Subgroups — C₁₃₁×#23234 #45709118

Order 33536.

722 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3131×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2962×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2935×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#1986×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1980×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	#1937×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#1929×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	#1866×C₁₃₁	C₂	1	2	yes	maximal, complemented
9	C₄×#108×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×C₄⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#1764×C₁₃₁	C₂	1	2	yes	maximal, complemented
12	C₂×#131×C₁₃₁	C₂²	1	2	yes	—
13	#533×C₁₃₁	C₂²	1	1	yes	characteristic
14	#532×C₁₃₁	C₂²	1	1	yes	characteristic
15	#488×C₁₃₁	C₄	1	2	yes	complemented
16	#488×C₁₃₁	C₄	1	2	yes	complemented
17	#487×C₁₃₁	C₄	1	1	yes	characteristic, complemented
18	#487×C₁₃₁	C₄	1	1	yes	characteristic, complemented
19	#486×C₁₃₁	C₄	1	1	yes	characteristic, complemented
20	#486×C₁₃₁	C₄	1	1	yes	characteristic, complemented
21	#446×C₁₃₁	—	2	2	no	—
22	#446×C₁₃₁	—	2	1	no	—
23	C₈×Q₈×C₁₃₁	—	2	1	no	—
24	#436×C₁₃₁	—	2	1	no	—
25	#435×C₁₃₁	—	2	2	no	—
26	C₈×D₄×C₁₃₁	—	2	1	no	—
27	C₂×#108×C₁₃₁	C₂²	1	1	yes	characteristic
28	C₂×#108×C₁₃₁	C₂²	1	1	yes	characteristic
29	#424×C₁₃₁	C₂²	1	2	yes	—
30	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
31	C₂×#104×C₁₃₁	C₂²	1	2	yes	—
32	C₂×#104×C₁₃₁	C₂²	1	2	yes	—
33	#416×C₁₃₁	C₂²	1	2	yes	—
34	#416×C₁₃₁	C₂²	1	2	yes	—
35	#408×C₁₃₁	C₂²	1	2	yes	—
36	#405×C₁₃₁	C₂²	1	2	yes	—
37	C₄×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
40	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
42	#341×C₁₃₁	C₂²	1	1	yes	characteristic
43	#341×C₁₃₁	C₂²	1	1	yes	characteristic
44	#339×C₁₃₁	C₂²	1	2	yes	—
45	#339×C₁₃₁	C₂²	1	2	yes	—
46	#336×C₁₃₁	C₂²	1	2	yes	—
47	C₂×#41×C₁₃₁	C₂³	1	2	yes	—
48	#131×C₁₃₁	—	2	4	no	—
49	#131×C₁₃₁	—	4	2	no	—
50	C₂×M₁₆×C₁₃₁	—	2	2	no	—
51	C₂×M₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
54	C₂×M₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₁₃₁	—	2	2	no	—
56	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
59	C₄⋊Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
60	C₄⋊Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
62	#127×C₁₃₁	C₂×C₄	1	1	yes	characteristic
63	#125×C₁₃₁	C₂×C₄	1	2	yes	—
64	#124×C₁₃₁	C₂×C₄	1	2	yes	—
65	C₂²⋊Q₈×C₁₃₁	—	2	1	no	—
66	C₂²⋊Q₈×C₁₃₁	—	2	1	no	—
67	#121×C₁₃₁	—	2	1	no	—
68	#121×C₁₃₁	—	2	1	no	—
69	C₄×Q₈×C₁₃₁	—	2	1	no	—
70	C₄×Q₈×C₁₃₁	—	2	1	no	—
71	C₄×D₄×C₁₃₁	—	2	1	no	—
72	C₄×D₄×C₁₃₁	—	2	1	no	—
73	#117×C₁₃₁	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
75	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
76	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
77	#108×C₁₃₁	—	2	2	no	—
78	#108×C₁₃₁	—	2	2	no	—
79	C₄⋊C₈×C₁₃₁	—	2	2	no	—
80	C₄⋊C₈×C₁₃₁	—	2	2	no	—
81	C₄⋊C₈×C₁₃₁	D₄	1	1	yes	characteristic
82	C₄⋊C₈×C₁₃₁	D₄	1	2	yes	—
83	C₄⋊C₈×C₁₃₁	D₄	1	1	yes	characteristic
84	#104×C₁₃₁	—	2	4	no	—
85	#104×C₁₃₁	—	4	2	no	—
86	#103×C₁₃₁	C₂×C₄	1	2	yes	—
87	#103×C₁₃₁	C₂×C₄	1	2	yes	—
88	#103×C₁₃₁	C₂×C₄	1	2	yes	—
89	#103×C₁₃₁	C₂×C₄	1	2	yes	—
90	#102×C₁₃₁	C₂×C₄	1	2	yes	—
91	#102×C₁₃₁	C₂×C₄	1	2	yes	—
92	#102×C₁₃₁	C₂×C₄	1	2	yes	—
93	#102×C₁₃₁	C₂×C₄	1	2	yes	—
94	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
95	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
96	#97×C₁₃₁	—	2	2	no	—
97	#97×C₁₃₁	—	2	1	no	—
98	#97×C₁₃₁	—	2	1	no	—
99	C₄×C₈×C₁₃₁	—	2	2	no	—
100	C₄×C₈×C₁₃₁	—	2	1	no	—
101	C₄×C₈×C₁₃₁	—	2	1	no	—
102	C₄×C₈×C₁₃₁	—	4	2	no	—
103	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
104	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
105	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
106	C₄×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
107	#95×C₁₃₁	—	2	1	no	—
108	#41×C₁₃₁	—	2	2	no	—
109	#41×C₁₃₁	—	2	2	no	—
110	#41×C₁₃₁	—	2	2	no	—
111	#41×C₁₃₁	—	2	2	no	—
112	C₂×Q₈×C₁₃₁	—	2	1	no	—
113	C₂×Q₈×C₁₃₁	—	2	1	no	—
114	C₂×Q₈×C₁₃₁	C₂²×C₄	1	2	yes	—
115	C₂×D₄×C₁₃₁	—	2	2	no	—
116	C₂×D₄×C₁₃₁	—	2	1	no	—
117	C₂×D₄×C₁₃₁	—	2	1	no	—
118	C₂×D₄×C₁₃₁	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₁₃₁	—	2	2	no	—
120	C₂²×C₄×C₁₃₁	—	2	2	no	—
121	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
122	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
123	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
124	M₁₆×C₁₃₁	—	2	1	no	—
125	M₁₆×C₁₃₁	—	2	1	no	—
126	M₁₆×C₁₃₁	—	2	1	no	—
127	M₁₆×C₁₃₁	—	2	1	no	—
128	M₁₆×C₁₃₁	—	2	1	no	—
129	M₁₆×C₁₃₁	—	2	1	no	—
130	M₁₆×C₁₃₁	—	4	4	no	—
131	M₁₆×C₁₃₁	#41	1	2	yes	—
132	M₁₆×C₁₃₁	#41	1	2	yes	—
133	C₂×C₈×C₁₃₁	—	2	1	no	—
134	C₂×C₈×C₁₃₁	—	2	1	no	—
135	C₂×C₈×C₁₃₁	—	2	1	no	—
136	C₂×C₈×C₁₃₁	—	2	1	no	—
137	C₂×C₈×C₁₃₁	—	2	2	no	—
138	C₂×C₈×C₁₃₁	—	2	1	no	—
139	C₂×C₈×C₁₃₁	—	2	2	no	—
140	C₂×C₈×C₁₃₁	—	2	1	no	—
141	C₂×C₈×C₁₃₁	—	2	1	no	—
142	C₂×C₈×C₁₃₁	—	2	1	no	—
143	C₂×C₈×C₁₃₁	—	4	4	no	—
144	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
146	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
149	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
150	C₄⋊C₄×C₁₃₁	—	2	4	no	—
151	C₄⋊C₄×C₁₃₁	—	2	1	no	—
152	C₄⋊C₄×C₁₃₁	—	2	1	no	—
153	C₄⋊C₄×C₁₃₁	—	2	2	no	—
154	C₄⋊C₄×C₁₃₁	—	2	2	no	—
155	C₄⋊C₄×C₁₃₁	—	2	1	no	—
156	C₄⋊C₄×C₁₃₁	—	2	1	no	—
157	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
158	C₄⋊C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
159	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
160	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
161	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
162	C₄²×C₁₃₁	—	2	2	no	—
163	C₄²×C₁₃₁	—	2	2	no	—
164	C₄²×C₁₃₁	—	2	2	no	—
165	C₄²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
166	C₄²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
167	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
169	C₂³×C₁₃₁	—	2	2	no	—
170	C₂³×C₁₃₁	C₂×#41	1	1	yes	characteristic
171	Q₈×C₁₃₁	—	2	2	no	—
172	Q₈×C₁₃₁	—	2	2	no	—
173	Q₈×C₁₃₁	—	4	2	no	—
174	Q₈×C₁₃₁	—	8	1	no	—
175	D₄×C₁₃₁	—	2	2	no	—
176	D₄×C₁₃₁	—	2	2	no	—
177	D₄×C₁₃₁	—	4	2	no	—
178	D₄×C₁₃₁	—	4	2	no	—
179	D₄×C₁₃₁	—	4	2	no	—
180	D₄×C₁₃₁	—	8	1	no	—
181	C₂×C₄×C₁₃₁	—	2	2	no	—
182	C₂×C₄×C₁₃₁	—	2	4	no	—
183	C₂×C₄×C₁₃₁	—	2	2	no	—
184	C₂×C₄×C₁₃₁	—	2	2	no	—
185	C₂×C₄×C₁₃₁	—	2	2	no	—
186	C₂×C₄×C₁₃₁	—	4	1	no	—
187	C₂×C₄×C₁₃₁	—	4	2	no	—
188	C₂×C₄×C₁₃₁	—	4	2	no	—
189	C₂×C₄×C₁₃₁	—	4	2	no	—
190	C₂×C₄×C₁₃₁	—	4	2	no	—
191	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
196	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
199	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
200	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
201	C₈×C₁₃₁	—	2	1	no	—
202	C₈×C₁₃₁	—	2	1	no	—
203	C₈×C₁₃₁	—	2	1	no	—
204	C₈×C₁₃₁	—	2	1	no	—
205	C₈×C₁₃₁	—	2	2	no	—
206	C₈×C₁₃₁	—	2	2	no	—
207	C₈×C₁₃₁	—	2	2	no	—
208	C₈×C₁₃₁	—	2	2	no	—
209	C₈×C₁₃₁	—	4	1	no	—
210	C₈×C₁₃₁	—	4	1	no	—
211	C₂²×C₁₃₁	—	2	1	no	—
212	C₂²×C₁₃₁	—	2	1	no	—
213	C₂²×C₁₃₁	—	4	2	no	—
214	C₂²×C₁₃₁	—	4	2	no	—
215	C₂²×C₁₃₁	—	4	2	no	—
216	C₂²×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
217	C₂²×C₁₃₁	#556	1	1	yes	characteristic
218	C₂²×C₁₃₁	#559	1	1	yes	characteristic
219	C₄×C₁₃₁	—	2	1	no	—
220	C₄×C₁₃₁	—	2	1	no	—
221	C₄×C₁₃₁	—	2	1	no	—
222	C₄×C₁₃₁	—	4	4	no	—
223	C₄×C₁₃₁	—	4	2	no	—
224	C₄×C₁₃₁	—	4	2	no	—
225	C₄×C₁₃₁	—	4	2	no	—
226	C₄×C₁₃₁	#443	1	2	yes	—
227	C₄×C₁₃₁	C₄×#41	1	1	yes	characteristic
228	C₄×C₁₃₁	#520	1	1	yes	characteristic
229	C₄×C₁₃₁	#546	1	1	yes	characteristic
230	C₄×C₁₃₁	#548	1	1	yes	characteristic
231	C₂×C₁₃₁	—	2	1	no	—
232	C₂×C₁₃₁	—	2	1	no	—
233	C₂×C₁₃₁	—	8	2	no	—
234	C₂×C₁₃₁	#2367	1	1	yes	characteristic
235	C₂×C₁₃₁	C₂×#443	1	1	yes	characteristic
236	C₂×C₁₃₁	#2957	1	1	yes	characteristic
237	#23234	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
238	C₁₃₁	#23234	1	1	yes	characteristic, direct product factor, Sylow
239	#3131	C₂×C₁₃₁	1	1	yes	characteristic
240	#2962	C₂×C₁₃₁	1	1	yes	characteristic, complemented
241	#2935	C₂×C₁₃₁	1	1	yes	characteristic
242	#1986	C₂×C₁₃₁	1	1	yes	characteristic, complemented
243	#1980	C₂×C₁₃₁	1	1	yes	characteristic
244	#1937	C₂×C₁₃₁	1	2	yes	complemented
245	#1929	C₂×C₁₃₁	1	2	yes	complemented
246	#1866	C₂×C₁₃₁	1	2	yes	complemented
247	C₄×#108	C₂×C₁₃₁	1	1	yes	characteristic, complemented
248	C₈×C₄⋊C₄	C₂×C₁₃₁	1	1	yes	characteristic, complemented
249	#1764	C₂×C₁₃₁	1	2	yes	complemented
250	C₂×#131	C₂²×C₁₃₁	1	2	yes	—
251	#533	C₂²×C₁₃₁	1	1	yes	characteristic
252	#532	C₂²×C₁₃₁	1	1	yes	characteristic
253	#488	C₄×C₁₃₁	1	2	yes	complemented
254	#488	C₄×C₁₃₁	1	2	yes	complemented
255	#487	C₄×C₁₃₁	1	1	yes	characteristic, complemented
256	#487	C₄×C₁₃₁	1	1	yes	characteristic, complemented
257	#486	C₄×C₁₃₁	1	1	yes	characteristic, complemented
258	#486	C₄×C₁₃₁	1	1	yes	characteristic, complemented
259	#446	—	2	2	no	—
260	#446	—	2	1	no	—
261	C₈×Q₈	—	2	1	no	—
262	#436	—	2	1	no	—
263	#435	—	2	2	no	—
264	C₈×D₄	—	2	1	no	—
265	C₂×#108	C₂²×C₁₃₁	1	1	yes	characteristic
266	C₂×#108	C₂²×C₁₃₁	1	1	yes	characteristic
267	#424	C₂²×C₁₃₁	1	2	yes	—
268	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
269	C₂×#104	C₂²×C₁₃₁	1	2	yes	—
270	C₂×#104	C₂²×C₁₃₁	1	2	yes	—
271	#416	C₂²×C₁₃₁	1	2	yes	—
272	#416	C₂²×C₁₃₁	1	2	yes	—
273	#408	C₂²×C₁₃₁	1	2	yes	—
274	#405	C₂²×C₁₃₁	1	2	yes	—
275	C₄×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
276	C₄×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
277	C₂×C₄×C₈	C₂²×C₁₃₁	1	2	yes	—
278	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
279	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
280	#341	C₂²×C₁₃₁	1	1	yes	characteristic
281	#341	C₂²×C₁₃₁	1	1	yes	characteristic
282	#339	C₂²×C₁₃₁	1	2	yes	—
283	#339	C₂²×C₁₃₁	1	2	yes	—
284	#336	C₂²×C₁₃₁	1	2	yes	—
285	C₂×#41	C₂³×C₁₃₁	1	2	yes	—
286	#131	—	2	4	no	—
287	#131	—	4	2	no	—
288	C₂×M₁₆	—	2	2	no	—
289	C₂×M₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
290	C₂×M₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
291	C₂×M₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
292	C₂×M₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
293	C₂²×C₈	—	2	2	no	—
294	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
295	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
297	C₄⋊Q₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
298	C₄⋊Q₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
299	C₄⋊Q₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
300	#127	C₂×C₄×C₁₃₁	1	1	yes	characteristic
301	#125	C₂×C₄×C₁₃₁	1	2	yes	—
302	#124	C₂×C₄×C₁₃₁	1	2	yes	—
303	C₂²⋊Q₈	—	2	1	no	—
304	C₂²⋊Q₈	—	2	1	no	—
305	#121	—	2	1	no	—
306	#121	—	2	1	no	—
307	C₄×Q₈	—	2	1	no	—
308	C₄×Q₈	—	2	1	no	—
309	C₄×D₄	—	2	1	no	—
310	C₄×D₄	—	2	1	no	—
311	#117	C₂³×C₁₃₁	1	2	yes	—
312	C₂×C₄⋊C₄	—	2	1	no	—
313	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
314	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
315	#108	—	2	2	no	—
316	#108	—	2	2	no	—
317	C₄⋊C₈	—	2	2	no	—
318	C₄⋊C₈	—	2	2	no	—
319	C₄⋊C₈	D₄×C₁₃₁	1	1	yes	characteristic
320	C₄⋊C₈	D₄×C₁₃₁	1	2	yes	—
321	C₄⋊C₈	D₄×C₁₃₁	1	1	yes	characteristic
322	#104	—	2	4	no	—
323	#104	—	4	2	no	—
324	#103	C₂×C₄×C₁₃₁	1	2	yes	—
325	#103	C₂×C₄×C₁₃₁	1	2	yes	—
326	#103	C₂×C₄×C₁₃₁	1	2	yes	—
327	#103	C₂×C₄×C₁₃₁	1	2	yes	—
328	#102	C₂×C₄×C₁₃₁	1	2	yes	—
329	#102	C₂×C₄×C₁₃₁	1	2	yes	—
330	#102	C₂×C₄×C₁₃₁	1	2	yes	—
331	#102	C₂×C₄×C₁₃₁	1	2	yes	—
332	C₂²⋊C₈	—	2	2	no	—
333	C₂²⋊C₈	—	2	2	no	—
334	#97	—	2	2	no	—
335	#97	—	2	1	no	—
336	#97	—	2	1	no	—
337	C₄×C₈	—	2	2	no	—
338	C₄×C₈	—	2	1	no	—
339	C₄×C₈	—	2	1	no	—
340	C₄×C₈	—	4	2	no	—
341	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
342	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
343	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
344	C₄×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
345	#95	—	2	1	no	—
346	#41	—	2	2	no	—
347	#41	—	2	2	no	—
348	#41	—	2	2	no	—
349	#41	—	2	2	no	—
350	C₂×Q₈	—	2	1	no	—
351	C₂×Q₈	—	2	1	no	—
352	C₂×Q₈	C₂²×C₄×C₁₃₁	1	2	yes	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	C₂²×C₄×C₁₃₁	1	2	yes	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
360	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
361	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
362	M₁₆	—	2	1	no	—
363	M₁₆	—	2	1	no	—
364	M₁₆	—	2	1	no	—
365	M₁₆	—	2	1	no	—
366	M₁₆	—	2	1	no	—
367	M₁₆	—	2	1	no	—
368	M₁₆	—	4	4	no	—
369	M₁₆	#41×C₁₃₁	1	2	yes	—
370	M₁₆	#41×C₁₃₁	1	2	yes	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	4	4	no	—
382	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
383	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
384	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
385	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
386	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
387	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
388	C₄⋊C₄	—	2	4	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
396	C₄⋊C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
397	C₂²⋊C₄	—	2	2	no	—
398	C₂²⋊C₄	—	2	2	no	—
399	C₂²⋊C₄	—	2	2	no	—
400	C₄²	—	2	2	no	—
401	C₄²	—	2	2	no	—
402	C₄²	—	2	2	no	—
403	C₄²	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
404	C₄²	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
405	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
406	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
407	C₂³	—	2	2	no	—
408	C₂³	C₂×#41×C₁₃₁	1	1	yes	characteristic
409	Q₈	—	2	2	no	—
410	Q₈	—	2	2	no	—
411	Q₈	—	4	2	no	—
412	Q₈	—	8	1	no	—
413	D₄	—	2	2	no	—
414	D₄	—	2	2	no	—
415	D₄	—	4	2	no	—
416	D₄	—	4	2	no	—
417	D₄	—	4	2	no	—
418	D₄	—	8	1	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
430	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
431	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
432	C₂×C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
433	C₂×C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
434	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
435	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
436	C₂×C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
437	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
438	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
439	C₈	—	2	1	no	—
440	C₈	—	2	1	no	—
441	C₈	—	2	1	no	—
442	C₈	—	2	1	no	—
443	C₈	—	2	2	no	—
444	C₈	—	2	2	no	—
445	C₈	—	2	2	no	—
446	C₈	—	2	2	no	—
447	C₈	—	4	1	no	—
448	C₈	—	4	1	no	—
449	C₂²	—	2	1	no	—
450	C₂²	—	2	1	no	—
451	C₂²	—	4	2	no	—
452	C₂²	—	4	2	no	—
453	C₂²	—	4	2	no	—
454	C₂²	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
455	C₂²	#556×C₁₃₁	1	1	yes	characteristic
456	C₂²	#559×C₁₃₁	1	1	yes	characteristic
457	C₄	—	2	1	no	—
458	C₄	—	2	1	no	—
459	C₄	—	2	1	no	—
460	C₄	—	4	4	no	—
461	C₄	—	4	2	no	—
462	C₄	—	4	2	no	—
463	C₄	—	4	2	no	—
464	C₄	#443×C₁₃₁	1	2	yes	—
465	C₄	C₄×#41×C₁₃₁	1	1	yes	characteristic
466	C₄	#520×C₁₃₁	1	1	yes	characteristic
467	C₄	#546×C₁₃₁	1	1	yes	characteristic
468	C₄	#548×C₁₃₁	1	1	yes	characteristic
469	C₂	—	2	1	no	—
470	C₂	—	2	1	no	—
471	C₂	—	8	2	no	—
472	C₂	#2367×C₁₃₁	1	1	yes	characteristic
473	C₂	C₂×#443×C₁₃₁	1	1	yes	characteristic
474	C₂	#2957×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	C₂×C₄×D₄
prime radical	#532×C₁₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂×C₄⋊C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2962×C₁₃₁	C₂	semidirect
4. #1986×C₁₃₁	C₂	semidirect
6. #1937×C₁₃₁	C₂	semidirect
7. #1929×C₁₃₁	C₂	semidirect
8. #1866×C₁₃₁	C₂	semidirect
9. C₄×#108×C₁₃₁	C₂	semidirect
10. C₈×C₄⋊C₄×C₁₃₁	C₂	semidirect
11. #1764×C₁₃₁	C₂	semidirect
15. #488×C₁₃₁	C₄	semidirect
16. #488×C₁₃₁	C₄	semidirect
17. #487×C₁₃₁	C₄	semidirect
18. #487×C₁₃₁	C₄	semidirect
19. #486×C₁₃₁	C₄	semidirect
20. #486×C₁₃₁	C₄	semidirect
237. #23234	C₁₃₁	direct (N × H)
238. C₁₃₁	#23234	direct (N × H)
240. #2962	C₂×C₁₃₁	semidirect
242. #1986	C₂×C₁₃₁	semidirect
244. #1937	C₂×C₁₃₁	semidirect
245. #1929	C₂×C₁₃₁	semidirect
246. #1866	C₂×C₁₃₁	semidirect
247. C₄×#108	C₂×C₁₃₁	semidirect
248. C₈×C₄⋊C₄	C₂×C₁₃₁	semidirect
249. #1764	C₂×C₁₃₁	semidirect
253. #488	C₄×C₁₃₁	semidirect
254. #488	C₄×C₁₃₁	semidirect
255. #487	C₄×C₁₃₁	semidirect
256. #487	C₄×C₁₃₁	semidirect
257. #486	C₄×C₁₃₁	semidirect
258. #486	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3131×C₁₃₁	2
#2962×C₁₃₁	2
#2935×C₁₃₁	2
#1986×C₁₃₁	2
#1980×C₁₃₁	2
#1937×C₁₃₁	2
#1929×C₁₃₁	2
#1866×C₁₃₁	2
C₄×#108×C₁₃₁	2
C₈×C₄⋊C₄×C₁₃₁	2
#1764×C₁₃₁	2
#23234	131

Subgroups — C131×#23234 #45709118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#23234 #45709118