
83	M₁₆×C₁₃₁	—	2	2	no	—
84	M₁₆×C₁₃₁	—	2	16	no	—
85	M₁₆×C₁₃₁	—	2	4	no	—
86	M₁₆×C₁₃₁	—	2	2	no	—
87	M₁₆×C₁₃₁	—	2	4	no	—
88	M₁₆×C₁₃₁	—	2	2	no	—
89	M₁₆×C₁₃₁	—	4	8	no	—
90	M₁₆×C₁₃₁	C₂×D₄	1	4	yes	—
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	—	2	1	no	—
95	C₂×C₈×C₁₃₁	—	2	1	no	—
96	C₂×C₈×C₁₃₁	—	2	2	no	—
97	C₂×C₈×C₁₃₁	—	2	4	no	—
98	C₂×C₈×C₁₃₁	—	4	8	no	—
99	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₁₃₁	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₁₃₁	#41	1	2	yes	—
104	C₄⋊C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
105	C₄⋊C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
106	C₂²⋊C₄×C₁₃₁	C₄⋊C₄	1	4	yes	—
107	C₂²⋊C₄×C₁₃₁	C₄⋊C₄	1	4	yes	—
108	C₄²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₁₃₁	—	2	2	no	—
113	C₂³×C₁₃₁	—	2	2	no	—
114	C₂³×C₁₃₁	—	4	2	no	—
115	C₂³×C₁₃₁	—	4	2	no	—
116	C₂³×C₁₃₁	C₂×C₄⋊C₄	1	4	yes	—
117	C₂³×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
118	D₄×C₁₃₁	—	2	4	no	—
119	D₄×C₁₃₁	—	4	2	no	—
120	C₂×C₄×C₁₃₁	—	2	2	no	—
121	C₂×C₄×C₁₃₁	—	2	2	no	—
122	C₂×C₄×C₁₃₁	—	2	4	no	—
123	C₂×C₄×C₁₃₁	—	2	4	no	—
124	C₂×C₄×C₁₃₁	—	2	4	no	—
125	C₂×C₄×C₁₃₁	—	2	4	no	—
126	C₂×C₄×C₁₃₁	—	2	2	no	—
127	C₂×C₄×C₁₃₁	—	4	1	no	—
128	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	4	yes	—
132	C₂×C₄×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
139	C₈×C₁₃₁	—	2	4	no	—
140	C₈×C₁₃₁	—	2	2	no	—
141	C₈×C₁₃₁	—	2	4	no	—
142	C₈×C₁₃₁	—	4	4	no	—
143	C₈×C₁₃₁	—	4	2	no	—
144	C₈×C₁₃₁	C₄×D₄	1	4	yes	—
145	C₂²×C₁₃₁	—	2	1	no	—
146	C₂²×C₁₃₁	—	2	1	no	—
147	C₂²×C₁₃₁	—	2	4	no	—
148	C₂²×C₁₃₁	—	2	4	no	—
149	C₂²×C₁₃₁	—	2	4	no	—
150	C₂²×C₁₃₁	—	8	2	no	—
151	C₂²×C₁₃₁	—	8	2	no	—
152	C₂²×C₁₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×C₁₃₁	D₄×Q₈	1	1	yes	characteristic
154	C₂²×C₁₃₁	#554	1	1	yes	characteristic
155	C₄×C₁₃₁	—	2	4	no	—
156	C₄×C₁₃₁	—	2	1	no	—
157	C₄×C₁₃₁	—	2	1	no	—
158	C₄×C₁₃₁	—	2	1	no	—
159	C₄×C₁₃₁	#430	1	2	yes	complemented
160	C₄×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
161	C₄×C₁₃₁	#520	1	1	yes	characteristic
162	C₄×C₁₃₁	D₄²	1	1	yes	characteristic
163	C₄×C₁₃₁	#547	1	1	yes	characteristic
164	C₂×C₁₃₁	—	2	1	no	—
165	C₂×C₁₃₁	—	2	1	no	—
166	C₂×C₁₃₁	—	4	4	no	—
167	C₂×C₁₃₁	D₄×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₁₃₁	C₂×#430	1	1	yes	characteristic
169	C₂×C₁₃₁	#2916	1	1	yes	characteristic
170	#22669	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₁₃₁	#22669	1	1	yes	characteristic, direct product factor, Sylow
172	D₄×M₁₆	C₂×C₁₃₁	1	2	yes	—
173	C₂×#436	C₂×C₁₃₁	1	1	yes	characteristic
174	C₂×#430	C₂×C₁₃₁	1	2	yes	complemented
175	#1950	C₂×C₁₃₁	1	2	yes	complemented
176	#1855	C₂×C₁₃₁	1	4	yes	complemented
177	#1848	C₂×C₁₃₁	1	1	yes	characteristic, complemented
178	#1811	C₂×C₁₃₁	1	2	yes	complemented
179	C₄×#108	C₂×C₁₃₁	1	1	yes	characteristic, complemented
180	C₂²×M₁₆	C₂²×C₁₃₁	1	4	yes	—
181	C₂²×M₁₆	C₂²×C₁₃₁	1	2	yes	—
182	C₂×C₄×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
183	#436	—	2	2	no	—
184	#436	C₄×C₁₃₁	1	4	yes	—
185	#436	C₄×C₁₃₁	1	4	yes	—
186	#435	—	2	4	no	—
187	C₈×D₄	—	2	2	no	—
188	#430	—	2	8	no	—
189	C₂×#108	C₂²×C₁₃₁	1	1	yes	characteristic
190	C₂×#108	C₂²×C₁₃₁	1	2	yes	—
191	C₂×#108	C₂²×C₁₃₁	1	4	yes	—
192	C₂×#108	C₂²×C₁₃₁	1	2	yes	—
193	C₂×#108	C₂²×C₁₃₁	1	1	yes	characteristic
194	#423	C₂²×C₁₃₁	1	2	yes	—
195	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
196	#407	C₂²×C₁₃₁	1	4	yes	—
197	C₂×C₂²⋊C₈	C₂²×C₁₃₁	1	2	yes	—
198	C₄×M₁₆	C₂²×C₁₃₁	1	2	yes	—
199	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
200	#341	C₂²×C₁₃₁	1	4	yes	—
201	#341	C₂²×C₁₃₁	1	2	yes	—
202	C₂²×D₄	Q₈×C₁₃₁	1	1	yes	characteristic
203	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
204	C₂×M₁₆	—	2	4	no	—
205	C₂×M₁₆	—	2	4	no	—
206	C₂×M₁₆	—	2	4	no	—
207	C₂×M₁₆	—	2	8	no	—
208	C₂×M₁₆	—	4	4	no	—
209	C₂×M₁₆	C₂×C₄×C₁₃₁	1	8	yes	—
210	C₂×M₁₆	C₂³×C₁₃₁	1	4	yes	—
211	C₂×M₁₆	C₂³×C₁₃₁	1	2	yes	—
212	C₂×M₁₆	C₂³×C₁₃₁	1	2	yes	—
213	C₂²×C₈	—	2	4	no	—
214	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
215	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
217	C₄×D₄	C₂×C₄×C₁₃₁	1	2	yes	—
218	C₄×D₄	C₂×C₄×C₁₃₁	1	2	yes	—
219	C₄×D₄	D₄×C₁₃₁	1	2	yes	—
220	C₄×D₄	D₄×C₁₃₁	1	2	yes	—
221	C₂×C₄⋊C₄	Q₈×C₁₃₁	1	1	yes	characteristic
222	C₂×C₂²⋊C₄	Q₈×C₁₃₁	1	2	yes	—
223	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
224	#108	—	2	8	no	—
225	#108	—	2	4	no	—
226	#108	—	2	2	no	—
227	#108	—	4	2	no	—
228	#108	D₄×C₁₃₁	1	4	yes	complemented
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
232	C₄⋊C₈	C₂×C₄×C₁₃₁	1	2	yes	—
233	C₂²⋊C₈	—	2	4	no	—
234	C₂²⋊C₈	—	2	4	no	—
235	C₂²⋊C₈	C₂×C₄×C₁₃₁	1	8	yes	—
236	#97	—	2	2	no	—
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
239	C₄×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
240	C₂⁴	C₂×Q₈×C₁₃₁	1	2	yes	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
243	C₂×D₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
249	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
250	C₂²×C₄	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
251	C₂²×C₄	C₂×Q₈×C₁₃₁	1	2	yes	—
252	C₂²×C₄	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
253	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
254	M₁₆	—	2	2	no	—
255	M₁₆	—	2	16	no	—
256	M₁₆	—	2	4	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	2	4	no	—
259	M₁₆	—	2	2	no	—
260	M₁₆	—	4	8	no	—
261	M₁₆	C₂×D₄×C₁₃₁	1	4	yes	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	4	8	no	—
270	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
271	C₂×C₈	C₂²×C₄×C₁₃₁	1	4	yes	—
272	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
273	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
274	C₂×C₈	#41×C₁₃₁	1	2	yes	—
275	C₄⋊C₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
276	C₄⋊C₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
277	C₂²⋊C₄	C₄⋊C₄×C₁₃₁	1	4	yes	—
278	C₂²⋊C₄	C₄⋊C₄×C₁₃₁	1	4	yes	—
279	C₄²	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
280	C₄²	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
283	C₂³	—	2	2	no	—
284	C₂³	—	2	2	no	—
285	C₂³	—	4	2	no	—
286	C₂³	—	4	2	no	—
287	C₂³	C₂×C₄⋊C₄×C₁₃₁	1	4	yes	—
288	C₂³	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
289	D₄	—	2	4	no	—
290	D₄	—	4	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
300	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
301	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
302	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	4	yes	—
303	C₂×C₄	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
308	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
309	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
310	C₈	—	2	4	no	—
311	C₈	—	2	2	no	—
312	C₈	—	2	4	no	—
313	C₈	—	4	4	no	—
314	C₈	—	4	2	no	—
315	C₈	C₄×D₄×C₁₃₁	1	4	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	2	4	no	—
320	C₂²	—	2	4	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	C₂²×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
324	C₂²	D₄×Q₈×C₁₃₁	1	1	yes	characteristic
325	C₂²	#554×C₁₃₁	1	1	yes	characteristic
326	C₄	—	2	4	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	#430×C₁₃₁	1	2	yes	complemented
331	C₄	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
332	C₄	#520×C₁₃₁	1	1	yes	characteristic
333	C₄	D₄²×C₁₃₁	1	1	yes	characteristic
334	C₄	#547×C₁₃₁	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	4	4	no	—
338	C₂	D₄×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
339	C₂	C₂×#430×C₁₃₁	1	1	yes	characteristic
340	C₂	#2916×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
socle	C₂²×C₁₃₁	C₂²×C₄⋊C₄
prime radical	C₂²×D₄×C₁₃₁	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂×C₄×D₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#430×C₁₃₁	C₂	semidirect
4. #1950×C₁₃₁	C₂	semidirect
5. #1855×C₁₃₁	C₂	semidirect
6. #1848×C₁₃₁	C₂	semidirect
7. #1811×C₁₃₁	C₂	semidirect
8. C₄×#108×C₁₃₁	C₂	semidirect
57. #108×C₁₃₁	D₄	semidirect
159. C₄×C₁₃₁	#430	semidirect
170. #22669	C₁₃₁	direct (N × H)
171. C₁₃₁	#22669	direct (N × H)
174. C₂×#430	C₂×C₁₃₁	semidirect
175. #1950	C₂×C₁₃₁	semidirect
176. #1855	C₂×C₁₃₁	semidirect
177. #1848	C₂×C₁₃₁	semidirect
178. #1811	C₂×C₁₃₁	semidirect
179. C₄×#108	C₂×C₁₃₁	semidirect
228. #108	D₄×C₁₃₁	semidirect
330. C₄	#430×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
D₄×M₁₆×C₁₃₁	2
C₂×#436×C₁₃₁	2
C₂×#430×C₁₃₁	2
#1950×C₁₃₁	2
#1855×C₁₃₁	2
#1848×C₁₃₁	2
#1811×C₁₃₁	2
C₄×#108×C₁₃₁	2
#22669	131

Subgroups — C131×#22669 #45708553

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#22669 #45708553