Subgroups — #31966750

Order 18176.

612 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597472	C₂	1	2	yes	maximal
2	#22587129	C₂	1	6	yes	maximal, complemented
3	#22587096	C₂	1	3	yes	maximal, complemented
4	#22587088	C₂	1	3	yes	maximal, complemented
5	C₈×#41×C₇₁	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₇₁⋊C₈	—	2	2	no	—
7	#17986077	—	2	6	no	—
8	#17986052	C₂²	1	6	yes	—
9	#17986023	C₂²	1	6	yes	—
10	#17985752	C₂²	1	6	yes	—
11	#17985750	C₂²	1	1	yes	characteristic
12	#17985750	C₂²	1	3	yes	—
13	#17985750	C₂²	1	6	yes	—
14	C₄×#41×C₇₁	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₇₁	C₄	1	1	yes	characteristic
16	C₈×Q₈×C₇₁	C₄	1	1	yes	characteristic
17	C₈×D₄×C₇₁	C₄	1	3	yes	—
18	C₈×D₄×C₇₁	C₄	1	3	yes	—
19	#431×C₇₁	C₂²	1	3	yes	—
20	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
21	C₂×#11765849	—	2	6	no	—
22	C₄⋊(C₇₁⋊C₈)	—	2	6	no	—
23	C₄⋊(C₇₁⋊C₈)	C₂³	1	2	yes	—
24	C₄⋊(C₇₁⋊C₈)	C₂³	1	6	yes	—
25	C₂²⋊(C₇₁⋊C₈)	—	2	6	no	—
26	C₄×C₇₁⋊C₈	—	2	6	no	—
27	C₂×#41×C₇₁	Q₈	1	1	yes	characteristic
28	C₂²×C₈×C₇₁	C₂×C₄	1	3	yes	—
29	C₂²×C₈×C₇₁	C₂×C₄	1	3	yes	—
30	C₄×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
32	C₄×D₄×C₇₁	C₂×C₄	1	3	yes	—
33	C₄×D₄×C₇₁	D₄	1	3	yes	—
34	#117×C₇₁	Q₈	1	3	yes	—
35	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
36	C₄⋊C₈×C₇₁	C₂×C₄	1	3	yes	—
37	C₄⋊C₈×C₇₁	C₂×C₄	1	3	yes	—
38	C₂²⋊C₈×C₇₁	C₂×C₄	1	6	yes	—
39	C₄×C₈×C₇₁	C₂×C₄	1	3	yes	—
40	C₄×C₈×C₇₁	C₂×C₄	1	3	yes	—
41	C₄×C₈×C₇₁	C₂³	1	3	yes	—
42	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
43	#11765849	—	8	6	no	—
44	C₂×C₇₁⋊C₈	—	2	2	no	—
45	C₂×C₇₁⋊C₈	—	2	6	no	—
46	#41×C₇₁	—	2	4	no	—
47	C₂×Q₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂×D₄×C₇₁	C₄⋊C₄	1	3	yes	—
49	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
50	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
51	C₂²×C₄×C₇₁	C₂×Q₈	1	3	yes	—
52	C₂×C₈×C₇₁	—	2	6	no	—
53	C₂×C₈×C₇₁	—	4	3	no	—
54	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₁	C₂²×C₄	1	3	yes	—
56	C₂×C₈×C₇₁	C₂²×C₄	1	3	yes	—
57	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₇₁	C₄⋊C₄	1	3	yes	—
59	C₄⋊C₄×C₇₁	C₄⋊C₄	1	3	yes	—
60	C₂²⋊C₄×C₇₁	C₄⋊C₄	1	6	yes	—
61	C₄²×C₇₁	C₂²×C₄	1	3	yes	—
62	C₄²×C₇₁	C₂×D₄	1	3	yes	—
63	C₄²×C₇₁	C₂×Q₈	1	3	yes	—
64	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
65	C₂³×C₇₁	C₂×C₄⋊C₄	1	3	yes	—
66	Q₈×C₇₁	#108	1	4	yes	—
67	D₄×C₇₁	—	2	6	no	—
68	C₂×C₄×C₇₁	—	2	3	no	—
69	C₂×C₄×C₇₁	—	2	3	no	—
70	C₂×C₄×C₇₁	—	2	6	no	—
71	C₂×C₄×C₇₁	—	2	6	no	—
72	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	3	yes	—
73	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
78	C₈×C₇₁	—	2	3	no	—
79	C₈×C₇₁	—	2	1	no	—
80	C₈×C₇₁	—	2	3	no	—
81	C₈×C₇₁	#131	1	2	yes	—
82	C₇₁⋊C₈	—	4	2	no	—
83	C₇₁⋊C₈	—	4	6	no	—
84	C₂²×C₇₁	—	2	3	no	—
85	C₂²×C₇₁	—	2	3	no	—
86	C₂²×C₇₁	—	2	3	no	—
87	C₂²×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₄×C₇₁	—	2	3	no	—
89	C₄×C₇₁	—	2	3	no	—
90	C₄×C₇₁	—	2	1	no	—
91	C₄×C₇₁	C₂×#108	1	6	yes	—
92	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
93	C₄×C₇₁	#568	1	1	yes	characteristic
94	C₄×C₇₁	#577	1	2	yes	—
95	#17835	—	71	1	no	maximal, Sylow
96	C₂×C₇₁	—	4	3	no	—
97	C₂×C₇₁	#2906	1	1	yes	characteristic
98	C₂×C₇₁	#2912	1	1	yes	characteristic
99	C₂×C₇₁	C₂²×#108	1	1	yes	characteristic
100	#2991	—	71	2	no	—
101	C₈×#41	D₇₁	1	1	yes	characteristic
102	#1574	—	71	6	no	—
103	#1566	—	71	3	no	—
104	#1564	—	71	3	no	—
105	C₇₁	#17835	1	1	yes	characteristic, complemented, Sylow
106	C₄×#41	C₂×D₇₁	1	1	yes	characteristic
107	C₈×Q₈	—	142	2	no	—
108	C₈×Q₈	Dic₇₁	1	1	yes	characteristic
109	C₈×Q₈	Dic₇₁	1	1	yes	characteristic
110	#436	—	142	6	no	—
111	C₈×D₄	Dic₇₁	1	3	yes	—
112	C₈×D₄	Dic₇₁	1	3	yes	—
113	#431	—	71	6	no	—
114	#431	C₂×D₇₁	1	3	yes	—
115	#423	—	71	6	no	—
116	C₂×C₄×C₈	C₂×D₇₁	1	3	yes	—
117	#335	—	71	6	no	—
118	#334	—	71	1	no	—
119	#334	—	71	3	no	—
120	#334	—	71	6	no	—
121	C₂×#41	Dic₁₄₂	1	1	yes	characteristic
122	C₂×M₁₆	—	142	6	no	—
123	C₂²×C₈	C₂×Dic₇₁	1	3	yes	—
124	C₂²×C₈	C₂×Dic₇₁	1	3	yes	—
125	C₄×Q₈	D₂₈₄	1	1	yes	characteristic
126	C₄×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
127	C₄×D₄	D₂₈₄	1	3	yes	—
128	C₄×D₄	C₂×Dic₇₁	1	3	yes	—
129	#117	Dic₁₄₂	1	3	yes	—
130	C₂×C₄²	C₂²×D₇₁	1	3	yes	—
131	C₄⋊C₈	—	71	6	no	—
132	C₄⋊C₈	—	71	2	no	—
133	C₄⋊C₈	—	142	6	no	—
134	C₄⋊C₈	C₂×Dic₇₁	1	3	yes	—
135	C₄⋊C₈	C₂×Dic₇₁	1	3	yes	—
136	C₂²⋊C₈	—	142	6	no	—
137	C₂²⋊C₈	C₂×Dic₇₁	1	6	yes	—
138	C₄×C₈	—	142	6	no	—
139	C₄×C₈	C₂×Dic₇₁	1	3	yes	—
140	C₄×C₈	C₂×Dic₇₁	1	3	yes	—
141	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
142	C₄×C₈	C₂²×D₇₁	1	3	yes	—
143	#41	—	2	4	no	—
144	C₂×Q₈	C₄⋊Dic₇₁	1	1	yes	characteristic
145	C₂×D₄	C₄⋊Dic₇₁	1	3	yes	—
146	C₂²×C₄	C₂²×Dic₇₁	1	3	yes	—
147	C₂²×C₄	C₂×D₂₈₄	1	3	yes	—
148	C₂²×C₄	C₂×Dic₁₄₂	1	3	yes	—
149	M₁₆	—	568	6	no	—
150	C₂×C₈	—	2	6	no	—
151	C₂×C₈	—	4	3	no	—
152	C₂×C₈	—	142	2	no	—
153	C₂×C₈	—	142	6	no	—
154	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
155	C₂×C₈	C₂²×Dic₇₁	1	3	yes	—
156	C₂×C₈	C₂²×Dic₇₁	1	3	yes	—
157	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
158	C₄⋊C₄	C₄⋊Dic₇₁	1	3	yes	—
159	C₄⋊C₄	C₄⋊Dic₇₁	1	3	yes	—
160	C₂²⋊C₄	C₄⋊Dic₇₁	1	6	yes	—
161	C₄²	C₂²×Dic₇₁	1	3	yes	—
162	C₄²	C₂×D₂₈₄	1	3	yes	—
163	C₄²	C₂×Dic₁₄₂	1	3	yes	—
164	C₄²	C₂³×D₇₁	1	1	yes	characteristic
165	C₂³	C₂×C₄⋊Dic₇₁	1	3	yes	—
166	Q₈	#14556064	1	4	yes	—
167	D₄	—	2	6	no	—
168	C₂×C₄	—	2	3	no	—
169	C₂×C₄	—	2	3	no	—
170	C₂×C₄	—	2	6	no	—
171	C₂×C₄	—	2	6	no	—
172	C₂×C₄	C₂×C₄⋊Dic₇₁	1	3	yes	—
173	C₂×C₄	C₂×C₄⋊Dic₇₁	1	3	yes	—
174	C₂×C₄	C₂×C₄⋊Dic₇₁	1	3	yes	—
175	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
176	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
177	C₂×C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
178	C₈	—	2	3	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	3	no	—
181	C₈	—	284	6	no	—
182	C₈	—	284	2	no	—
183	C₈	#14556135	1	2	yes	—
184	C₂²	—	2	3	no	—
185	C₂²	—	2	3	no	—
186	C₂²	—	2	3	no	—
187	C₂²	C₂²×C₄⋊Dic₇₁	1	1	yes	characteristic
188	C₄	—	2	3	no	—
189	C₄	—	2	3	no	—
190	C₄	—	2	1	no	—
191	C₄	C₂×#14556064	1	6	yes	—
192	C₄	C₂×#14556135	1	1	yes	characteristic
193	C₄	#17986766	1	1	yes	characteristic
194	C₄	#17986811	1	2	yes	—
195	C₂	—	4	3	no	—
196	C₂	#22596908	1	1	yes	characteristic
197	C₂	#22596935	1	1	yes	characteristic
198	C₂	C₂²×#14556064	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₄²	C₂³×D₇₁
Fitting	C₈×#41×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₇₁	Q₈
cosocle	C₄²×C₇₁	C₂⁴
hypercenter	C₈×#41	D₇₁
hypocenter	C₇₁	#17835
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #22587129	C₂	semidirect
3. #22587096	C₂	semidirect
4. #22587088	C₂	semidirect
105. C₇₁	#17835	semidirect

Maximal subgroups

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

subgroup	index
#22597472	2
#22587129	2
#22587096	2
#22587088	2
C₈×#41×C₇₁	2
#17835	71