Subgroups — C₇₁×#23184 #31844056

Order 18176.

958 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₇₁	C₂	1	2	yes	maximal
3	C₇₁×#2365	C₂	1	2	yes	maximal
4	C₇₁×#1981	C₂	1	2	yes	maximal
5	C₇₁×#1969	C₂	1	2	yes	maximal
6	C₇₁×#1878	C₂	1	4	yes	maximal
7	C₄×#107×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₇₁	C₂²	1	2	yes	—
10	C₂²×Q₁₆×C₇₁	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₇₁	C₂²	1	4	yes	—
13	#494×C₇₁	C₄	1	8	yes	complemented
14	C₄×Q₁₆×C₇₁	—	2	8	no	—
15	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×#103×C₇₁	C₂²	1	4	yes	—
19	C₂×#103×C₇₁	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
22	#391×C₇₁	C₂²	1	4	yes	—
23	#389×C₇₁	—	2	2	no	—
24	#387×C₇₁	—	2	4	no	—
25	C₄×C₄⋊C₄×C₇₁	—	2	2	no	—
26	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
28	#340×C₇₁	C₂²	1	2	yes	—
29	#336×C₇₁	C₂²	1	2	yes	—
30	C₂²×Q₈×C₇₁	C₂³	1	4	yes	—
31	C₂×Q₁₆×C₇₁	—	2	4	no	—
32	C₂×Q₁₆×C₇₁	—	2	4	no	—
33	C₂×Q₁₆×C₇₁	—	4	2	no	—
34	C₂×Q₁₆×C₇₁	—	4	2	no	—
35	C₂×Q₁₆×C₇₁	C₂×C₄	1	8	yes	—
36	C₂×Q₁₆×C₇₁	C₂×C₄	1	8	yes	—
37	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₇₁	—	2	4	no	—
41	C₄⋊Q₈×C₇₁	C₂×C₄	1	8	yes	—
42	C₄×Q₈×C₇₁	—	2	8	no	—
43	C₄×Q₈×C₇₁	—	4	4	no	—
44	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
47	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
51	C₂×C₄²×C₇₁	—	2	4	no	—
52	C₂×C₄²×C₇₁	—	2	2	no	—
53	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
54	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
55	#107×C₇₁	—	2	2	no	—
56	#107×C₇₁	D₄	1	4	yes	—
57	C₄⋊C₈×C₇₁	—	2	2	no	—
58	#103×C₇₁	—	2	8	no	—
59	#103×C₇₁	—	4	4	no	—
60	C₄×C₈×C₇₁	—	2	4	no	—
61	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
62	#95×C₇₁	—	2	4	no	—
63	#95×C₇₁	—	2	2	no	—
64	#95×C₇₁	—	2	4	no	—
65	#95×C₇₁	D₄	1	2	yes	—
66	C₂×Q₈×C₇₁	—	2	4	no	—
67	C₂×Q₈×C₇₁	—	2	4	no	—
68	C₂×Q₈×C₇₁	—	2	4	no	—
69	C₂×Q₈×C₇₁	—	2	4	no	—
70	C₂×Q₈×C₇₁	—	2	4	no	—
71	C₂×Q₈×C₇₁	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₇₁	—	2	4	no	—
73	C₂²×C₄×C₇₁	—	2	2	no	—
74	C₂²×C₄×C₇₁	—	2	2	no	—
75	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
80	Q₁₆×C₇₁	—	2	16	no	—
81	Q₁₆×C₇₁	—	4	8	no	—
82	C₂×C₈×C₇₁	—	2	1	no	—
83	C₂×C₈×C₇₁	—	2	1	no	—
84	C₂×C₈×C₇₁	—	2	2	no	—
85	C₂×C₈×C₇₁	—	2	2	no	—
86	C₂×C₈×C₇₁	—	2	2	no	—
87	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₇₁	#41	1	2	yes	—
91	C₄⋊C₄×C₇₁	—	2	8	no	—
92	C₄⋊C₄×C₇₁	—	2	8	no	—
93	C₄⋊C₄×C₇₁	—	2	2	no	—
94	C₄⋊C₄×C₇₁	—	2	2	no	—
95	C₄⋊C₄×C₇₁	—	2	2	no	—
96	C₄⋊C₄×C₇₁	—	2	2	no	—
97	C₄⋊C₄×C₇₁	—	4	8	no	—
98	C₄⋊C₄×C₇₁	Q₁₆	1	4	yes	complemented
99	C₄⋊C₄×C₇₁	C₂×D₄	1	4	yes	—
100	C₄²×C₇₁	—	2	2	no	—
101	C₄²×C₇₁	—	2	2	no	—
102	C₄²×C₇₁	—	4	8	no	—
103	C₄²×C₇₁	—	4	4	no	—
104	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
105	C₄²×C₇₁	#41	1	2	yes	—
106	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₇₁	—	2	8	no	—
108	Q₈×C₇₁	—	2	8	no	—
109	Q₈×C₇₁	—	4	4	no	—
110	Q₈×C₇₁	—	4	4	no	—
111	C₂×C₄×C₇₁	—	2	2	no	—
112	C₂×C₄×C₇₁	—	2	1	no	—
113	C₂×C₄×C₇₁	—	2	1	no	—
114	C₂×C₄×C₇₁	—	2	2	no	—
115	C₂×C₄×C₇₁	—	2	8	no	—
116	C₂×C₄×C₇₁	—	2	2	no	—
117	C₂×C₄×C₇₁	—	2	4	no	—
118	C₂×C₄×C₇₁	—	2	2	no	—
119	C₂×C₄×C₇₁	—	2	2	no	—
120	C₂×C₄×C₇₁	—	4	4	no	—
121	C₂×C₄×C₇₁	—	4	4	no	—
122	C₂×C₄×C₇₁	—	4	2	no	—
123	C₂×C₄×C₇₁	—	4	2	no	—
124	C₂×C₄×C₇₁	—	4	2	no	—
125	C₂×C₄×C₇₁	—	4	2	no	—
126	C₂×C₄×C₇₁	—	4	2	no	—
127	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₇₁	C₂×Q₁₆	1	4	yes	—
130	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
131	C₂×C₄×C₇₁	#135	1	2	yes	—
132	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
138	C₈×C₇₁	—	2	2	no	—
139	C₈×C₇₁	—	4	2	no	—
140	C₈×C₇₁	C₄×D₄	1	4	yes	—
141	C₂²×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²×C₇₁	D₄²	1	1	yes	characteristic
143	C₂²×C₇₁	#550	1	1	yes	characteristic
144	C₂²×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
145	C₂²×C₇₁	C₂×#135	1	1	yes	characteristic
146	C₂²×C₇₁	#578	1	1	yes	characteristic
147	C₂²×C₇₁	#578	1	1	yes	characteristic
148	C₄×C₇₁	—	2	2	no	—
149	C₄×C₇₁	—	2	4	no	—
150	C₄×C₇₁	—	4	8	no	—
151	C₄×C₇₁	—	4	2	no	—
152	C₄×C₇₁	—	4	4	no	—
153	C₄×C₇₁	—	8	2	no	—
154	C₄×C₇₁	C₄×Q₁₆	1	4	yes	complemented
155	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
156	C₄×C₇₁	#518	1	1	yes	characteristic
157	C₄×C₇₁	#550	1	1	yes	characteristic
158	C₄×C₇₁	#559	1	1	yes	characteristic
159	#23184	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂×C₇₁	#2366	1	1	yes	characteristic
161	C₂×C₇₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
162	C₂×C₇₁	#2950	1	1	yes	characteristic
163	C₂×C₇₁	#3283	1	1	yes	characteristic
164	C₂×C₇₁	D₄×Q₁₆	1	1	yes	characteristic
165	C₂×C₇₁	#3393	1	1	yes	characteristic
166	C₂×C₇₁	#3395	1	1	yes	characteristic
167	C₂×#494	C₂×C₇₁	1	1	yes	characteristic
168	C₂×C₄×Q₁₆	C₂×C₇₁	1	2	yes	—
169	#2365	C₂×C₇₁	1	2	yes	—
170	#1981	C₂×C₇₁	1	2	yes	—
171	#1969	C₂×C₇₁	1	2	yes	—
172	#1878	C₂×C₇₁	1	4	yes	—
173	C₄×#107	C₂×C₇₁	1	1	yes	characteristic
174	C₈×C₄⋊C₄	C₂×C₇₁	1	1	yes	characteristic
175	C₇₁	#23184	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²×Q₁₆	C₂²×C₇₁	1	2	yes	—
177	C₂²×Q₁₆	C₂²×C₇₁	1	2	yes	—
178	C₂×C₄⋊Q₈	C₂²×C₇₁	1	2	yes	—
179	C₂×C₄×Q₈	C₂²×C₇₁	1	4	yes	—
180	#494	C₄×C₇₁	1	8	yes	complemented
181	C₄×Q₁₆	—	2	8	no	—
182	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
183	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
184	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
185	C₂×#103	C₂²×C₇₁	1	4	yes	—
186	C₂×#103	C₂²×C₇₁	1	4	yes	—
187	C₂×C₄×C₈	C₂²×C₇₁	1	2	yes	—
188	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
189	#391	C₂²×C₇₁	1	4	yes	—
190	#389	—	2	2	no	—
191	#387	—	2	4	no	—
192	C₄×C₄⋊C₄	—	2	2	no	—
193	C₄×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
194	C₄×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
195	#340	C₂²×C₇₁	1	2	yes	—
196	#336	C₂²×C₇₁	1	2	yes	—
197	C₂²×Q₈	C₂³×C₇₁	1	4	yes	—
198	C₂×Q₁₆	—	2	4	no	—
199	C₂×Q₁₆	—	2	4	no	—
200	C₂×Q₁₆	—	4	2	no	—
201	C₂×Q₁₆	—	4	2	no	—
202	C₂×Q₁₆	C₂×C₄×C₇₁	1	8	yes	—
203	C₂×Q₁₆	C₂×C₄×C₇₁	1	8	yes	—
204	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
205	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
207	C₄⋊Q₈	—	2	4	no	—
208	C₄⋊Q₈	C₂×C₄×C₇₁	1	8	yes	—
209	C₄×Q₈	—	2	8	no	—
210	C₄×Q₈	—	4	4	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	4	no	—
214	C₂×C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
215	C₂×C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
217	C₂×C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
218	C₂×C₄²	—	2	4	no	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
221	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
222	#107	—	2	2	no	—
223	#107	D₄×C₇₁	1	4	yes	—
224	C₄⋊C₈	—	2	2	no	—
225	#103	—	2	8	no	—
226	#103	—	4	4	no	—
227	C₄×C₈	—	2	4	no	—
228	C₄×C₈	C₂×C₄×C₇₁	1	4	yes	—
229	#95	—	2	4	no	—
230	#95	—	2	2	no	—
231	#95	—	2	4	no	—
232	#95	D₄×C₇₁	1	2	yes	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	—	2	4	no	—
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	C₂²×C₄×C₇₁	1	8	yes	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
245	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
246	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
247	Q₁₆	—	2	16	no	—
248	Q₁₆	—	4	8	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	C₂²×C₄×C₇₁	1	2	yes	—
255	C₂×C₈	C₂²×C₄×C₇₁	1	2	yes	—
256	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
257	C₂×C₈	#41×C₇₁	1	2	yes	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	2	8	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	8	no	—
265	C₄⋊C₄	Q₁₆×C₇₁	1	4	yes	complemented
266	C₄⋊C₄	C₂×D₄×C₇₁	1	4	yes	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	4	8	no	—
270	C₄²	—	4	4	no	—
271	C₄²	C₂²×C₄×C₇₁	1	2	yes	—
272	C₄²	#41×C₇₁	1	2	yes	—
273	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
274	Q₈	—	2	8	no	—
275	Q₈	—	2	8	no	—
276	Q₈	—	4	4	no	—
277	Q₈	—	4	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
295	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
296	C₂×C₄	C₂×Q₁₆×C₇₁	1	4	yes	—
297	C₂×C₄	C₂×Q₁₆×C₇₁	1	2	yes	—
298	C₂×C₄	#135×C₇₁	1	2	yes	—
299	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
301	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
302	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
303	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
304	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	4	2	no	—
307	C₈	C₄×D₄×C₇₁	1	4	yes	—
308	C₂²	C₂×C₄×D₄×C₇₁	1	1	yes	characteristic
309	C₂²	D₄²×C₇₁	1	1	yes	characteristic
310	C₂²	#550×C₇₁	1	1	yes	characteristic
311	C₂²	C₂²×Q₁₆×C₇₁	1	1	yes	characteristic
312	C₂²	C₂×#135×C₇₁	1	1	yes	characteristic
313	C₂²	#578×C₇₁	1	1	yes	characteristic
314	C₂²	#578×C₇₁	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	2	4	no	—
317	C₄	—	4	8	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	4	no	—
320	C₄	—	8	2	no	—
321	C₄	C₄×Q₁₆×C₇₁	1	4	yes	complemented
322	C₄	C₂×C₄×D₄×C₇₁	1	1	yes	characteristic
323	C₄	#518×C₇₁	1	1	yes	characteristic
324	C₄	#550×C₇₁	1	1	yes	characteristic
325	C₄	#559×C₇₁	1	1	yes	characteristic
326	C₂	C₇₁×#2366	1	1	yes	characteristic
327	C₂	C₂×C₄×Q₁₆×C₇₁	1	1	yes	characteristic
328	C₂	C₇₁×#2950	1	1	yes	characteristic
329	C₂	C₇₁×#3283	1	1	yes	characteristic
330	C₂	D₄×Q₁₆×C₇₁	1	1	yes	characteristic
331	C₂	C₇₁×#3393	1	1	yes	characteristic
332	C₂	C₇₁×#3395	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₄×C₄⋊C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. #494×C₇₁	C₄	semidirect
98. C₄⋊C₄×C₇₁	Q₁₆	semidirect
154. C₄×C₇₁	C₄×Q₁₆	semidirect
159. #23184	C₇₁	direct (N × H)
175. C₇₁	#23184	direct (N × H)
180. #494	C₄×C₇₁	semidirect
265. C₄⋊C₄	Q₁₆×C₇₁	semidirect
321. C₄	C₄×Q₁₆×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#494×C₇₁	2
C₂×C₄×Q₁₆×C₇₁	2
C₇₁×#2365	2
C₇₁×#1981	2
C₇₁×#1969	2
C₇₁×#1878	2
C₄×#107×C₇₁	2
C₈×C₄⋊C₄×C₇₁	2
#23184	71

Subgroups — C71×#23184 #31844056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23184 #31844056