Subgroups — #31896012

Order 18176.

378 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22589056	C₂	1	2	yes	maximal
2	C₄×#14556062	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985721	C₂	1	1	yes	maximal, characteristic
4	#22586179	C₂	1	2	yes	maximal
5	C₂×#335×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556062	C₄	1	2	yes	—
7	C₂×Dic₇₁⋊C₈	—	2	2	no	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊Dic₇₁	C₂²	1	2	yes	—
11	#17985762	C₄	1	1	yes	characteristic
12	#17985762	C₄	1	1	yes	characteristic
13	#17985753	—	2	2	no	—
14	#17985721	—	2	2	no	—
15	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#335×C₇₁	—	2	2	no	—
18	C₂²×C₇₁⋊C₈	—	2	1	no	—
19	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
20	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
21	C₂×C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
22	C₂×C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
23	C₂×C₄×Dic₇₁	—	2	2	no	—
24	#14556062	—	2	4	no	—
25	Dic₇₁⋊C₈	—	4	4	no	—
26	C₄×C₇₁⋊C₈	—	2	1	no	—
27	C₄×C₇₁⋊C₈	—	2	1	no	—
28	C₄×C₇₁⋊C₈	—	2	1	no	—
29	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
30	C₇₁⋊#95	—	2	2	no	—
31	C₂²×C₈×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	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₇₁	—	2	2	no	—
36	C₄⋊C₈×C₇₁	—	2	2	no	—
37	C₄×C₈×C₇₁	D₄	1	2	yes	—
38	C₄×C₈×C₇₁	Q₈	1	2	yes	—
39	C₂²×Dic₇₁	—	2	2	no	—
40	C₂²×Dic₇₁	—	2	2	no	—
41	C₂×C₇₁⋊C₈	—	2	1	no	—
42	C₂×C₇₁⋊C₈	—	2	1	no	—
43	C₂×C₇₁⋊C₈	—	2	1	no	—
44	C₂×C₇₁⋊C₈	—	2	1	no	—
45	C₂×C₇₁⋊C₈	—	2	2	no	—
46	C₂×C₇₁⋊C₈	—	4	2	no	—
47	C₂×C₇₁⋊C₈	D₈	1	2	yes	—
48	C₂×C₇₁⋊C₈	Q₁₆	1	2	yes	—
49	C₄⋊Dic₇₁	—	2	2	no	—
50	C₄⋊Dic₇₁	—	2	2	no	—
51	C₄⋊Dic₇₁	—	2	2	no	—
52	C₄⋊Dic₇₁	M₁₆	1	4	yes	—
53	C₄×Dic₇₁	—	4	2	no	—
54	C₄×Dic₇₁	—	4	2	no	—
55	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
58	C₂×C₈×C₇₁	—	2	2	no	—
59	C₂×C₈×C₇₁	—	2	2	no	—
60	C₂×C₈×C₇₁	—	4	2	no	—
61	C₂×C₈×C₇₁	—	4	2	no	—
62	C₂×C₈×C₇₁	—	4	2	no	—
63	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
64	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
65	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₇₁	#41	1	1	yes	characteristic
69	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₁	—	2	2	no	—
71	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₁	#125	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₇₁	#131	1	2	yes	—
79	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
82	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
83	C₂×Dic₇₁	—	2	4	no	—
84	C₂×Dic₇₁	—	4	2	no	—
85	C₂×Dic₇₁	—	4	2	no	—
86	C₂×Dic₇₁	—	4	2	no	—
87	C₂×Dic₇₁	—	4	2	no	—
88	C₂×Dic₇₁	—	4	2	no	—
89	C₈×C₇₁	—	2	2	no	—
90	C₈×C₇₁	—	2	2	no	—
91	C₈×C₇₁	—	8	2	no	—
92	C₇₁⋊C₈	—	2	2	no	—
93	C₇₁⋊C₈	—	2	2	no	—
94	C₇₁⋊C₈	—	4	2	no	—
95	C₂²×C₇₁	#389	1	1	yes	characteristic
96	C₂²×C₇₁	#435	1	2	yes	—
97	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
98	C₂²×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
99	C₂²×C₇₁	#444	1	2	yes	—
100	C₄×C₇₁	—	2	1	no	—
101	C₄×C₇₁	—	2	1	no	—
102	C₄×C₇₁	#423	1	1	yes	characteristic
103	C₄×C₇₁	#424	1	1	yes	characteristic
104	C₄×C₇₁	#446	1	2	yes	—
105	C₄×C₇₁	#466	1	1	yes	characteristic
106	C₄×C₇₁	#467	1	1	yes	characteristic
107	C₄×C₇₁	#499	1	1	yes	characteristic
108	C₄×C₇₁	#500	1	1	yes	characteristic
109	Dic₇₁	—	4	4	no	—
110	Dic₇₁	—	8	2	no	—
111	#9598	—	71	1	no	maximal, Sylow
112	C₂×C₇₁	#1581	1	2	yes	—
113	C₂×C₇₁	#1584	1	2	yes	—
114	C₂×C₇₁	#1940	1	1	yes	characteristic
115	C₂×C₇₁	#1942	1	1	yes	characteristic
116	C₂×C₇₁	#1946	1	1	yes	characteristic
117	#1939	—	71	2	no	—
118	C₄×#107	—	71	1	no	—
119	C₂×#335	D₇₁	1	1	yes	characteristic
120	C₂×#322	—	71	1	no	—
121	#1279	—	71	2	no	—
122	C₇₁	#9598	1	1	yes	characteristic, complemented, Sylow
123	C₂×#107	—	71	2	no	—
124	C₂×C₄⋊C₈	—	142	2	no	—
125	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
126	C₂×C₄×C₈	—	71	1	no	—
127	C₂×C₄×C₈	—	71	1	no	—
128	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
129	C₄×C₄⋊C₄	—	71	2	no	—
130	#340	—	71	1	no	—
131	#340	—	71	1	no	—
132	#336	—	142	2	no	—
133	#335	—	2	2	no	—
134	#322	—	142	2	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	71	1	no	—
137	C₂²×C₈	—	71	1	no	—
138	C₂²×C₈	—	142	1	no	—
139	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
140	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	71	2	no	—
142	C₂×C₄⋊C₄	—	71	2	no	—
143	C₂×C₄²	—	142	2	no	—
144	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
145	#107	—	142	4	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	284	4	no	—
149	C₄×C₈	—	71	2	no	—
150	C₄×C₈	—	142	1	no	—
151	C₄×C₈	—	142	1	no	—
152	C₄×C₈	—	142	1	no	—
153	C₄×C₈	Dic₁₄₂	1	2	yes	—
154	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
155	#95	—	142	2	no	—
156	C₂²×C₄	—	142	2	no	—
157	C₂²×C₄	—	142	2	no	—
158	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
159	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
160	C₂²×C₄	#11765866	1	1	yes	characteristic
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	284	2	no	—
167	C₂×C₈	—	71	2	no	—
168	C₂×C₈	—	71	2	no	—
169	C₂×C₈	—	142	2	no	—
170	C₂×C₈	—	142	1	no	—
171	C₂×C₈	—	142	1	no	—
172	C₂×C₈	—	142	1	no	—
173	C₂×C₈	—	142	1	no	—
174	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
175	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
176	C₄⋊C₄	—	71	4	no	—
177	C₄⋊C₄	—	142	2	no	—
178	C₄⋊C₄	—	142	2	no	—
179	C₄⋊C₄	—	142	2	no	—
180	C₄²	—	284	2	no	—
181	C₄²	—	284	2	no	—
182	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
183	C₄²	Q₈×D₇₁	1	1	yes	characteristic
184	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
185	C₄²	#11765864	1	1	yes	characteristic
186	C₂³	#14556090	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	284	2	no	—
189	C₂×C₄	—	284	2	no	—
190	C₂×C₄	—	284	2	no	—
191	C₂×C₄	—	284	2	no	—
192	C₂×C₄	—	284	2	no	—
193	C₂×C₄	—	142	4	no	—
194	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
195	C₂×C₄	#14556093	1	1	yes	characteristic
196	C₂×C₄	#14556099	1	1	yes	characteristic
197	C₂×C₄	#14556104	1	1	yes	characteristic
198	C₂×C₄	#14556119	1	1	yes	characteristic
199	C₂×C₄	#14556127	1	1	yes	characteristic
200	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
201	C₂×C₄	#14556134	1	2	yes	—
202	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
203	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
204	C₂×C₄	#14556147	1	1	yes	characteristic
205	C₂×C₄	#14556151	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	8	2	no	—
209	C₈	—	284	2	no	—
210	C₈	—	142	2	no	—
211	C₈	—	142	2	no	—
212	C₂²	#17985895	1	1	yes	characteristic
213	C₂²	#17986069	1	2	yes	—
214	C₂²	#17986081	1	1	yes	characteristic
215	C₂²	#17986093	1	1	yes	characteristic
216	C₂²	#17986120	1	2	yes	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	284	4	no	—
220	C₄	—	568	2	no	—
221	C₄	#17986022	1	1	yes	characteristic
222	C₄	#17986025	1	1	yes	characteristic
223	C₄	#17986128	1	2	yes	—
224	C₄	#17986232	1	1	yes	characteristic
225	C₄	#17986237	1	1	yes	characteristic
226	C₄	#17986400	1	1	yes	characteristic
227	C₄	#17986403	1	1	yes	characteristic
228	C₂	#22587168	1	2	yes	—
229	C₂	#22587189	1	2	yes	—
230	C₂	#22589061	1	1	yes	characteristic
231	C₂	#22589071	1	1	yes	characteristic
232	C₂	#22589091	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×#335×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#335	D₇₁
hypocenter	C₇₁	#9598
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
122. C₇₁	#9598	semidirect

Maximal subgroups

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

subgroup	index
#22589056	2
C₄×#14556062	2
C₂×#17985721	2
#22586179	2
C₂×#335×C₇₁	2
#9598	71