Subgroups — #31896355

Order 18176.

482 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22589277	C₂	1	2	yes	maximal
2	C₂×#17985750	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₇₁⋊C₈)	C₂	1	2	yes	maximal
4	C₂×#17985721	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	#17985754	—	2	4	no	—
10	#17985750	—	2	2	no	—
11	Q₈⋊(C₇₁⋊C₈)	—	2	4	no	—
12	#17985721	—	2	2	no	—
13	C₂²×Q₁₆×C₇₁	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₇₁	C₂²	1	2	yes	—
15	C₄×Q₁₆×C₇₁	—	2	2	no	—
16	C₄×Q₁₆×C₇₁	—	2	2	no	—
17	C₂×#107×C₇₁	C₄	1	1	yes	characteristic
18	C₂×#103×C₇₁	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂²×C₇₁⋊C₈	—	2	2	no	—
21	C₂²×C₇₁⋊C₈	—	2	1	no	—
22	C₂²×C₇₁⋊C₈	D₄	1	2	yes	—
23	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
25	C₄×C₇₁⋊C₈	—	2	1	no	—
26	C₄×C₇₁⋊C₈	—	2	1	no	—
27	C₄×C₇₁⋊C₈	—	2	1	no	—
28	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
29	C₂²×Q₈×C₇₁	C₂×C₄	1	2	yes	—
30	C₂×Q₁₆×C₇₁	—	2	4	no	—
31	C₂×Q₁₆×C₇₁	—	2	2	no	—
32	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₇₁	—	2	2	no	—
35	C₄×Q₈×C₇₁	—	2	2	no	—
36	C₄×Q₈×C₇₁	—	2	2	no	—
37	C₄×Q₈×C₇₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₇₁	—	2	2	no	—
41	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
42	#107×C₇₁	—	2	2	no	—
43	#103×C₇₁	—	2	4	no	—
44	C₄×C₈×C₇₁	D₄	1	2	yes	—
45	C₄×C₈×C₇₁	D₄	1	2	yes	—
46	C₂×C₇₁⋊C₈	—	2	4	no	—
47	C₂×C₇₁⋊C₈	—	2	2	no	—
48	C₂×C₇₁⋊C₈	—	2	2	no	—
49	C₂×C₇₁⋊C₈	—	2	4	no	—
50	C₂×C₇₁⋊C₈	—	4	2	no	—
51	C₂×C₇₁⋊C₈	—	4	4	no	—
52	C₂×Q₈×C₇₁	—	2	2	no	—
53	C₂×Q₈×C₇₁	—	2	2	no	—
54	C₂×Q₈×C₇₁	—	2	2	no	—
55	C₂×Q₈×C₇₁	—	2	2	no	—
56	C₂×Q₈×C₇₁	—	2	2	no	—
57	C₂×Q₈×C₇₁	M₁₆	1	4	yes	—
58	C₂²×C₄×C₇₁	—	2	2	no	—
59	C₂²×C₄×C₇₁	—	2	2	no	—
60	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
63	Q₁₆×C₇₁	—	2	8	no	—
64	C₂×C₈×C₇₁	—	2	2	no	—
65	C₂×C₈×C₇₁	—	2	2	no	—
66	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
68	C₄⋊C₄×C₇₁	—	2	2	no	—
69	C₄⋊C₄×C₇₁	—	2	2	no	—
70	C₄⋊C₄×C₇₁	—	4	2	no	—
71	C₄⋊C₄×C₇₁	—	4	2	no	—
72	C₄²×C₇₁	—	4	2	no	—
73	C₄²×C₇₁	—	4	2	no	—
74	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₇₁	#41	1	1	yes	characteristic
78	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
79	Q₈×C₇₁	—	2	4	no	—
80	Q₈×C₇₁	—	2	4	no	—
81	Q₈×C₇₁	—	4	2	no	—
82	Q₈×C₇₁	—	4	2	no	—
83	C₂×C₄×C₇₁	—	2	2	no	—
84	C₂×C₄×C₇₁	—	2	4	no	—
85	C₂×C₄×C₇₁	—	4	2	no	—
86	C₂×C₄×C₇₁	—	4	2	no	—
87	C₂×C₄×C₇₁	—	4	2	no	—
88	C₂×C₄×C₇₁	—	4	4	no	—
89	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
92	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
93	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
94	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
95	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
96	C₂×C₄×C₇₁	#131	1	2	yes	—
97	C₂×C₄×C₇₁	#137	1	4	yes	—
98	C₈×C₇₁	—	2	2	no	—
99	C₈×C₇₁	—	2	2	no	—
100	C₇₁⋊C₈	—	2	4	no	—
101	C₇₁⋊C₈	—	4	2	no	—
102	C₇₁⋊C₈	—	8	2	no	—
103	C₂²×C₇₁	#390	1	1	yes	characteristic
104	C₂²×C₇₁	#435	1	2	yes	—
105	C₂²×C₇₁	#441	1	2	yes	—
106	C₂²×C₇₁	#443	1	2	yes	—
107	C₄×C₇₁	—	2	1	no	—
108	C₄×C₇₁	—	2	1	no	—
109	C₄×C₇₁	—	4	4	no	—
110	C₄×C₇₁	—	8	2	no	—
111	C₄×C₇₁	#407	1	1	yes	characteristic
112	C₄×C₇₁	#408	1	1	yes	characteristic
113	C₄×C₇₁	#435	1	2	yes	—
114	C₄×C₇₁	#470	1	2	yes	—
115	C₄×C₇₁	#497	1	2	yes	—
116	#9664	—	71	1	no	maximal, Sylow
117	C₂×C₇₁	#1587	1	4	yes	—
118	C₂×C₇₁	#1955	1	1	yes	characteristic
119	C₂×C₇₁	#1971	1	1	yes	characteristic
120	C₂×C₇₁	#1974	1	1	yes	characteristic
121	C₂×C₄×Q₁₆	D₇₁	1	1	yes	characteristic
122	#1984	—	71	2	no	—
123	C₂×#334	—	71	1	no	—
124	C₂×Q₈⋊C₈	—	71	2	no	—
125	C₂×#322	—	71	1	no	—
126	C₇₁	#9664	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₁₆	Dic₇₁	1	1	yes	characteristic
128	C₂×C₄×Q₈	C₂×D₇₁	1	2	yes	—
129	C₄×Q₁₆	—	2	2	no	—
130	C₄×Q₁₆	—	2	2	no	—
131	C₂×#107	Dic₇₁	1	1	yes	characteristic
132	C₂×C₄⋊C₈	—	71	2	no	—
133	C₂×#103	Dic₇₁	1	2	yes	—
134	C₂×C₄×C₈	—	71	1	no	—
135	C₂×C₄×C₈	—	71	1	no	—
136	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
137	#336	—	142	4	no	—
138	#334	—	142	2	no	—
139	Q₈⋊C₈	—	142	4	no	—
140	#322	—	142	2	no	—
141	C₂²×Q₈	C₂×Dic₇₁	1	2	yes	—
142	C₂×Q₁₆	—	2	4	no	—
143	C₂×Q₁₆	—	2	2	no	—
144	C₂²×C₈	—	71	2	no	—
145	C₂²×C₈	—	142	2	no	—
146	C₂²×C₈	—	142	1	no	—
147	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
148	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
149	C₄×Q₈	—	2	2	no	—
150	C₄×Q₈	—	2	2	no	—
151	C₄×Q₈	—	2	2	no	—
152	C₄×Q₈	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
155	C₂×C₄²	—	2	2	no	—
156	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
157	#107	—	2	2	no	—
158	C₄⋊C₈	—	142	2	no	—
159	C₄⋊C₈	—	142	2	no	—
160	#103	—	2	4	no	—
161	C₄×C₈	—	71	2	no	—
162	C₄×C₈	—	142	1	no	—
163	C₄×C₈	—	142	1	no	—
164	C₄×C₈	—	142	1	no	—
165	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
166	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	#11765849	1	4	yes	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
176	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
177	C₂²×C₄	#11765864	1	1	yes	characteristic
178	Q₁₆	—	2	8	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	284	4	no	—
182	C₂×C₈	—	284	2	no	—
183	C₂×C₈	—	142	4	no	—
184	C₂×C₈	—	142	2	no	—
185	C₂×C₈	—	142	4	no	—
186	C₂×C₈	—	142	2	no	—
187	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
188	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄²	—	4	2	no	—
194	C₄²	—	4	2	no	—
195	C₄²	D₄×D₇₁	1	1	yes	characteristic
196	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
197	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
198	C₄²	#11765864	1	1	yes	characteristic
199	C₂³	D₄×Dic₇₁	1	1	yes	characteristic
200	Q₈	—	2	4	no	—
201	Q₈	—	2	4	no	—
202	Q₈	—	4	2	no	—
203	Q₈	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
211	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
212	C₂×C₄	#14556099	1	1	yes	characteristic
213	C₂×C₄	#14556099	1	1	yes	characteristic
214	C₂×C₄	#14556115	1	1	yes	characteristic
215	C₂×C₄	#14556124	1	1	yes	characteristic
216	C₂×C₄	C₂×#11765849	1	2	yes	—
217	C₂×C₄	#14556135	1	2	yes	—
218	C₂×C₄	#14556160	1	4	yes	—
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	—	284	2	no	—
222	C₈	—	568	2	no	—
223	C₈	—	142	4	no	—
224	C₂²	#17985902	1	1	yes	characteristic
225	C₂²	#17986071	1	2	yes	—
226	C₂²	#17986106	1	2	yes	—
227	C₂²	#17986116	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	4	no	—
231	C₄	—	8	2	no	—
232	C₄	#17985963	1	1	yes	characteristic
233	C₄	#17985966	1	1	yes	characteristic
234	C₄	#17986071	1	2	yes	—
235	C₄	#17986254	1	2	yes	—
236	C₄	#17986392	1	2	yes	—
237	C₂	#22587212	1	4	yes	—
238	C₂	#22589139	1	1	yes	characteristic
239	C₂	#22589212	1	1	yes	characteristic
240	C₂	#22589229	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₁
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×C₄×Q₁₆×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×C₄×Q₁₆	D₇₁
hypocenter	C₇₁	#9664
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
126. C₇₁	#9664	semidirect

Maximal subgroups

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

subgroup	index
#22589277	2
C₂×#17985750	2
C₂×Q₈⋊(C₇₁⋊C₈)	2
C₂×#17985721	2
C₂×C₄×Q₁₆×C₇₁	2
#9664	71