Subgroups — C₁₃₁×#33844 #45719728

Order 33536.

674 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#510×C₁₃₁	C₂	1	2	yes	maximal, complemented
2	#3287×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	D₄×Q₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#2248×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#2220×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#2209×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#2207×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#2189×C₁₃₁	C₂	1	2	yes	maximal, complemented
9	#2185×C₁₃₁	C₂	1	2	yes	maximal, complemented
10	C₄×Q₃₂×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#2169×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₁₃₁	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₁₃₁	C₂²	1	2	yes	—
14	#550×C₁₃₁	—	2	1	no	—
15	D₄×Q₈×C₁₃₁	—	2	1	no	—
16	#510×C₁₃₁	—	2	8	no	—
17	C₂×Q₃₂×C₁₃₁	C₂²	1	2	yes	—
18	C₂×Q₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×SD₃₂×C₁₃₁	C₂²	1	2	yes	—
20	C₂×SD₃₂×C₁₃₁	C₂²	1	2	yes	—
21	C₂×SD₃₂×C₁₃₁	C₂²	1	2	yes	—
22	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	—
23	#494×C₁₃₁	C₂²	1	1	yes	characteristic
24	#493×C₁₃₁	C₂²	1	1	yes	characteristic
25	#467×C₁₃₁	C₂²	1	2	yes	—
26	#466×C₁₃₁	C₂²	1	2	yes	—
27	#462×C₁₃₁	—	2	1	no	—
28	#460×C₁₃₁	—	2	1	no	—
29	#451×C₁₃₁	—	2	2	no	—
30	#449×C₁₃₁	—	2	2	no	—
31	C₄×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
32	C₄×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
33	C₈×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
34	#366×C₁₃₁	C₂²	1	1	yes	characteristic
35	C₄⋊C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
36	#358×C₁₃₁	C₂²	1	2	yes	—
37	#358×C₁₃₁	C₂²	1	1	yes	characteristic
38	#358×C₁₃₁	C₂²	1	1	yes	characteristic
39	#357×C₁₃₁	C₂²	1	2	yes	—
40	C₂²⋊C₁₆×C₁₃₁	C₂²	1	2	yes	—
41	C₄×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
42	C₂×#41×C₁₃₁	—	2	2	no	—
43	C₂²×Q₈×C₁₃₁	—	2	2	no	—
44	#135×C₁₃₁	—	2	4	no	—
45	#135×C₁₃₁	—	4	2	no	—
46	C₂×Q₁₆×C₁₃₁	—	2	4	no	—
47	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
48	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
49	C₂×Q₁₆×C₁₃₁	—	4	2	no	—
50	C₂×Q₁₆×C₁₃₁	C₂³	1	2	yes	—
51	C₂×Q₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
54	C₂×D₈×C₁₃₁	—	2	1	no	—
55	C₂×D₈×C₁₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₁₃₁	C₂³	1	2	yes	—
57	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
58	C₄⋊Q₈×C₁₃₁	—	4	1	no	—
59	#127×C₁₃₁	—	2	1	no	—
60	#127×C₁₃₁	—	4	1	no	—
61	C₂²⋊Q₈×C₁₃₁	—	2	2	no	—
62	C₂²⋊Q₈×C₁₃₁	—	4	2	no	—
63	#121×C₁₃₁	—	2	2	no	—
64	#121×C₁₃₁	—	4	2	no	—
65	C₄×Q₈×C₁₃₁	—	2	1	no	—
66	C₄×Q₈×C₁₃₁	—	2	1	no	—
67	C₄×D₄×C₁₃₁	—	4	1	no	—
68	C₄×D₄×C₁₃₁	—	4	1	no	—
69	C₄×D₄×C₁₃₁	D₄	1	1	yes	characteristic
70	Q₃₂×C₁₃₁	—	2	4	no	—
71	Q₃₂×C₁₃₁	D₄	1	4	yes	complemented
72	SD₃₂×C₁₃₁	—	2	4	no	—
73	SD₃₂×C₁₃₁	—	2	4	no	—
74	SD₃₂×C₁₃₁	—	4	2	no	—
75	M₃₂×C₁₃₁	—	2	4	no	—
76	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
77	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
79	#107×C₁₃₁	C₂³	1	1	yes	characteristic
80	#107×C₁₃₁	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₁₃₁	D₄	1	1	yes	characteristic
82	#103×C₁₃₁	—	2	1	no	—
83	#103×C₁₃₁	—	2	1	no	—
84	#103×C₁₃₁	—	2	2	no	—
85	#102×C₁₃₁	—	2	2	no	—
86	C₂²⋊C₈×C₁₃₁	D₄	1	2	yes	—
87	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
88	#41×C₁₃₁	—	4	4	no	—
89	#41×C₁₃₁	—	8	2	no	—
90	C₂×Q₈×C₁₃₁	—	2	1	no	—
91	C₂×Q₈×C₁₃₁	—	2	1	no	—
92	C₂×Q₈×C₁₃₁	—	2	2	no	—
93	C₂×Q₈×C₁₃₁	—	4	4	no	—
94	C₂×Q₈×C₁₃₁	—	4	2	no	—
95	C₂×Q₈×C₁₃₁	—	8	2	no	—
96	C₂×D₄×C₁₃₁	—	2	2	no	—
97	C₂×D₄×C₁₃₁	—	4	2	no	—
98	C₂×D₄×C₁₃₁	—	4	2	no	—
99	C₂×D₄×C₁₃₁	—	4	1	no	—
100	C₂×D₄×C₁₃₁	—	4	1	no	—
101	C₂×D₄×C₁₃₁	D₈	1	1	yes	characteristic
102	C₂²×C₄×C₁₃₁	—	4	2	no	—
103	C₂²×C₄×C₁₃₁	—	4	2	no	—
104	C₂²×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
105	Q₁₆×C₁₃₁	—	2	2	no	—
106	Q₁₆×C₁₃₁	—	2	1	no	—
107	Q₁₆×C₁₃₁	—	2	1	no	—
108	Q₁₆×C₁₃₁	—	2	2	no	—
109	Q₁₆×C₁₃₁	—	4	4	no	—
110	Q₁₆×C₁₃₁	—	8	1	no	—
111	Q₁₆×C₁₃₁	C₂×D₄	1	2	yes	—
112	Q₁₆×C₁₃₁	C₂×D₄	1	2	yes	—
113	SD₁₆×C₁₃₁	—	4	4	no	—
114	D₈×C₁₃₁	—	2	2	no	—
115	D₈×C₁₃₁	—	2	2	no	—
116	D₈×C₁₃₁	—	8	1	no	—
117	C₂×C₈×C₁₃₁	—	2	2	no	—
118	C₂×C₈×C₁₃₁	—	2	2	no	—
119	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
122	C₄⋊C₄×C₁₃₁	—	2	1	no	—
123	C₄⋊C₄×C₁₃₁	—	2	1	no	—
124	C₄⋊C₄×C₁₃₁	—	4	1	no	—
125	C₄⋊C₄×C₁₃₁	—	4	1	no	—
126	C₄⋊C₄×C₁₃₁	—	4	1	no	—
127	C₄⋊C₄×C₁₃₁	—	4	2	no	—
128	C₄⋊C₄×C₁₃₁	—	4	1	no	—
129	C₄⋊C₄×C₁₃₁	D₈	1	1	yes	characteristic
130	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
131	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
132	C₂²⋊C₄×C₁₃₁	D₈	1	2	yes	—
133	C₄²×C₁₃₁	—	4	1	no	—
134	C₄²×C₁₃₁	—	4	1	no	—
135	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
136	C₁₆×C₁₃₁	—	2	2	no	—
137	C₁₆×C₁₃₁	—	2	1	no	—
138	C₁₆×C₁₃₁	C₂×D₄	1	2	yes	—
139	C₂³×C₁₃₁	—	4	2	no	—
140	C₂³×C₁₃₁	C₂×D₈	1	2	yes	—
141	Q₈×C₁₃₁	—	2	2	no	—
142	Q₈×C₁₃₁	—	2	2	no	—
143	Q₈×C₁₃₁	—	4	1	no	—
144	Q₈×C₁₃₁	—	4	1	no	—
145	Q₈×C₁₃₁	—	4	2	no	—
146	Q₈×C₁₃₁	—	4	2	no	—
147	Q₈×C₁₃₁	—	8	4	no	—
148	D₄×C₁₃₁	—	2	2	no	—
149	D₄×C₁₃₁	—	4	2	no	—
150	D₄×C₁₃₁	—	4	2	no	—
151	D₄×C₁₃₁	—	8	4	no	—
152	D₄×C₁₃₁	—	8	2	no	—
153	D₄×C₁₃₁	—	16	1	no	—
154	D₄×C₁₃₁	—	16	2	no	—
155	C₂×C₄×C₁₃₁	—	2	2	no	—
156	C₂×C₄×C₁₃₁	—	2	2	no	—
157	C₂×C₄×C₁₃₁	—	4	1	no	—
158	C₂×C₄×C₁₃₁	—	4	1	no	—
159	C₂×C₄×C₁₃₁	—	4	1	no	—
160	C₂×C₄×C₁₃₁	—	4	1	no	—
161	C₂×C₄×C₁₃₁	—	4	2	no	—
162	C₂×C₄×C₁₃₁	—	8	4	no	—
163	C₂×C₄×C₁₃₁	—	8	4	no	—
164	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
165	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
166	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
167	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
168	C₈×C₁₃₁	—	2	2	no	—
169	C₈×C₁₃₁	—	2	1	no	—
170	C₈×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
171	C₈×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
172	C₂²×C₁₃₁	—	2	2	no	—
173	C₂²×C₁₃₁	—	2	2	no	—
174	C₂²×C₁₃₁	—	2	2	no	—
175	C₂²×C₁₃₁	—	8	2	no	—
176	C₂²×C₁₃₁	—	8	2	no	—
177	C₂²×C₁₃₁	—	8	2	no	—
178	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
179	C₄×C₁₃₁	—	2	1	no	—
180	C₄×C₁₃₁	—	2	2	no	—
181	C₄×C₁₃₁	—	4	2	no	—
182	C₄×C₁₃₁	—	4	2	no	—
183	C₄×C₁₃₁	—	8	1	no	—
184	C₄×C₁₃₁	—	8	1	no	—
185	C₄×C₁₃₁	—	8	2	no	—
186	C₄×C₁₃₁	#510	1	2	yes	complemented
187	C₄×C₁₃₁	D₄²	1	1	yes	characteristic
188	C₄×C₁₃₁	#576	1	1	yes	characteristic
189	C₂×C₁₃₁	—	4	2	no	—
190	C₂×C₁₃₁	—	16	2	no	—
191	C₂×C₁₃₁	D₄×D₈	1	1	yes	characteristic
192	C₂×C₁₃₁	C₂×#510	1	1	yes	characteristic
193	C₂×C₁₃₁	#3416	1	1	yes	characteristic
194	#33844	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₁₃₁	#33844	1	1	yes	characteristic, direct product factor, Sylow
196	C₂×#510	C₂×C₁₃₁	1	2	yes	complemented
197	#3287	C₂×C₁₃₁	1	1	yes	characteristic
198	D₄×Q₁₆	C₂×C₁₃₁	1	1	yes	characteristic, complemented
199	#2248	C₂×C₁₃₁	1	1	yes	characteristic, complemented
200	#2220	C₂×C₁₃₁	1	2	yes	complemented
201	#2209	C₂×C₁₃₁	1	1	yes	characteristic, complemented
202	#2207	C₂×C₁₃₁	1	1	yes	characteristic, complemented
203	#2189	C₂×C₁₃₁	1	2	yes	complemented
204	#2185	C₂×C₁₃₁	1	2	yes	complemented
205	C₄×Q₃₂	C₂×C₁₃₁	1	1	yes	characteristic, complemented
206	#2169	C₂×C₁₃₁	1	1	yes	characteristic, complemented
207	C₂×#135	C₂²×C₁₃₁	1	2	yes	—
208	C₂²×Q₁₆	C₂²×C₁₃₁	1	2	yes	—
209	#550	—	2	1	no	—
210	D₄×Q₈	—	2	1	no	—
211	#510	—	2	8	no	—
212	C₂×Q₃₂	C₂²×C₁₃₁	1	2	yes	—
213	C₂×Q₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
214	C₂×SD₃₂	C₂²×C₁₃₁	1	2	yes	—
215	C₂×SD₃₂	C₂²×C₁₃₁	1	2	yes	—
216	C₂×SD₃₂	C₂²×C₁₃₁	1	2	yes	—
217	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	—
218	#494	C₂²×C₁₃₁	1	1	yes	characteristic
219	#493	C₂²×C₁₃₁	1	1	yes	characteristic
220	#467	C₂²×C₁₃₁	1	2	yes	—
221	#466	C₂²×C₁₃₁	1	2	yes	—
222	#462	—	2	1	no	—
223	#460	—	2	1	no	—
224	#451	—	2	2	no	—
225	#449	—	2	2	no	—
226	C₄×Q₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
227	C₄×Q₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
228	C₈×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
229	#366	C₂²×C₁₃₁	1	1	yes	characteristic
230	C₄⋊C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
231	#358	C₂²×C₁₃₁	1	1	yes	characteristic
232	#358	C₂²×C₁₃₁	1	1	yes	characteristic
233	#358	C₂²×C₁₃₁	1	2	yes	—
234	#357	C₂²×C₁₃₁	1	2	yes	—
235	C₂²⋊C₁₆	C₂²×C₁₃₁	1	2	yes	—
236	C₄×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
237	C₂×#41	—	2	2	no	—
238	C₂²×Q₈	—	2	2	no	—
239	#135	—	2	4	no	—
240	#135	—	4	2	no	—
241	C₂×Q₁₆	—	2	4	no	—
242	C₂×Q₁₆	—	2	2	no	—
243	C₂×Q₁₆	—	2	1	no	—
244	C₂×Q₁₆	—	4	2	no	—
245	C₂×Q₁₆	C₂³×C₁₃₁	1	2	yes	—
246	C₂×Q₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
247	C₂×Q₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
248	C₂×SD₁₆	—	2	2	no	—
249	C₂×D₈	—	2	1	no	—
250	C₂×D₈	C₂³×C₁₃₁	1	2	yes	—
251	C₂²×C₈	C₂³×C₁₃₁	1	2	yes	—
252	C₄⋊Q₈	—	2	1	no	—
253	C₄⋊Q₈	—	4	1	no	—
254	#127	—	2	1	no	—
255	#127	—	4	1	no	—
256	C₂²⋊Q₈	—	2	2	no	—
257	C₂²⋊Q₈	—	4	2	no	—
258	#121	—	2	2	no	—
259	#121	—	4	2	no	—
260	C₄×Q₈	—	2	1	no	—
261	C₄×Q₈	—	2	1	no	—
262	C₄×D₄	—	4	1	no	—
263	C₄×D₄	—	4	1	no	—
264	C₄×D₄	D₄×C₁₃₁	1	1	yes	characteristic
265	Q₃₂	—	2	4	no	—
266	Q₃₂	D₄×C₁₃₁	1	4	yes	complemented
267	SD₃₂	—	2	4	no	—
268	SD₃₂	—	2	4	no	—
269	SD₃₂	—	4	2	no	—
270	M₃₂	—	2	4	no	—
271	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
272	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
273	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
274	#107	C₂³×C₁₃₁	1	1	yes	characteristic
275	#107	C₂³×C₁₃₁	1	1	yes	characteristic
276	C₄⋊C₈	D₄×C₁₃₁	1	1	yes	characteristic
277	#103	—	2	1	no	—
278	#103	—	2	1	no	—
279	#103	—	2	2	no	—
280	#102	—	2	2	no	—
281	C₂²⋊C₈	D₄×C₁₃₁	1	2	yes	—
282	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
283	#41	—	4	4	no	—
284	#41	—	8	2	no	—
285	C₂×Q₈	—	2	1	no	—
286	C₂×Q₈	—	2	1	no	—
287	C₂×Q₈	—	2	2	no	—
288	C₂×Q₈	—	4	4	no	—
289	C₂×Q₈	—	4	2	no	—
290	C₂×Q₈	—	8	2	no	—
291	C₂×D₄	—	2	2	no	—
292	C₂×D₄	—	4	2	no	—
293	C₂×D₄	—	4	2	no	—
294	C₂×D₄	—	4	1	no	—
295	C₂×D₄	—	4	1	no	—
296	C₂×D₄	D₈×C₁₃₁	1	1	yes	characteristic
297	C₂²×C₄	—	4	2	no	—
298	C₂²×C₄	—	4	2	no	—
299	C₂²×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
300	Q₁₆	—	2	2	no	—
301	Q₁₆	—	2	1	no	—
302	Q₁₆	—	2	1	no	—
303	Q₁₆	—	2	2	no	—
304	Q₁₆	—	4	4	no	—
305	Q₁₆	—	8	1	no	—
306	Q₁₆	C₂×D₄×C₁₃₁	1	2	yes	—
307	Q₁₆	C₂×D₄×C₁₃₁	1	2	yes	—
308	SD₁₆	—	4	4	no	—
309	D₈	—	2	2	no	—
310	D₈	—	2	2	no	—
311	D₈	—	8	1	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
315	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
316	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
317	C₄⋊C₄	—	2	1	no	—
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	4	1	no	—
320	C₄⋊C₄	—	4	1	no	—
321	C₄⋊C₄	—	4	1	no	—
322	C₄⋊C₄	—	4	2	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	D₈×C₁₃₁	1	1	yes	characteristic
325	C₂²⋊C₄	—	4	2	no	—
326	C₂²⋊C₄	—	4	2	no	—
327	C₂²⋊C₄	D₈×C₁₃₁	1	2	yes	—
328	C₄²	—	4	1	no	—
329	C₄²	—	4	1	no	—
330	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
331	C₁₆	—	2	2	no	—
332	C₁₆	—	2	1	no	—
333	C₁₆	C₂×D₄×C₁₃₁	1	2	yes	—
334	C₂³	—	4	2	no	—
335	C₂³	C₂×D₈×C₁₃₁	1	2	yes	—
336	Q₈	—	2	2	no	—
337	Q₈	—	2	2	no	—
338	Q₈	—	4	1	no	—
339	Q₈	—	4	1	no	—
340	Q₈	—	4	2	no	—
341	Q₈	—	4	2	no	—
342	Q₈	—	8	4	no	—
343	D₄	—	2	2	no	—
344	D₄	—	4	2	no	—
345	D₄	—	4	2	no	—
346	D₄	—	8	4	no	—
347	D₄	—	8	2	no	—
348	D₄	—	16	1	no	—
349	D₄	—	16	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	1	no	—
355	C₂×C₄	—	4	1	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	8	4	no	—
358	C₂×C₄	—	8	4	no	—
359	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
360	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
361	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
362	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
363	C₈	—	2	2	no	—
364	C₈	—	2	1	no	—
365	C₈	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
366	C₈	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
367	C₂²	—	2	2	no	—
368	C₂²	—	2	2	no	—
369	C₂²	—	2	2	no	—
370	C₂²	—	8	2	no	—
371	C₂²	—	8	2	no	—
372	C₂²	—	8	2	no	—
373	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
374	C₄	—	2	2	no	—
375	C₄	—	2	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	4	2	no	—
378	C₄	—	8	2	no	—
379	C₄	—	8	1	no	—
380	C₄	—	8	1	no	—
381	C₄	#510×C₁₃₁	1	2	yes	complemented
382	C₄	D₄²×C₁₃₁	1	1	yes	characteristic
383	C₄	#576×C₁₃₁	1	1	yes	characteristic
384	C₂	—	4	2	no	—
385	C₂	—	16	2	no	—
386	C₂	D₄×D₈×C₁₃₁	1	1	yes	characteristic
387	C₂	C₂×#510×C₁₃₁	1	1	yes	characteristic
388	C₂	#3416×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
prime radical	#3287×C₁₃₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₄×D₄×C₁₃₁ ◁ C₈×D₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#510×C₁₃₁	C₂	semidirect
3. D₄×Q₁₆×C₁₃₁	C₂	semidirect
4. #2248×C₁₃₁	C₂	semidirect
5. #2220×C₁₃₁	C₂	semidirect
6. #2209×C₁₃₁	C₂	semidirect
7. #2207×C₁₃₁	C₂	semidirect
8. #2189×C₁₃₁	C₂	semidirect
9. #2185×C₁₃₁	C₂	semidirect
10. C₄×Q₃₂×C₁₃₁	C₂	semidirect
11. #2169×C₁₃₁	C₂	semidirect
71. Q₃₂×C₁₃₁	D₄	semidirect
186. C₄×C₁₃₁	#510	semidirect
194. #33844	C₁₃₁	direct (N × H)
195. C₁₃₁	#33844	direct (N × H)
196. C₂×#510	C₂×C₁₃₁	semidirect
198. D₄×Q₁₆	C₂×C₁₃₁	semidirect
199. #2248	C₂×C₁₃₁	semidirect
200. #2220	C₂×C₁₃₁	semidirect
201. #2209	C₂×C₁₃₁	semidirect
202. #2207	C₂×C₁₃₁	semidirect
203. #2189	C₂×C₁₃₁	semidirect
204. #2185	C₂×C₁₃₁	semidirect
205. C₄×Q₃₂	C₂×C₁₃₁	semidirect
206. #2169	C₂×C₁₃₁	semidirect
266. Q₃₂	D₄×C₁₃₁	semidirect
381. C₄	#510×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#510×C₁₃₁	2
#3287×C₁₃₁	2
D₄×Q₁₆×C₁₃₁	2
#2248×C₁₃₁	2
#2220×C₁₃₁	2
#2209×C₁₃₁	2
#2207×C₁₃₁	2
#2189×C₁₃₁	2
#2185×C₁₃₁	2
C₄×Q₃₂×C₁₃₁	2
#2169×C₁₃₁	2
#33844	131

Subgroups — C131×#33844 #45719728

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33844 #45719728