Subgroups — C₇₁×#23249 #31844121

Order 18176.

696 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1985	C₂	1	2	yes	maximal
3	C₇₁×#1984	C₂	1	2	yes	maximal
4	C₇₁×#1940	C₂	1	4	yes	maximal
5	C₇₁×#1771	C₂	1	4	yes	maximal
6	C₇₁×#1767	C₂	1	2	yes	maximal
7	C₂×C₄⋊Q₈×C₇₁	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
14	C₂×#97×C₇₁	C₂²	1	4	yes	—
15	C₂×#97×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
17	#391×C₇₁	C₄	1	1	yes	characteristic
18	#391×C₇₁	C₄	1	1	yes	characteristic
19	#389×C₇₁	C₄	1	4	yes	—
20	#384×C₇₁	C₄	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₇₁	C₂²	1	4	yes	—
22	C₄³×C₇₁	C₂²	1	1	yes	characteristic
23	#336×C₇₁	C₂²	1	4	yes	—
24	#336×C₇₁	C₂²	1	4	yes	—
25	#336×C₇₁	C₂²	1	4	yes	—
26	C₂²×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₁	C₂³	1	4	yes	—
29	C₂²×C₈×C₇₁	C₂³	1	4	yes	—
30	C₄⋊Q₈×C₇₁	—	2	4	no	—
31	C₄×Q₈×C₇₁	—	4	1	no	—
32	C₄×Q₈×C₇₁	—	4	1	no	—
33	C₄×Q₈×C₇₁	—	4	1	no	—
34	C₄×Q₈×C₇₁	—	4	1	no	—
35	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
45	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
46	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
48	C₄⋊C₈×C₇₁	—	2	4	no	—
49	C₄⋊C₈×C₇₁	—	4	2	no	—
50	C₄⋊C₈×C₇₁	—	4	2	no	—
51	C₄⋊C₈×C₇₁	Q₈	1	8	yes	complemented
52	#97×C₇₁	—	2	8	no	—
53	#97×C₇₁	—	4	2	no	—
54	C₄×C₈×C₇₁	—	4	2	no	—
55	#95×C₇₁	C₂×C₄	1	4	yes	—
56	#95×C₇₁	C₂×C₄	1	4	yes	—
57	C₂×Q₈×C₇₁	—	2	2	no	—
58	C₂×Q₈×C₇₁	—	4	1	no	—
59	C₂×Q₈×C₇₁	—	4	1	no	—
60	C₂×Q₈×C₇₁	—	4	1	no	—
61	C₂×Q₈×C₇₁	—	4	1	no	—
62	C₂×Q₈×C₇₁	M₁₆	1	4	yes	—
63	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₇₁	—	2	4	no	—
73	C₂×C₈×C₇₁	—	2	4	no	—
74	C₂×C₈×C₇₁	—	2	4	no	—
75	C₂×C₈×C₇₁	—	2	4	no	—
76	C₂×C₈×C₇₁	—	2	4	no	—
77	C₂×C₈×C₇₁	C₂×Q₈	1	8	yes	—
78	C₄⋊C₄×C₇₁	—	2	8	no	—
79	C₄⋊C₄×C₇₁	—	4	4	no	—
80	C₄⋊C₄×C₇₁	—	4	1	no	—
81	C₄⋊C₄×C₇₁	—	4	1	no	—
82	C₄⋊C₄×C₇₁	—	4	2	no	—
83	C₄⋊C₄×C₇₁	—	4	2	no	—
84	C₄²×C₇₁	—	2	2	no	—
85	C₄²×C₇₁	—	2	2	no	—
86	C₄²×C₇₁	—	2	2	no	—
87	C₄²×C₇₁	—	2	2	no	—
88	C₄²×C₇₁	—	2	2	no	—
89	C₄²×C₇₁	—	2	1	no	—
90	C₄²×C₇₁	—	2	1	no	—
91	C₄²×C₇₁	—	4	2	no	—
92	C₄²×C₇₁	—	4	1	no	—
93	C₄²×C₇₁	C₂×Q₈	1	4	yes	—
94	C₄²×C₇₁	#41	1	4	yes	—
95	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
96	Q₈×C₇₁	—	4	4	no	—
97	Q₈×C₇₁	—	4	4	no	—
98	C₂×C₄×C₇₁	—	2	1	no	—
99	C₂×C₄×C₇₁	—	2	2	no	—
100	C₂×C₄×C₇₁	—	2	2	no	—
101	C₂×C₄×C₇₁	—	2	2	no	—
102	C₂×C₄×C₇₁	—	2	2	no	—
103	C₂×C₄×C₇₁	—	2	2	no	—
104	C₂×C₄×C₇₁	—	2	1	no	—
105	C₂×C₄×C₇₁	—	2	2	no	—
106	C₂×C₄×C₇₁	—	2	1	no	—
107	C₂×C₄×C₇₁	—	2	2	no	—
108	C₂×C₄×C₇₁	—	2	2	no	—
109	C₂×C₄×C₇₁	—	2	2	no	—
110	C₂×C₄×C₇₁	—	2	2	no	—
111	C₂×C₄×C₇₁	—	2	2	no	—
112	C₂×C₄×C₇₁	—	2	4	no	—
113	C₂×C₄×C₇₁	—	2	4	no	—
114	C₂×C₄×C₇₁	—	2	2	no	—
115	C₂×C₄×C₇₁	C₄×Q₈	1	4	yes	—
116	C₂×C₄×C₇₁	C₄×Q₈	1	4	yes	—
117	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
118	C₂×C₄×C₇₁	C₂×M₁₆	1	4	yes	—
119	C₂×C₄×C₇₁	C₂²×Q₈	1	2	yes	—
120	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
121	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
123	C₈×C₇₁	—	2	8	no	—
124	C₈×C₇₁	—	4	4	no	—
125	C₂²×C₇₁	C₂×C₄×Q₈	1	2	yes	—
126	C₂²×C₇₁	#518	1	1	yes	characteristic
127	C₂²×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
128	C₂²×C₇₁	#568	1	1	yes	characteristic
129	C₂²×C₇₁	#568	1	2	yes	—
130	C₄×C₇₁	—	2	4	no	—
131	C₄×C₇₁	—	2	4	no	—
132	C₄×C₇₁	—	2	4	no	—
133	C₄×C₇₁	—	2	2	no	—
134	C₄×C₇₁	—	4	2	no	—
135	C₄×C₇₁	—	4	1	no	—
136	C₄×C₇₁	—	4	2	no	—
137	C₄×C₇₁	—	4	2	no	—
138	C₄×C₇₁	#554	1	2	yes	—
139	C₄×C₇₁	#555	1	1	yes	characteristic
140	C₄×C₇₁	Q₈²	1	1	yes	characteristic
141	#23249	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂×C₇₁	#2369	1	1	yes	characteristic
143	C₂×C₇₁	#2961	1	2	yes	—
144	C₂×C₇₁	Q₈×M₁₆	1	2	yes	—
145	C₂×C₇₁	#2973	1	1	yes	characteristic
146	C₂×C₇₁	#2980	1	1	yes	characteristic
147	C₄×C₄⋊Q₈	C₂×C₇₁	1	1	yes	characteristic
148	#1985	C₂×C₇₁	1	2	yes	—
149	#1984	C₂×C₇₁	1	2	yes	—
150	#1940	C₂×C₇₁	1	4	yes	—
151	#1771	C₂×C₇₁	1	4	yes	—
152	#1767	C₂×C₇₁	1	2	yes	—
153	C₇₁	#23249	1	1	yes	characteristic, direct product factor, Sylow
154	C₂×C₄⋊Q₈	C₄×C₇₁	1	1	yes	characteristic
155	C₂×C₄×Q₈	C₂²×C₇₁	1	1	yes	characteristic
156	C₂×C₄×Q₈	C₂²×C₇₁	1	1	yes	characteristic
157	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
158	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
159	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
160	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
161	C₂×#97	C₂²×C₇₁	1	4	yes	—
162	C₂×#97	C₂²×C₇₁	1	2	yes	—
163	C₂×C₄×C₈	C₂²×C₇₁	1	2	yes	—
164	#391	C₄×C₇₁	1	1	yes	characteristic
165	#391	C₄×C₇₁	1	1	yes	characteristic
166	#389	C₄×C₇₁	1	4	yes	—
167	#384	C₄×C₇₁	1	1	yes	characteristic
168	C₄×C₄⋊C₄	C₂²×C₇₁	1	4	yes	—
169	C₄³	C₂²×C₇₁	1	1	yes	characteristic
170	#336	C₂²×C₇₁	1	4	yes	—
171	#336	C₂²×C₇₁	1	4	yes	—
172	#336	C₂²×C₇₁	1	4	yes	—
173	C₂²×Q₈	C₂×C₄×C₇₁	1	1	yes	characteristic
174	C₂²×Q₈	C₂×C₄×C₇₁	1	1	yes	characteristic
175	C₂²×C₈	C₂³×C₇₁	1	4	yes	—
176	C₂²×C₈	C₂³×C₇₁	1	4	yes	—
177	C₄⋊Q₈	—	2	4	no	—
178	C₄×Q₈	—	4	1	no	—
179	C₄×Q₈	—	4	1	no	—
180	C₄×Q₈	—	4	1	no	—
181	C₄×Q₈	—	4	1	no	—
182	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	4	yes	—
183	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
186	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
187	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	4	yes	—
188	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
189	C₂×C₄²	C₂×C₄×C₇₁	1	2	yes	—
190	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
193	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
194	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
195	C₄⋊C₈	—	2	4	no	—
196	C₄⋊C₈	—	4	2	no	—
197	C₄⋊C₈	—	4	2	no	—
198	C₄⋊C₈	Q₈×C₇₁	1	8	yes	complemented
199	#97	—	2	8	no	—
200	#97	—	4	2	no	—
201	C₄×C₈	—	4	2	no	—
202	#95	C₂×C₄×C₇₁	1	4	yes	—
203	#95	C₂×C₄×C₇₁	1	4	yes	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	4	1	no	—
206	C₂×Q₈	—	4	1	no	—
207	C₂×Q₈	—	4	1	no	—
208	C₂×Q₈	—	4	1	no	—
209	C₂×Q₈	M₁₆×C₇₁	1	4	yes	—
210	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
213	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
215	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
216	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
217	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
218	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	C₂×Q₈×C₇₁	1	8	yes	—
225	C₄⋊C₄	—	2	8	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	4	2	no	—
239	C₄²	—	4	1	no	—
240	C₄²	C₂×Q₈×C₇₁	1	4	yes	—
241	C₄²	#41×C₇₁	1	4	yes	—
242	C₂³	C₂³×C₄×C₇₁	1	1	yes	characteristic
243	Q₈	—	4	4	no	—
244	Q₈	—	4	4	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	C₄×Q₈×C₇₁	1	4	yes	—
263	C₂×C₄	C₄×Q₈×C₇₁	1	4	yes	—
264	C₂×C₄	C₂×M₁₆×C₇₁	1	2	yes	—
265	C₂×C₄	C₂×M₁₆×C₇₁	1	4	yes	—
266	C₂×C₄	C₂²×Q₈×C₇₁	1	2	yes	—
267	C₂×C₄	C₂×#41×C₇₁	1	2	yes	—
268	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
269	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
270	C₈	—	2	8	no	—
271	C₈	—	4	4	no	—
272	C₂²	C₂×C₄×Q₈×C₇₁	1	2	yes	—
273	C₂²	#518×C₇₁	1	1	yes	characteristic
274	C₂²	C₂²×M₁₆×C₇₁	1	1	yes	characteristic
275	C₂²	#568×C₇₁	1	1	yes	characteristic
276	C₂²	#568×C₇₁	1	2	yes	—
277	C₄	—	2	4	no	—
278	C₄	—	2	4	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	4	no	—
281	C₄	—	4	2	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	2	no	—
284	C₄	—	4	2	no	—
285	C₄	#554×C₇₁	1	2	yes	—
286	C₄	#555×C₇₁	1	1	yes	characteristic
287	C₄	Q₈²×C₇₁	1	1	yes	characteristic
288	C₂	C₇₁×#2369	1	1	yes	characteristic
289	C₂	C₇₁×#2961	1	2	yes	—
290	C₂	Q₈×M₁₆×C₇₁	1	2	yes	—
291	C₂	C₇₁×#2973	1	1	yes	characteristic
292	C₂	C₇₁×#2980	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
51. C₄⋊C₈×C₇₁	Q₈	semidirect
141. #23249	C₇₁	direct (N × H)
153. C₇₁	#23249	direct (N × H)
198. C₄⋊C₈	Q₈×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₇₁	2
C₇₁×#1985	2
C₇₁×#1984	2
C₇₁×#1940	2
C₇₁×#1771	2
C₇₁×#1767	2
#23249	71

Subgroups — C71×#23249 #31844121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23249 #31844121