Subgroups — C₂×C₇₁×#2368 #31855014

Order 18176.

3094 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#2368	C₂	1	16	yes	maximal, direct product factor
3	C₂×#391×C₇₁	C₂	1	2	yes	maximal, complemented
4	C₂×#387×C₇₁	C₂	1	2	yes	maximal, complemented
5	C₂×#387×C₇₁	C₂	1	2	yes	maximal, complemented
6	C₂×#385×C₇₁	C₂	1	2	yes	maximal
7	C₂×C₄×C₄⋊C₄×C₇₁	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₂²⋊C₄×C₇₁	C₂	1	2	yes	maximal
9	C₂×C₄×C₂²⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
10	C₂²×#95×C₇₁	C₂	1	1	yes	maximal, characteristic
11	C₂³×Q₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₇₁	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Q₈×C₇₁	C₄	1	16	yes	complemented
15	C₂×C₂²⋊Q₈×C₇₁	C₂²	1	8	yes	—
16	C₂²×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
20	C₂²×C₂²⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊C₄×C₇₁	C₂²	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₇₁	C₂²	1	2	yes	—
24	C₂²×C₄²×C₇₁	C₂²	1	2	yes	—
25	C₂²×C₄²×C₇₁	C₂²	1	2	yes	—
26	#391×C₇₁	C₂²	1	16	yes	complemented
27	#387×C₇₁	C₂²	1	16	yes	complemented
28	#387×C₇₁	C₂²	1	16	yes	complemented
29	#385×C₇₁	C₂²	1	16	yes	—
30	C₄×C₄⋊C₄×C₇₁	C₂²	1	16	yes	complemented
31	C₄×C₂²⋊C₄×C₇₁	C₂²	1	16	yes	—
32	C₄×C₂²⋊C₄×C₇₁	C₂²	1	8	yes	—
33	C₂×#95×C₇₁	C₂²	1	2	yes	—
34	C₂×#95×C₇₁	C₂²	1	2	yes	—
35	C₂×#95×C₇₁	C₂²	1	1	yes	characteristic
36	C₂×#95×C₇₁	C₂²	1	2	yes	—
37	C₂×#95×C₇₁	C₂²	1	2	yes	—
38	C₂×#95×C₇₁	C₂²	1	2	yes	—
39	C₂×#95×C₇₁	C₂²	1	2	yes	—
40	C₂×#95×C₇₁	C₂²	1	8	yes	—
41	C₂×#95×C₇₁	C₂²	1	1	yes	characteristic
42	C₂⁵×C₇₁	C₂³	1	1	yes	characteristic
43	C₂²×Q₈×C₇₁	—	4	4	no	—
44	C₂²×Q₈×C₇₁	C₂×C₄	1	8	yes	complemented
45	C₂²×Q₈×C₇₁	C₂³	1	4	yes	—
46	C₂³×C₄×C₇₁	—	2	8	no	—
47	C₂³×C₄×C₇₁	—	2	8	no	—
48	C₂³×C₄×C₇₁	—	4	4	no	—
49	C₂³×C₄×C₇₁	—	4	4	no	—
50	C₂³×C₄×C₇₁	C₂×C₄	1	8	yes	—
51	C₂³×C₄×C₇₁	C₂³	1	2	yes	—
52	C₂³×C₄×C₇₁	C₂³	1	2	yes	—
53	C₂³×C₄×C₇₁	C₂³	1	8	yes	—
54	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₇₁	C₂³	1	2	yes	—
56	C₂³×C₄×C₇₁	C₂³	1	2	yes	—
57	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₇₁	C₂³	1	2	yes	—
59	C₂³×C₄×C₇₁	C₂³	1	2	yes	—
60	C₂³×C₄×C₇₁	C₂³	1	4	yes	—
61	C₂²⋊Q₈×C₇₁	C₂×C₄	1	64	yes	complemented
62	C₂×C₄⋊C₄×C₇₁	—	2	8	no	—
63	C₂×C₄⋊C₄×C₇₁	—	4	4	no	—
64	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	8	yes	complemented
65	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	16	yes	complemented
66	C₂×C₄⋊C₄×C₇₁	C₂³	1	8	yes	—
67	C₂×C₄⋊C₄×C₇₁	C₂³	1	8	yes	—
68	C₂×C₄⋊C₄×C₇₁	C₂³	1	8	yes	—
69	C₂×C₄⋊C₄×C₇₁	C₂³	1	4	yes	—
70	C₂×C₂²⋊C₄×C₇₁	—	2	4	no	—
71	C₂×C₂²⋊C₄×C₇₁	C₂×C₄	1	16	yes	—
72	C₂×C₂²⋊C₄×C₇₁	Q₈	1	8	yes	complemented
73	C₂×C₂²⋊C₄×C₇₁	C₂³	1	4	yes	—
74	C₂×C₂²⋊C₄×C₇₁	C₂³	1	8	yes	—
75	C₂×C₂²⋊C₄×C₇₁	C₂³	1	4	yes	—
76	C₂×C₄²×C₇₁	—	2	8	no	—
77	C₂×C₄²×C₇₁	—	4	4	no	—
78	C₂×C₄²×C₇₁	—	4	4	no	—
79	C₂×C₄²×C₇₁	C₂³	1	8	yes	—
80	C₂×C₄²×C₇₁	C₂³	1	8	yes	—
81	C₂×C₄²×C₇₁	C₂³	1	8	yes	—
82	#95×C₇₁	C₂³	1	4	yes	—
83	#95×C₇₁	C₂³	1	4	yes	—
84	#95×C₇₁	C₂³	1	8	yes	—
85	#95×C₇₁	C₂³	1	8	yes	—
86	#95×C₇₁	C₂³	1	8	yes	—
87	#95×C₇₁	C₂³	1	8	yes	—
88	#95×C₇₁	C₂³	1	8	yes	—
89	#95×C₇₁	C₂³	1	8	yes	—
90	C₂⁴×C₇₁	—	2	4	no	—
91	C₂⁴×C₇₁	—	2	4	no	—
92	C₂⁴×C₇₁	C₂²×C₄	1	4	yes	—
93	C₂⁴×C₇₁	C₂×Q₈	1	4	yes	—
94	C₂⁴×C₇₁	#41	1	4	yes	—
95	C₂⁴×C₇₁	C₂⁴	1	2	yes	—
96	C₂⁴×C₇₁	C₂⁴	1	1	yes	characteristic
97	C₂×Q₈×C₇₁	—	4	16	no	—
98	C₂×Q₈×C₇₁	—	4	8	no	—
99	C₂×Q₈×C₇₁	C₂²×C₄	1	16	yes	complemented
100	C₂²×C₄×C₇₁	—	2	4	no	—
101	C₂²×C₄×C₇₁	—	2	4	no	—
102	C₂²×C₄×C₇₁	—	2	4	no	—
103	C₂²×C₄×C₇₁	—	2	4	no	—
104	C₂²×C₄×C₇₁	—	2	4	no	—
105	C₂²×C₄×C₇₁	—	2	4	no	—
106	C₂²×C₄×C₇₁	—	2	4	no	—
107	C₂²×C₄×C₇₁	—	2	4	no	—
108	C₂²×C₄×C₇₁	—	2	4	no	—
109	C₂²×C₄×C₇₁	—	2	4	no	—
110	C₂²×C₄×C₇₁	—	2	4	no	—
111	C₂²×C₄×C₇₁	—	2	4	no	—
112	C₂²×C₄×C₇₁	—	2	16	no	—
113	C₂²×C₄×C₇₁	—	2	4	no	—
114	C₂²×C₄×C₇₁	—	2	4	no	—
115	C₂²×C₄×C₇₁	—	2	4	no	—
116	C₂²×C₄×C₇₁	—	2	4	no	—
117	C₂²×C₄×C₇₁	—	2	16	no	—
118	C₂²×C₄×C₇₁	—	4	16	no	—
119	C₂²×C₄×C₇₁	—	4	8	no	—
120	C₂²×C₄×C₇₁	—	4	8	no	—
121	C₂²×C₄×C₇₁	—	4	32	no	—
122	C₂²×C₄×C₇₁	C₂²×C₄	1	8	yes	—
123	C₂²×C₄×C₇₁	C₂²×C₄	1	8	yes	—
124	C₂²×C₄×C₇₁	C₂²×C₄	1	4	yes	—
125	C₂²×C₄×C₇₁	C₂²×C₄	1	4	yes	—
126	C₂²×C₄×C₇₁	C₂²×C₄	1	16	yes	—
127	C₂²×C₄×C₇₁	C₂×Q₈	1	8	yes	—
128	C₂²×C₄×C₇₁	#41	1	8	yes	—
129	C₂²×C₄×C₇₁	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₇₁	C₂⁴	1	4	yes	—
131	C₂²×C₄×C₇₁	C₂⁴	1	2	yes	—
132	C₂²×C₄×C₇₁	C₂⁴	1	2	yes	—
133	C₂²×C₄×C₇₁	C₂⁴	1	4	yes	—
134	C₂²×C₄×C₇₁	C₂⁴	1	4	yes	—
135	C₂²×C₄×C₇₁	C₂⁴	1	4	yes	—
136	C₂²×C₄×C₇₁	C₂⁴	1	4	yes	—
137	C₄⋊C₄×C₇₁	—	2	16	no	—
138	C₄⋊C₄×C₇₁	—	4	8	no	—
139	C₄⋊C₄×C₇₁	C₂²×C₄	1	16	yes	complemented
140	C₄⋊C₄×C₇₁	C₂²×C₄	1	32	yes	complemented
141	C₂²⋊C₄×C₇₁	—	2	8	no	—
142	C₂²⋊C₄×C₇₁	C₂²×C₄	1	32	yes	—
143	C₂²⋊C₄×C₇₁	C₂×Q₈	1	16	yes	complemented
144	C₄²×C₇₁	—	2	16	no	—
145	C₄²×C₇₁	—	4	8	no	—
146	C₄²×C₇₁	—	4	8	no	—
147	C₂³×C₇₁	—	2	8	no	—
148	C₂³×C₇₁	—	2	8	no	—
149	C₂³×C₇₁	—	2	16	no	—
150	C₂³×C₇₁	—	2	16	no	—
151	C₂³×C₇₁	—	2	4	no	—
152	C₂³×C₇₁	—	2	4	no	—
153	C₂³×C₇₁	C₄×Q₈	1	16	yes	—
154	C₂³×C₇₁	C₂³×C₄	1	4	yes	—
155	C₂³×C₇₁	C₂³×C₄	1	2	yes	—
156	C₂³×C₇₁	C₂²×Q₈	1	4	yes	—
157	C₂³×C₇₁	C₂²×Q₈	1	2	yes	—
158	C₂³×C₇₁	C₂×#41	1	4	yes	—
159	C₂³×C₇₁	C₂×#41	1	2	yes	—
160	C₂³×C₇₁	C₂×#41	1	4	yes	—
161	C₂³×C₇₁	H(5,𝔽₂)	1	4	yes	—
162	C₂³×C₇₁	C₂⁵	1	1	yes	characteristic
163	Q₈×C₇₁	—	4	16	no	—
164	C₂×C₄×C₇₁	—	2	16	no	—
165	C₂×C₄×C₇₁	—	2	8	no	—
166	C₂×C₄×C₇₁	—	2	8	no	—
167	C₂×C₄×C₇₁	—	2	16	no	—
168	C₂×C₄×C₇₁	—	2	16	no	—
169	C₂×C₄×C₇₁	—	2	16	no	—
170	C₂×C₄×C₇₁	—	2	4	no	—
171	C₂×C₄×C₇₁	—	2	4	no	—
172	C₂×C₄×C₇₁	—	2	4	no	—
173	C₂×C₄×C₇₁	—	2	4	no	—
174	C₂×C₄×C₇₁	—	2	4	no	—
175	C₂×C₄×C₇₁	—	2	4	no	—
176	C₂×C₄×C₇₁	—	2	4	no	—
177	C₂×C₄×C₇₁	—	2	4	no	—
178	C₂×C₄×C₇₁	—	2	4	no	—
179	C₂×C₄×C₇₁	—	2	4	no	—
180	C₂×C₄×C₇₁	—	2	4	no	—
181	C₂×C₄×C₇₁	—	2	4	no	—
182	C₂×C₄×C₇₁	—	2	4	no	—
183	C₂×C₄×C₇₁	—	2	4	no	—
184	C₂×C₄×C₇₁	—	2	4	no	—
185	C₂×C₄×C₇₁	—	2	4	no	—
186	C₂×C₄×C₇₁	—	4	16	no	—
187	C₂×C₄×C₇₁	—	4	32	no	—
188	C₂×C₄×C₇₁	—	4	8	no	—
189	C₂×C₄×C₇₁	—	4	8	no	—
190	C₂×C₄×C₇₁	C₂³×C₄	1	8	yes	—
191	C₂×C₄×C₇₁	C₂³×C₄	1	8	yes	—
192	C₂×C₄×C₇₁	C₂³×C₄	1	4	yes	—
193	C₂×C₄×C₇₁	C₂³×C₄	1	4	yes	—
194	C₂×C₄×C₇₁	C₂²×Q₈	1	8	yes	—
195	C₂×C₄×C₇₁	C₂×#41	1	8	yes	—
196	C₂²×C₇₁	—	2	32	no	—
197	C₂²×C₇₁	—	2	4	no	—
198	C₂²×C₇₁	—	2	4	no	—
199	C₂²×C₇₁	—	2	8	no	—
200	C₂²×C₇₁	—	2	8	no	—
201	C₂²×C₇₁	C₂×C₄×Q₈	1	8	yes	—
202	C₂²×C₇₁	C₂×C₄×Q₈	1	4	yes	—
203	C₂²×C₇₁	C₄×#41	1	4	yes	—
204	C₂²×C₇₁	#518	1	4	yes	—
205	C₂²×C₇₁	#551	1	8	yes	—
206	C₂²×C₇₁	#554	1	8	yes	—
207	C₂²×C₇₁	C₂⁴×C₄	1	1	yes	characteristic
208	C₂²×C₇₁	C₂³×Q₈	1	1	yes	characteristic
209	C₂²×C₇₁	C₂²×#41	1	1	yes	characteristic
210	C₂²×C₇₁	C₂²×#41	1	2	yes	—
211	C₂²×C₇₁	C₂×H(5,𝔽₂)	1	2	yes	—
212	C₄×C₇₁	—	2	8	no	—
213	C₄×C₇₁	—	2	4	no	—
214	C₄×C₇₁	—	2	4	no	—
215	C₄×C₇₁	—	2	8	no	—
216	C₄×C₇₁	—	2	8	no	—
217	C₄×C₇₁	—	2	8	no	—
218	C₄×C₇₁	—	4	4	no	—
219	C₄×C₇₁	—	4	4	no	—
220	C₂×#2368	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂×C₇₁	—	2	8	no	—
222	C₂×C₇₁	#2368	1	8	yes	direct product factor
223	C₂×C₇₁	C₂²×C₄×Q₈	1	1	yes	characteristic
224	C₂×C₇₁	C₂×C₄×#41	1	1	yes	characteristic
225	C₂×C₇₁	C₂×#518	1	1	yes	characteristic
226	C₂×C₇₁	C₂×#551	1	2	yes	—
227	C₂×C₇₁	C₂×#554	1	2	yes	—
228	C₂²×C₂²⋊Q₈	C₂×C₇₁	1	1	yes	characteristic
229	#2368	C₂×C₇₁	1	16	yes	direct product factor
230	C₂×#391	C₂×C₇₁	1	2	yes	complemented
231	C₂×#387	C₂×C₇₁	1	2	yes	complemented
232	C₂×#387	C₂×C₇₁	1	2	yes	complemented
233	C₂×#385	C₂×C₇₁	1	2	yes	—
234	C₂×C₄×C₄⋊C₄	C₂×C₇₁	1	2	yes	complemented
235	C₂×C₄×C₂²⋊C₄	C₂×C₇₁	1	1	yes	characteristic
236	C₂×C₄×C₂²⋊C₄	C₂×C₇₁	1	2	yes	—
237	C₂²×#95	C₂×C₇₁	1	1	yes	characteristic
238	C₇₁	C₂×#2368	1	1	yes	characteristic, direct product factor, Sylow
239	C₂³×Q₈	C₂²×C₇₁	1	1	yes	characteristic
240	C₂⁴×C₄	C₂²×C₇₁	1	2	yes	—
241	C₂⁴×C₄	C₂²×C₇₁	1	1	yes	characteristic
242	C₂×C₂²⋊Q₈	C₄×C₇₁	1	16	yes	complemented
243	C₂×C₂²⋊Q₈	C₂²×C₇₁	1	8	yes	—
244	C₂²×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
245	C₂²×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
246	C₂²×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
247	C₂²×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
248	C₂²×C₂²⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
249	C₂²×C₂²⋊C₄	C₂²×C₇₁	1	2	yes	—
250	C₂²×C₂²⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
251	C₂²×C₄²	C₂²×C₇₁	1	2	yes	—
252	C₂²×C₄²	C₂²×C₇₁	1	2	yes	—
253	C₂²×C₄²	C₂²×C₇₁	1	2	yes	—
254	#391	C₂²×C₇₁	1	16	yes	complemented
255	#387	C₂²×C₇₁	1	16	yes	complemented
256	#387	C₂²×C₇₁	1	16	yes	complemented
257	#385	C₂²×C₇₁	1	16	yes	—
258	C₄×C₄⋊C₄	C₂²×C₇₁	1	16	yes	complemented
259	C₄×C₂²⋊C₄	C₂²×C₇₁	1	16	yes	—
260	C₄×C₂²⋊C₄	C₂²×C₇₁	1	8	yes	—
261	C₂×#95	C₂²×C₇₁	1	8	yes	—
262	C₂×#95	C₂²×C₇₁	1	1	yes	characteristic
263	C₂×#95	C₂²×C₇₁	1	1	yes	characteristic
264	C₂×#95	C₂²×C₇₁	1	2	yes	—
265	C₂×#95	C₂²×C₇₁	1	2	yes	—
266	C₂×#95	C₂²×C₇₁	1	2	yes	—
267	C₂×#95	C₂²×C₇₁	1	2	yes	—
268	C₂×#95	C₂²×C₇₁	1	2	yes	—
269	C₂×#95	C₂²×C₇₁	1	2	yes	—
270	C₂⁵	C₂³×C₇₁	1	1	yes	characteristic
271	C₂²×Q₈	—	4	4	no	—
272	C₂²×Q₈	C₂×C₄×C₇₁	1	8	yes	complemented
273	C₂²×Q₈	C₂³×C₇₁	1	4	yes	—
274	C₂³×C₄	—	2	8	no	—
275	C₂³×C₄	—	2	8	no	—
276	C₂³×C₄	—	4	4	no	—
277	C₂³×C₄	—	4	4	no	—
278	C₂³×C₄	C₂×C₄×C₇₁	1	8	yes	—
279	C₂³×C₄	C₂³×C₇₁	1	2	yes	—
280	C₂³×C₄	C₂³×C₇₁	1	2	yes	—
281	C₂³×C₄	C₂³×C₇₁	1	8	yes	—
282	C₂³×C₄	C₂³×C₇₁	1	2	yes	—
283	C₂³×C₄	C₂³×C₇₁	1	2	yes	—
284	C₂³×C₄	C₂³×C₇₁	1	1	yes	characteristic
285	C₂³×C₄	C₂³×C₇₁	1	1	yes	characteristic
286	C₂³×C₄	C₂³×C₇₁	1	2	yes	—
287	C₂³×C₄	C₂³×C₇₁	1	2	yes	—
288	C₂³×C₄	C₂³×C₇₁	1	4	yes	—
289	C₂²⋊Q₈	C₂×C₄×C₇₁	1	64	yes	complemented
290	C₂×C₄⋊C₄	—	2	8	no	—
291	C₂×C₄⋊C₄	—	4	4	no	—
292	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	8	yes	complemented
293	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	16	yes	complemented
294	C₂×C₄⋊C₄	C₂³×C₇₁	1	8	yes	—
295	C₂×C₄⋊C₄	C₂³×C₇₁	1	8	yes	—
296	C₂×C₄⋊C₄	C₂³×C₇₁	1	8	yes	—
297	C₂×C₄⋊C₄	C₂³×C₇₁	1	4	yes	—
298	C₂×C₂²⋊C₄	—	2	4	no	—
299	C₂×C₂²⋊C₄	C₂×C₄×C₇₁	1	16	yes	—
300	C₂×C₂²⋊C₄	Q₈×C₇₁	1	8	yes	complemented
301	C₂×C₂²⋊C₄	C₂³×C₇₁	1	4	yes	—
302	C₂×C₂²⋊C₄	C₂³×C₇₁	1	8	yes	—
303	C₂×C₂²⋊C₄	C₂³×C₇₁	1	4	yes	—
304	C₂×C₄²	—	2	8	no	—
305	C₂×C₄²	—	4	4	no	—
306	C₂×C₄²	—	4	4	no	—
307	C₂×C₄²	C₂³×C₇₁	1	8	yes	—
308	C₂×C₄²	C₂³×C₇₁	1	8	yes	—
309	C₂×C₄²	C₂³×C₇₁	1	8	yes	—
310	#95	C₂³×C₇₁	1	4	yes	—
311	#95	C₂³×C₇₁	1	4	yes	—
312	#95	C₂³×C₇₁	1	8	yes	—
313	#95	C₂³×C₇₁	1	8	yes	—
314	#95	C₂³×C₇₁	1	8	yes	—
315	#95	C₂³×C₇₁	1	8	yes	—
316	#95	C₂³×C₇₁	1	8	yes	—
317	#95	C₂³×C₇₁	1	8	yes	—
318	C₂⁴	—	2	4	no	—
319	C₂⁴	—	2	4	no	—
320	C₂⁴	C₂²×C₄×C₇₁	1	4	yes	—
321	C₂⁴	C₂×Q₈×C₇₁	1	4	yes	—
322	C₂⁴	#41×C₇₁	1	4	yes	—
323	C₂⁴	C₂⁴×C₇₁	1	2	yes	—
324	C₂⁴	C₂⁴×C₇₁	1	1	yes	characteristic
325	C₂×Q₈	—	4	16	no	—
326	C₂×Q₈	—	4	8	no	—
327	C₂×Q₈	C₂²×C₄×C₇₁	1	16	yes	complemented
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	2	4	no	—
330	C₂²×C₄	—	2	4	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	4	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	4	no	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	4	no	—
339	C₂²×C₄	—	2	4	no	—
340	C₂²×C₄	—	2	16	no	—
341	C₂²×C₄	—	2	4	no	—
342	C₂²×C₄	—	2	4	no	—
343	C₂²×C₄	—	2	4	no	—
344	C₂²×C₄	—	2	4	no	—
345	C₂²×C₄	—	2	16	no	—
346	C₂²×C₄	—	4	16	no	—
347	C₂²×C₄	—	4	8	no	—
348	C₂²×C₄	—	4	8	no	—
349	C₂²×C₄	—	4	32	no	—
350	C₂²×C₄	C₂²×C₄×C₇₁	1	8	yes	—
351	C₂²×C₄	C₂²×C₄×C₇₁	1	8	yes	—
352	C₂²×C₄	C₂²×C₄×C₇₁	1	4	yes	—
353	C₂²×C₄	C₂²×C₄×C₇₁	1	4	yes	—
354	C₂²×C₄	C₂²×C₄×C₇₁	1	16	yes	—
355	C₂²×C₄	C₂×Q₈×C₇₁	1	8	yes	—
356	C₂²×C₄	#41×C₇₁	1	8	yes	—
357	C₂²×C₄	C₂⁴×C₇₁	1	4	yes	—
358	C₂²×C₄	C₂⁴×C₇₁	1	4	yes	—
359	C₂²×C₄	C₂⁴×C₇₁	1	2	yes	—
360	C₂²×C₄	C₂⁴×C₇₁	1	2	yes	—
361	C₂²×C₄	C₂⁴×C₇₁	1	4	yes	—
362	C₂²×C₄	C₂⁴×C₇₁	1	4	yes	—
363	C₂²×C₄	C₂⁴×C₇₁	1	4	yes	—
364	C₂²×C₄	C₂⁴×C₇₁	1	4	yes	—
365	C₄⋊C₄	—	2	16	no	—
366	C₄⋊C₄	—	4	8	no	—
367	C₄⋊C₄	C₂²×C₄×C₇₁	1	16	yes	complemented
368	C₄⋊C₄	C₂²×C₄×C₇₁	1	32	yes	complemented
369	C₂²⋊C₄	—	2	8	no	—
370	C₂²⋊C₄	C₂²×C₄×C₇₁	1	32	yes	—
371	C₂²⋊C₄	C₂×Q₈×C₇₁	1	16	yes	complemented
372	C₄²	—	2	16	no	—
373	C₄²	—	4	8	no	—
374	C₄²	—	4	8	no	—
375	C₂³	—	2	8	no	—
376	C₂³	—	2	8	no	—
377	C₂³	—	2	16	no	—
378	C₂³	—	2	16	no	—
379	C₂³	—	2	4	no	—
380	C₂³	—	2	4	no	—
381	C₂³	C₄×Q₈×C₇₁	1	16	yes	—
382	C₂³	C₂³×C₄×C₇₁	1	4	yes	—
383	C₂³	C₂³×C₄×C₇₁	1	2	yes	—
384	C₂³	C₂²×Q₈×C₇₁	1	4	yes	—
385	C₂³	C₂²×Q₈×C₇₁	1	2	yes	—
386	C₂³	C₂×#41×C₇₁	1	4	yes	—
387	C₂³	C₂×#41×C₇₁	1	2	yes	—
388	C₂³	C₂×#41×C₇₁	1	4	yes	—
389	C₂³	H(5,𝔽₂)×C₇₁	1	4	yes	—
390	C₂³	C₂⁵×C₇₁	1	1	yes	characteristic
391	Q₈	—	4	16	no	—
392	C₂×C₄	—	2	16	no	—
393	C₂×C₄	—	2	8	no	—
394	C₂×C₄	—	2	8	no	—
395	C₂×C₄	—	2	16	no	—
396	C₂×C₄	—	2	16	no	—
397	C₂×C₄	—	2	16	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	2	4	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	4	16	no	—
415	C₂×C₄	—	4	32	no	—
416	C₂×C₄	—	4	8	no	—
417	C₂×C₄	—	4	8	no	—
418	C₂×C₄	C₂³×C₄×C₇₁	1	8	yes	—
419	C₂×C₄	C₂³×C₄×C₇₁	1	8	yes	—
420	C₂×C₄	C₂³×C₄×C₇₁	1	4	yes	—
421	C₂×C₄	C₂³×C₄×C₇₁	1	4	yes	—
422	C₂×C₄	C₂²×Q₈×C₇₁	1	8	yes	—
423	C₂×C₄	C₂×#41×C₇₁	1	8	yes	—
424	C₂²	—	2	32	no	—
425	C₂²	—	2	4	no	—
426	C₂²	—	2	4	no	—
427	C₂²	—	2	8	no	—
428	C₂²	—	2	8	no	—
429	C₂²	C₂×C₄×Q₈×C₇₁	1	8	yes	—
430	C₂²	C₂×C₄×Q₈×C₇₁	1	4	yes	—
431	C₂²	C₄×#41×C₇₁	1	4	yes	—
432	C₂²	#518×C₇₁	1	4	yes	—
433	C₂²	#551×C₇₁	1	8	yes	—
434	C₂²	#554×C₇₁	1	8	yes	—
435	C₂²	C₂⁴×C₄×C₇₁	1	1	yes	characteristic
436	C₂²	C₂³×Q₈×C₇₁	1	1	yes	characteristic
437	C₂²	C₂²×#41×C₇₁	1	1	yes	characteristic
438	C₂²	C₂²×#41×C₇₁	1	2	yes	—
439	C₂²	C₂×H(5,𝔽₂)×C₇₁	1	2	yes	—
440	C₄	—	2	8	no	—
441	C₄	—	2	4	no	—
442	C₄	—	2	4	no	—
443	C₄	—	2	8	no	—
444	C₄	—	2	8	no	—
445	C₄	—	2	8	no	—
446	C₄	—	4	4	no	—
447	C₄	—	4	4	no	—
448	C₂	—	2	8	no	—
449	C₂	C₇₁×#2368	1	8	yes	direct product factor
450	C₂	C₂²×C₄×Q₈×C₇₁	1	1	yes	characteristic
451	C₂	C₂×C₄×#41×C₇₁	1	1	yes	characteristic
452	C₂	C₂×#518×C₇₁	1	1	yes	characteristic
453	C₂	C₂×#551×C₇₁	1	2	yes	—
454	C₂	C₂×#554×C₇₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇₁	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₇₁
Frattini, cosocle	C₂³	C₂⁵×C₇₁
prime radical	C₂⁵×C₇₁	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₁×#2368	C₂	direct (N × H)
3. C₂×#391×C₇₁	C₂	semidirect
4. C₂×#387×C₇₁	C₂	semidirect
5. C₂×#387×C₇₁	C₂	semidirect
7. C₂×C₄×C₄⋊C₄×C₇₁	C₂	semidirect
14. C₂×C₂²⋊Q₈×C₇₁	C₄	semidirect
26. #391×C₇₁	C₂²	semidirect
27. #387×C₇₁	C₂²	semidirect
28. #387×C₇₁	C₂²	semidirect
30. C₄×C₄⋊C₄×C₇₁	C₂²	semidirect
44. C₂²×Q₈×C₇₁	C₂×C₄	semidirect
61. C₂²⋊Q₈×C₇₁	C₂×C₄	semidirect
64. C₂×C₄⋊C₄×C₇₁	C₂×C₄	semidirect
65. C₂×C₄⋊C₄×C₇₁	C₂×C₄	semidirect
72. C₂×C₂²⋊C₄×C₇₁	Q₈	semidirect
99. C₂×Q₈×C₇₁	C₂²×C₄	semidirect
139. C₄⋊C₄×C₇₁	C₂²×C₄	semidirect
140. C₄⋊C₄×C₇₁	C₂²×C₄	semidirect
143. C₂²⋊C₄×C₇₁	C₂×Q₈	semidirect
220. C₂×#2368	C₇₁	direct (N × H)
222. C₂×C₇₁	#2368	direct (N × H)
229. #2368	C₂×C₇₁	direct (N × H)
230. C₂×#391	C₂×C₇₁	semidirect
231. C₂×#387	C₂×C₇₁	semidirect
232. C₂×#387	C₂×C₇₁	semidirect
234. C₂×C₄×C₄⋊C₄	C₂×C₇₁	semidirect
238. C₇₁	C₂×#2368	direct (N × H)
242. C₂×C₂²⋊Q₈	C₄×C₇₁	semidirect
254. #391	C₂²×C₇₁	semidirect
255. #387	C₂²×C₇₁	semidirect
256. #387	C₂²×C₇₁	semidirect
258. C₄×C₄⋊C₄	C₂²×C₇₁	semidirect
272. C₂²×Q₈	C₂×C₄×C₇₁	semidirect
289. C₂²⋊Q₈	C₂×C₄×C₇₁	semidirect
292. C₂×C₄⋊C₄	C₂×C₄×C₇₁	semidirect
293. C₂×C₄⋊C₄	C₂×C₄×C₇₁	semidirect
300. C₂×C₂²⋊C₄	Q₈×C₇₁	semidirect
327. C₂×Q₈	C₂²×C₄×C₇₁	semidirect
367. C₄⋊C₄	C₂²×C₄×C₇₁	semidirect
368. C₄⋊C₄	C₂²×C₄×C₇₁	semidirect
371. C₂²⋊C₄	C₂×Q₈×C₇₁	semidirect
449. C₂	C₇₁×#2368	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊Q₈×C₇₁	2
C₇₁×#2368	2
C₂×#391×C₇₁	2
C₂×#387×C₇₁	2
C₂×#385×C₇₁	2
C₂×C₄×C₄⋊C₄×C₇₁	2
C₂×C₄×C₂²⋊C₄×C₇₁	2
C₂²×#95×C₇₁	2
C₂×#2368	71

Subgroups — C2×C71×#2368 #31855014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇₁×#2368 #31855014