Subgroups — C₇₁×#29155 #31850027

Order 18176.

608 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#3162	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#444×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#2031	C₂	1	4	yes	maximal, complemented
4	C₇₁×#1976	C₂	1	2	yes	maximal
5	C₇₁×#1833	C₂	1	2	yes	maximal, complemented
6	C₇₁×#1814	C₂	1	2	yes	maximal, complemented
7	C₇₁×#1812	C₂	1	2	yes	maximal, complemented
8	C₇₁×#1773	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₇₁	C₂²	1	1	yes	characteristic
10	C₂×#131×C₇₁	C₂²	1	2	yes	—
11	#533×C₇₁	C₂²	1	2	yes	—
12	#501×C₇₁	—	2	4	no	—
13	#444×C₇₁	—	2	4	no	—
14	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
16	#427×C₇₁	C₂²	1	1	yes	characteristic
17	#427×C₇₁	C₂²	1	2	yes	—
18	#427×C₇₁	C₂²	1	2	yes	—
19	#421×C₇₁	C₂²	1	4	yes	—
20	C₂×#104×C₇₁	C₂²	1	2	yes	—
21	#410×C₇₁	C₂²	1	4	yes	—
22	#405×C₇₁	C₂²	1	2	yes	—
23	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
24	#344×C₇₁	C₂²	1	4	yes	—
25	#339×C₇₁	C₂²	1	2	yes	—
26	H(3,ℤ/4)×C₇₁	C₂²	1	4	yes	—
27	C₂×#41×C₇₁	C₂³	1	2	yes	—
28	#135×C₇₁	—	2	2	no	—
29	#135×C₇₁	—	2	2	no	—
30	C₂×Q₁₆×C₇₁	Q₈	1	1	yes	characteristic, complemented
31	C₂×SD₁₆×C₇₁	Q₈	1	2	yes	complemented
32	C₂×D₈×C₇₁	Q₈	1	1	yes	characteristic, complemented
33	#131×C₇₁	—	2	2	no	—
34	#131×C₇₁	—	2	2	no	—
35	#131×C₇₁	—	2	2	no	—
36	#131×C₇₁	—	2	2	no	—
37	C₂×M₁₆×C₇₁	—	2	2	no	—
38	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
39	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₇₁	—	2	2	no	—
41	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₇₁	—	2	2	no	—
44	#125×C₇₁	—	2	2	no	—
45	C₂²⋊Q₈×C₇₁	—	4	2	no	—
46	C₄×Q₈×C₇₁	—	4	2	no	—
47	#117×C₇₁	C₂³	1	2	yes	—
48	#117×C₇₁	C₂³	1	4	yes	—
49	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
50	#108×C₇₁	—	2	2	no	—
51	#108×C₇₁	—	2	2	no	—
52	#107×C₇₁	—	2	2	no	—
53	#107×C₇₁	—	2	2	no	—
54	#107×C₇₁	—	2	1	no	—
55	#106×C₇₁	—	2	2	no	—
56	#106×C₇₁	—	2	2	no	—
57	#106×C₇₁	—	2	1	no	—
58	#104×C₇₁	—	2	4	no	—
59	#104×C₇₁	—	4	4	no	—
60	#99×C₇₁	—	4	4	no	—
61	#97×C₇₁	—	2	2	no	—
62	#97×C₇₁	D₄	1	4	yes	—
63	C₄×C₈×C₇₁	—	2	2	no	—
64	#41×C₇₁	—	2	2	no	—
65	#41×C₇₁	—	2	2	no	—
66	#41×C₇₁	—	2	2	no	—
67	#41×C₇₁	—	2	2	no	—
68	C₂×Q₈×C₇₁	—	4	2	no	—
69	C₂×Q₈×C₇₁	C₂×Q₈	1	2	yes	—
70	C₂×D₄×C₇₁	—	2	2	no	—
71	C₂×D₄×C₇₁	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₇₁	—	2	1	no	—
73	C₂²×C₄×C₇₁	—	2	2	no	—
74	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
75	Q₁₆×C₇₁	—	2	2	no	—
76	SD₁₆×C₇₁	—	2	4	no	—
77	D₈×C₇₁	—	2	2	no	—
78	M₁₆×C₇₁	—	2	2	no	—
79	M₁₆×C₇₁	—	2	2	no	—
80	M₁₆×C₇₁	—	2	2	no	—
81	M₁₆×C₇₁	—	2	2	no	—
82	M₁₆×C₇₁	—	4	2	no	—
83	M₁₆×C₇₁	—	4	2	no	—
84	C₂×C₈×C₇₁	—	2	2	no	—
85	C₂×C₈×C₇₁	—	2	2	no	—
86	C₂×C₈×C₇₁	—	4	2	no	—
87	C₂×C₈×C₇₁	—	4	2	no	—
88	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
91	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₈×C₇₁	#41	1	2	yes	—
93	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
94	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
95	C₂×C₈×C₇₁	#41	1	2	yes	—
96	C₄⋊C₄×C₇₁	—	2	4	no	—
97	C₄⋊C₄×C₇₁	—	2	2	no	—
98	C₄⋊C₄×C₇₁	—	4	4	no	—
99	C₄⋊C₄×C₇₁	—	4	2	no	—
100	C₂²⋊C₄×C₇₁	—	2	2	no	—
101	C₂²⋊C₄×C₇₁	—	2	4	no	—
102	C₄²×C₇₁	—	2	1	no	—
103	C₄²×C₇₁	—	2	2	no	—
104	C₄²×C₇₁	—	2	4	no	—
105	C₄²×C₇₁	C₂×D₄	1	2	yes	—
106	C₂³×C₇₁	—	2	2	no	—
107	C₂³×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
108	Q₈×C₇₁	—	2	2	no	—
109	Q₈×C₇₁	—	2	2	no	—
110	Q₈×C₇₁	—	8	4	no	—
111	D₄×C₇₁	—	2	2	no	—
112	D₄×C₇₁	—	2	2	no	—
113	D₄×C₇₁	—	4	2	no	—
114	D₄×C₇₁	—	4	2	no	—
115	C₂×C₄×C₇₁	—	2	2	no	—
116	C₂×C₄×C₇₁	—	2	2	no	—
117	C₂×C₄×C₇₁	—	4	2	no	—
118	C₂×C₄×C₇₁	—	4	2	no	—
119	C₂×C₄×C₇₁	—	4	4	no	—
120	C₂×C₄×C₇₁	—	4	2	no	—
121	C₂×C₄×C₇₁	—	8	1	no	—
122	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
128	C₈×C₇₁	—	4	2	no	—
129	C₈×C₇₁	—	4	2	no	—
130	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
131	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
132	C₈×C₇₁	#124	1	2	yes	—
133	C₈×C₇₁	#124	1	2	yes	—
134	C₂²×C₇₁	—	4	1	no	—
135	C₂²×C₇₁	—	4	2	no	—
136	C₂²×C₇₁	—	8	2	no	—
137	C₂²×C₇₁	#535	1	1	yes	characteristic
138	C₂²×C₇₁	#554	1	1	yes	characteristic
139	C₂²×C₇₁	#555	1	1	yes	characteristic
140	C₄×C₇₁	—	4	2	no	—
141	C₄×C₇₁	—	4	2	no	—
142	C₄×C₇₁	—	8	2	no	—
143	C₄×C₇₁	—	8	4	no	—
144	C₄×C₇₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
145	C₄×C₇₁	C₂×#124	1	1	yes	characteristic
146	C₄×C₇₁	#554	1	1	yes	characteristic
147	C₄×C₇₁	#555	1	1	yes	characteristic
148	#29155	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂×C₇₁	—	2	1	no	—
150	C₂×C₇₁	—	2	1	no	—
151	C₂×C₇₁	—	2	1	no	—
152	C₂×C₇₁	—	8	2	no	—
153	C₂×C₇₁	#2563	1	1	yes	characteristic
154	#3162	C₂×C₇₁	1	1	yes	characteristic, complemented
155	C₂×#444	C₂×C₇₁	1	1	yes	characteristic
156	#2031	C₂×C₇₁	1	4	yes	complemented
157	#1976	C₂×C₇₁	1	2	yes	—
158	#1833	C₂×C₇₁	1	2	yes	complemented
159	#1814	C₂×C₇₁	1	2	yes	complemented
160	#1812	C₂×C₇₁	1	2	yes	complemented
161	#1773	C₂×C₇₁	1	1	yes	characteristic, complemented
162	C₇₁	#29155	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×#135	C₂²×C₇₁	1	1	yes	characteristic
164	C₂×#131	C₂²×C₇₁	1	2	yes	—
165	#533	C₂²×C₇₁	1	2	yes	—
166	#501	—	2	4	no	—
167	#444	—	2	4	no	—
168	C₂×#108	C₂²×C₇₁	1	1	yes	characteristic
169	C₂×#108	C₂²×C₇₁	1	1	yes	characteristic
170	#427	C₂²×C₇₁	1	1	yes	characteristic
171	#427	C₂²×C₇₁	1	2	yes	—
172	#427	C₂²×C₇₁	1	2	yes	—
173	#421	C₂²×C₇₁	1	4	yes	—
174	C₂×#104	C₂²×C₇₁	1	2	yes	—
175	#410	C₂²×C₇₁	1	4	yes	—
176	#405	C₂²×C₇₁	1	2	yes	—
177	C₂×#97	C₂²×C₇₁	1	1	yes	characteristic
178	#344	C₂²×C₇₁	1	4	yes	—
179	#339	C₂²×C₇₁	1	2	yes	—
180	H(3,ℤ/4)	C₂²×C₇₁	1	4	yes	—
181	C₂×#41	C₂³×C₇₁	1	2	yes	—
182	#135	—	2	2	no	—
183	#135	—	2	2	no	—
184	C₂×Q₁₆	Q₈×C₇₁	1	1	yes	characteristic, complemented
185	C₂×SD₁₆	Q₈×C₇₁	1	2	yes	complemented
186	C₂×D₈	Q₈×C₇₁	1	1	yes	characteristic, complemented
187	#131	—	2	2	no	—
188	#131	—	2	2	no	—
189	#131	—	2	2	no	—
190	#131	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	C₂³×C₇₁	1	2	yes	—
193	C₂×M₁₆	C₂³×C₇₁	1	2	yes	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
197	C₄⋊Q₈	—	2	2	no	—
198	#125	—	2	2	no	—
199	C₂²⋊Q₈	—	4	2	no	—
200	C₄×Q₈	—	4	2	no	—
201	#117	C₂³×C₇₁	1	4	yes	—
202	#117	C₂³×C₇₁	1	2	yes	—
203	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
204	#108	—	2	2	no	—
205	#108	—	2	2	no	—
206	#107	—	2	2	no	—
207	#107	—	2	2	no	—
208	#107	—	2	1	no	—
209	#106	—	2	2	no	—
210	#106	—	2	2	no	—
211	#106	—	2	1	no	—
212	#104	—	2	4	no	—
213	#104	—	4	4	no	—
214	#99	—	4	4	no	—
215	#97	—	2	2	no	—
216	#97	D₄×C₇₁	1	4	yes	—
217	C₄×C₈	—	2	2	no	—
218	#41	—	2	2	no	—
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	#41	—	2	2	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	C₂×Q₈×C₇₁	1	2	yes	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂×Q₈×C₇₁	1	2	yes	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
229	Q₁₆	—	2	2	no	—
230	SD₁₆	—	2	4	no	—
231	D₈	—	2	2	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	2	no	—
236	M₁₆	—	4	2	no	—
237	M₁₆	—	4	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
243	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
244	C₂×C₈	C₂×Q₈×C₇₁	1	1	yes	characteristic
245	C₂×C₈	C₂×Q₈×C₇₁	1	1	yes	characteristic
246	C₂×C₈	#41×C₇₁	1	2	yes	—
247	C₂×C₈	#41×C₇₁	1	1	yes	characteristic
248	C₂×C₈	#41×C₇₁	1	1	yes	characteristic
249	C₂×C₈	#41×C₇₁	1	2	yes	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	4	no	—
259	C₄²	C₂×D₄×C₇₁	1	2	yes	—
260	C₂³	—	2	2	no	—
261	C₂³	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
262	Q₈	—	2	2	no	—
263	Q₈	—	2	2	no	—
264	Q₈	—	8	4	no	—
265	D₄	—	2	2	no	—
266	D₄	—	2	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
277	C₂×C₄	C₂²×Q₈×C₇₁	1	1	yes	characteristic
278	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
280	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
281	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
282	C₈	—	4	2	no	—
283	C₈	—	4	2	no	—
284	C₈	C₂²⋊Q₈×C₇₁	1	2	yes	—
285	C₈	C₂²⋊Q₈×C₇₁	1	2	yes	—
286	C₈	#124×C₇₁	1	2	yes	—
287	C₈	#124×C₇₁	1	2	yes	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	#535×C₇₁	1	1	yes	characteristic
292	C₂²	#554×C₇₁	1	1	yes	characteristic
293	C₂²	#555×C₇₁	1	1	yes	characteristic
294	C₄	—	4	2	no	—
295	C₄	—	4	2	no	—
296	C₄	—	8	4	no	—
297	C₄	—	8	2	no	—
298	C₄	C₂×C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
299	C₄	C₂×#124×C₇₁	1	1	yes	characteristic
300	C₄	#554×C₇₁	1	1	yes	characteristic
301	C₄	#555×C₇₁	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	8	2	no	—
306	C₂	C₇₁×#2563	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇₁	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
socle	C₂×C₇₁	#2563
prime radical	C₂×#135×C₇₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇₁×#3162	C₂	semidirect
3. C₇₁×#2031	C₂	semidirect
5. C₇₁×#1833	C₂	semidirect
6. C₇₁×#1814	C₂	semidirect
7. C₇₁×#1812	C₂	semidirect
8. C₇₁×#1773	C₂	semidirect
30. C₂×Q₁₆×C₇₁	Q₈	semidirect
31. C₂×SD₁₆×C₇₁	Q₈	semidirect
32. C₂×D₈×C₇₁	Q₈	semidirect
148. #29155	C₇₁	direct (N × H)
154. #3162	C₂×C₇₁	semidirect
156. #2031	C₂×C₇₁	semidirect
158. #1833	C₂×C₇₁	semidirect
159. #1814	C₂×C₇₁	semidirect
160. #1812	C₂×C₇₁	semidirect
161. #1773	C₂×C₇₁	semidirect
162. C₇₁	#29155	direct (N × H)
184. C₂×Q₁₆	Q₈×C₇₁	semidirect
185. C₂×SD₁₆	Q₈×C₇₁	semidirect
186. C₂×D₈	Q₈×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×#3162	2
C₂×#444×C₇₁	2
C₇₁×#2031	2
C₇₁×#1976	2
C₇₁×#1833	2
C₇₁×#1814	2
C₇₁×#1812	2
C₇₁×#1773	2
#29155	71

Subgroups — C71×#29155 #31850027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#29155 #31850027