Subgroups — C₇₁×#60099 #31880971

Order 18176.

2738 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568×C₇₁	C₂	1	2	yes	maximal
2	C₂×C₄×#41×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#2906	C₂	1	16	yes	maximal, complemented
4	C₇₁×#2905	C₂	1	6	yes	maximal, complemented
5	C₂×#423×C₇₁	C₂	1	6	yes	maximal, complemented
6	C₂²×#41×C₇₁	C₂²	1	1	yes	characteristic
7	#568×C₇₁	—	2	16	no	—
8	C₂×#131×C₇₁	C₂²	1	16	yes	—
9	C₂²×M₁₆×C₇₁	C₂²	1	12	yes	—
10	C₄×#41×C₇₁	C₄	1	4	yes	—
11	C₄×#41×C₇₁	C₄	1	4	yes	—
12	C₄×#41×C₇₁	C₂²	1	8	yes	—
13	C₂×C₄×Q₈×C₇₁	C₄	1	2	yes	—
14	C₂×C₄×D₄×C₇₁	C₄	1	6	yes	—
15	C₂×#117×C₇₁	C₂²	1	3	yes	—
16	C₂²×C₄²×C₇₁	C₂²	1	3	yes	—
17	#424×C₇₁	C₂²	1	48	yes	complemented
18	#423×C₇₁	C₂²	1	4	yes	—
19	#423×C₇₁	C₂²	1	12	yes	—
20	#423×C₇₁	C₂²	1	24	yes	complemented
21	C₂×C₄⋊C₈×C₇₁	C₂²	1	24	yes	complemented
22	C₂×#41×C₇₁	C₂×C₄	1	4	yes	—
23	C₂×#41×C₇₁	D₄	1	8	yes	—
24	C₂×#41×C₇₁	Q₈	1	8	yes	—
25	C₂×#41×C₇₁	C₂³	1	4	yes	—
26	C₂²×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×D₄×C₇₁	C₂×C₄	1	3	yes	—
28	C₂³×C₄×C₇₁	C₂×C₄	1	6	yes	—
29	C₂³×C₄×C₇₁	C₂³	1	3	yes	—
30	#131×C₇₁	—	2	64	no	—
31	C₂×M₁₆×C₇₁	—	2	48	no	—
32	C₂×M₁₆×C₇₁	C₂³	1	8	yes	—
33	C₂×M₁₆×C₇₁	C₂³	1	24	yes	—
34	C₂²×C₈×C₇₁	C₂³	1	24	yes	—
35	C₄×Q₈×C₇₁	C₂×C₄	1	8	yes	—
36	C₄×Q₈×C₇₁	C₂×C₄	1	8	yes	—
37	C₄×D₄×C₇₁	C₂×C₄	1	24	yes	—
38	C₄×D₄×C₇₁	C₂×C₄	1	24	yes	—
39	#117×C₇₁	C₂×C₄	1	12	yes	—
40	#117×C₇₁	C₂³	1	12	yes	—
41	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	6	yes	—
42	C₂×C₂²⋊C₄×C₇₁	C₂×C₄	1	6	yes	—
43	C₂×C₄²×C₇₁	C₂×C₄	1	6	yes	—
44	C₂×C₄²×C₇₁	C₂×C₄	1	6	yes	—
45	C₂×C₄²×C₇₁	C₂×C₄	1	6	yes	—
46	C₂×C₄²×C₇₁	C₂³	1	12	yes	—
47	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
49	C₄⋊C₈×C₇₁	C₂³	1	16	yes	—
50	C₄⋊C₈×C₇₁	C₂³	1	48	yes	—
51	C₂⁴×C₇₁	C₂²×C₄	1	3	yes	—
52	#41×C₇₁	C₄⋊C₄	1	32	yes	—
53	#41×C₇₁	C₂×D₄	1	16	yes	—
54	#41×C₇₁	C₂×Q₈	1	16	yes	—
55	C₂×Q₈×C₇₁	C₄⋊C₄	1	8	yes	—
56	C₂×Q₈×C₇₁	C₂²×C₄	1	4	yes	—
57	C₂×D₄×C₇₁	C₄⋊C₄	1	24	yes	—
58	C₂×D₄×C₇₁	C₂²×C₄	1	12	yes	—
59	C₂²×C₄×C₇₁	—	4	12	no	—
60	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₇₁	C₂²×C₄	1	12	yes	—
62	C₂²×C₄×C₇₁	C₂²×C₄	1	6	yes	—
63	C₂²×C₄×C₇₁	C₂²×C₄	1	12	yes	—
64	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
65	C₂²×C₄×C₇₁	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₇₁	C₂×D₄	1	12	yes	—
67	C₂²×C₄×C₇₁	C₂×Q₈	1	12	yes	—
68	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
69	C₂²×C₄×C₇₁	C₂⁴	1	6	yes	—
70	M₁₆×C₇₁	—	2	16	no	—
71	M₁₆×C₇₁	—	2	48	no	—
72	C₂×C₈×C₇₁	—	2	48	no	—
73	C₂×C₈×C₇₁	C₂⁴	1	16	yes	—
74	C₄⋊C₄×C₇₁	C₂²×C₄	1	24	yes	—
75	C₂²⋊C₄×C₇₁	C₂²×C₄	1	24	yes	—
76	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₇₁	C₂²×C₄	1	6	yes	—
79	C₄²×C₇₁	C₂²×C₄	1	12	yes	—
80	C₄²×C₇₁	C₂²×C₄	1	12	yes	—
81	C₄²×C₇₁	C₂⁴	1	2	yes	—
82	C₄²×C₇₁	C₂⁴	1	6	yes	—
83	C₂³×C₇₁	—	4	6	no	—
84	C₂³×C₇₁	C₂×C₄⋊C₄	1	12	yes	—
85	C₂³×C₇₁	C₂³×C₄	1	6	yes	—
86	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
87	Q₈×C₇₁	C₂×C₄⋊C₄	1	16	yes	—
88	D₄×C₇₁	C₂×C₄⋊C₄	1	48	yes	—
89	C₂×C₄×C₇₁	—	2	4	no	—
90	C₂×C₄×C₇₁	—	2	24	no	—
91	C₂×C₄×C₇₁	—	4	12	no	—
92	C₂×C₄×C₇₁	—	4	6	no	—
93	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	12	yes	—
94	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	24	yes	—
95	C₂×C₄×C₇₁	C₂³×C₄	1	2	yes	—
96	C₂×C₄×C₇₁	C₂³×C₄	1	12	yes	—
97	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₇₁	C₂³×C₄	1	6	yes	—
99	C₂×C₄×C₇₁	C₂³×C₄	1	2	yes	—
100	C₂×C₄×C₇₁	C₂²×D₄	1	12	yes	—
101	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
102	C₂×C₄×C₇₁	C₂²×Q₈	1	2	yes	—
103	C₂×C₄×C₇₁	C₂²×Q₈	1	12	yes	—
104	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
105	C₈×C₇₁	—	2	16	no	—
106	C₂²×C₇₁	—	2	2	no	—
107	C₂²×C₇₁	—	2	12	no	—
108	C₂²×C₇₁	—	4	12	no	—
109	C₂²×C₇₁	C₂²×C₄⋊C₄	1	2	yes	—
110	C₂²×C₇₁	C₂²×C₄⋊C₄	1	12	yes	—
111	C₂²×C₇₁	C₂⁴×C₄	1	1	yes	characteristic
112	C₄×C₇₁	—	2	2	no	—
113	C₄×C₇₁	—	2	12	no	—
114	C₄×C₇₁	C₂²×C₄⋊C₄	1	2	yes	—
115	C₄×C₇₁	C₂²×C₄⋊C₄	1	12	yes	—
116	C₄×C₇₁	#568	1	4	yes	complemented
117	C₄×C₇₁	C₂³×D₄	1	1	yes	characteristic
118	C₄×C₇₁	C₂³×Q₈	1	1	yes	characteristic
119	#60099	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂×C₇₁	—	2	2	no	—
121	C₂×C₇₁	—	2	12	no	—
122	C₂×C₇₁	C₂³×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₇₁	C₂×#568	1	2	yes	—
124	C₂×#568	C₂×C₇₁	1	2	yes	—
125	C₂×C₄×#41	C₂×C₇₁	1	1	yes	characteristic
126	#2906	C₂×C₇₁	1	16	yes	complemented
127	#2905	C₂×C₇₁	1	6	yes	complemented
128	C₂×#423	C₂×C₇₁	1	6	yes	complemented
129	C₇₁	#60099	1	1	yes	characteristic, direct product factor, Sylow
130	C₂²×#41	C₂²×C₇₁	1	1	yes	characteristic
131	#568	—	2	16	no	—
132	C₂×#131	C₂²×C₇₁	1	16	yes	—
133	C₂²×M₁₆	C₂²×C₇₁	1	12	yes	—
134	C₄×#41	C₄×C₇₁	1	4	yes	—
135	C₄×#41	C₄×C₇₁	1	4	yes	—
136	C₄×#41	C₂²×C₇₁	1	8	yes	—
137	C₂×C₄×Q₈	C₄×C₇₁	1	2	yes	—
138	C₂×C₄×D₄	C₄×C₇₁	1	6	yes	—
139	C₂×#117	C₂²×C₇₁	1	3	yes	—
140	C₂²×C₄²	C₂²×C₇₁	1	3	yes	—
141	#424	C₂²×C₇₁	1	48	yes	complemented
142	#423	C₂²×C₇₁	1	4	yes	—
143	#423	C₂²×C₇₁	1	12	yes	—
144	#423	C₂²×C₇₁	1	24	yes	complemented
145	C₂×C₄⋊C₈	C₂²×C₇₁	1	24	yes	complemented
146	C₂×#41	C₂×C₄×C₇₁	1	4	yes	—
147	C₂×#41	D₄×C₇₁	1	8	yes	—
148	C₂×#41	Q₈×C₇₁	1	8	yes	—
149	C₂×#41	C₂³×C₇₁	1	4	yes	—
150	C₂²×Q₈	C₂×C₄×C₇₁	1	1	yes	characteristic
151	C₂²×D₄	C₂×C₄×C₇₁	1	3	yes	—
152	C₂³×C₄	C₂×C₄×C₇₁	1	6	yes	—
153	C₂³×C₄	C₂³×C₇₁	1	3	yes	—
154	#131	—	2	64	no	—
155	C₂×M₁₆	—	2	48	no	—
156	C₂×M₁₆	C₂³×C₇₁	1	8	yes	—
157	C₂×M₁₆	C₂³×C₇₁	1	24	yes	—
158	C₂²×C₈	C₂³×C₇₁	1	24	yes	—
159	C₄×Q₈	C₂×C₄×C₇₁	1	8	yes	—
160	C₄×Q₈	C₂×C₄×C₇₁	1	8	yes	—
161	C₄×D₄	C₂×C₄×C₇₁	1	24	yes	—
162	C₄×D₄	C₂×C₄×C₇₁	1	24	yes	—
163	#117	C₂×C₄×C₇₁	1	12	yes	—
164	#117	C₂³×C₇₁	1	12	yes	—
165	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	6	yes	—
166	C₂×C₂²⋊C₄	C₂×C₄×C₇₁	1	6	yes	—
167	C₂×C₄²	C₂×C₄×C₇₁	1	6	yes	—
168	C₂×C₄²	C₂×C₄×C₇₁	1	6	yes	—
169	C₂×C₄²	C₂×C₄×C₇₁	1	6	yes	—
170	C₂×C₄²	C₂³×C₇₁	1	3	yes	—
171	C₂×C₄²	C₂³×C₇₁	1	12	yes	—
172	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
173	C₄⋊C₈	C₂³×C₇₁	1	16	yes	—
174	C₄⋊C₈	C₂³×C₇₁	1	48	yes	—
175	C₂⁴	C₂²×C₄×C₇₁	1	3	yes	—
176	#41	C₄⋊C₄×C₇₁	1	32	yes	—
177	#41	C₂×D₄×C₇₁	1	16	yes	—
178	#41	C₂×Q₈×C₇₁	1	16	yes	—
179	C₂×Q₈	C₄⋊C₄×C₇₁	1	8	yes	—
180	C₂×Q₈	C₂²×C₄×C₇₁	1	4	yes	—
181	C₂×D₄	C₄⋊C₄×C₇₁	1	24	yes	—
182	C₂×D₄	C₂²×C₄×C₇₁	1	12	yes	—
183	C₂²×C₄	—	4	12	no	—
184	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
185	C₂²×C₄	C₂²×C₄×C₇₁	1	12	yes	—
186	C₂²×C₄	C₂²×C₄×C₇₁	1	6	yes	—
187	C₂²×C₄	C₂²×C₄×C₇₁	1	12	yes	—
188	C₂²×C₄	C₂²×C₄×C₇₁	1	3	yes	—
189	C₂²×C₄	C₂²×C₄×C₇₁	1	6	yes	—
190	C₂²×C₄	C₂×D₄×C₇₁	1	12	yes	—
191	C₂²×C₄	C₂×Q₈×C₇₁	1	12	yes	—
192	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
193	C₂²×C₄	C₂⁴×C₇₁	1	6	yes	—
194	M₁₆	—	2	16	no	—
195	M₁₆	—	2	48	no	—
196	C₂×C₈	—	2	48	no	—
197	C₂×C₈	C₂⁴×C₇₁	1	16	yes	—
198	C₄⋊C₄	C₂²×C₄×C₇₁	1	24	yes	—
199	C₂²⋊C₄	C₂²×C₄×C₇₁	1	24	yes	—
200	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
201	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
202	C₄²	C₂²×C₄×C₇₁	1	6	yes	—
203	C₄²	C₂²×C₄×C₇₁	1	12	yes	—
204	C₄²	C₂²×C₄×C₇₁	1	12	yes	—
205	C₄²	C₂⁴×C₇₁	1	2	yes	—
206	C₄²	C₂⁴×C₇₁	1	6	yes	—
207	C₂³	—	4	6	no	—
208	C₂³	C₂×C₄⋊C₄×C₇₁	1	12	yes	—
209	C₂³	C₂³×C₄×C₇₁	1	6	yes	—
210	C₂³	C₂³×C₄×C₇₁	1	1	yes	characteristic
211	Q₈	C₂×C₄⋊C₄×C₇₁	1	16	yes	—
212	D₄	C₂×C₄⋊C₄×C₇₁	1	48	yes	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	24	no	—
215	C₂×C₄	—	4	12	no	—
216	C₂×C₄	—	4	6	no	—
217	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	12	yes	—
218	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	24	yes	—
219	C₂×C₄	C₂³×C₄×C₇₁	1	2	yes	—
220	C₂×C₄	C₂³×C₄×C₇₁	1	12	yes	—
221	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
222	C₂×C₄	C₂³×C₄×C₇₁	1	6	yes	—
223	C₂×C₄	C₂³×C₄×C₇₁	1	2	yes	—
224	C₂×C₄	C₂²×D₄×C₇₁	1	12	yes	—
225	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
226	C₂×C₄	C₂²×Q₈×C₇₁	1	2	yes	—
227	C₂×C₄	C₂²×Q₈×C₇₁	1	12	yes	—
228	C₂×C₄	C₂⁵×C₇₁	1	1	yes	characteristic
229	C₈	—	2	16	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	12	no	—
232	C₂²	—	4	12	no	—
233	C₂²	C₂²×C₄⋊C₄×C₇₁	1	2	yes	—
234	C₂²	C₂²×C₄⋊C₄×C₇₁	1	12	yes	—
235	C₂²	C₂⁴×C₄×C₇₁	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	12	no	—
238	C₄	C₂²×C₄⋊C₄×C₇₁	1	2	yes	—
239	C₄	C₂²×C₄⋊C₄×C₇₁	1	12	yes	—
240	C₄	#568×C₇₁	1	4	yes	complemented
241	C₄	C₂³×D₄×C₇₁	1	1	yes	characteristic
242	C₄	C₂³×Q₈×C₇₁	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	2	12	no	—
245	C₂	C₂³×C₄⋊C₄×C₇₁	1	1	yes	characteristic
246	C₂	C₂×#568×C₇₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₇₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₇₁
socle	C₂²×C₇₁	C₂⁴×C₄
prime radical	C₂²×#41×C₇₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇₁×#2906	C₂	semidirect
4. C₇₁×#2905	C₂	semidirect
5. C₂×#423×C₇₁	C₂	semidirect
17. #424×C₇₁	C₂²	semidirect
20. #423×C₇₁	C₂²	semidirect
21. C₂×C₄⋊C₈×C₇₁	C₂²	semidirect
116. C₄×C₇₁	#568	semidirect
119. #60099	C₇₁	direct (N × H)
126. #2906	C₂×C₇₁	semidirect
127. #2905	C₂×C₇₁	semidirect
128. C₂×#423	C₂×C₇₁	semidirect
129. C₇₁	#60099	direct (N × H)
141. #424	C₂²×C₇₁	semidirect
144. #423	C₂²×C₇₁	semidirect
145. C₂×C₄⋊C₈	C₂²×C₇₁	semidirect
240. C₄	#568×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#568×C₇₁	2
C₂×C₄×#41×C₇₁	2
C₇₁×#2906	2
C₇₁×#2905	2
C₂×#423×C₇₁	2
#60099	71

Subgroups — C71×#60099 #31880971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#60099 #31880971