Subgroups — #31913620

Order 18176.

438 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985825	C₂	1	2	yes	maximal, complemented
2	#22588369	C₂	1	2	yes	maximal
3	#22586432	C₂	1	2	yes	maximal, complemented
4	C₂³×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556062	C₂²	1	4	yes	—
6	#17985859	—	2	2	no	—
7	#17985825	—	2	4	no	—
8	#17985762	—	2	2	no	—
9	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₇₁	—	2	4	no	—
11	C₂²×C₁₆×C₇₁	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₇₁	—	2	4	no	—
15	C₂×C₂²⋊Dic₇₁	—	4	2	no	—
16	#14556062	—	2	4	no	—
17	#14556062	—	2	4	no	—
18	C₇₁⋊#95	—	4	2	no	—
19	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₇₁	—	2	2	no	—
21	C₂²×C₈×C₇₁	—	2	2	no	—
22	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
25	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
26	C₂×C₁₆×C₇₁	—	2	2	no	—
27	C₂×C₁₆×C₇₁	—	2	8	no	—
28	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₇₁	D₄	1	4	yes	—
31	C₂²×Dic₇₁	—	4	4	no	—
32	C₄⋊Dic₇₁	—	4	4	no	—
33	C₄⋊Dic₇₁	—	4	4	no	—
34	C₂²⋊Dic₇₁	—	8	4	no	—
35	C₂⁴×C₇₁	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₇₁	—	2	2	no	—
37	C₂²×C₄×C₇₁	—	2	2	no	—
38	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	Q₁₆	1	2	yes	—
41	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	—	2	2	no	—
43	C₂×C₈×C₇₁	—	2	2	no	—
44	C₂×C₈×C₇₁	—	2	2	no	—
45	C₂×C₈×C₇₁	—	2	4	no	—
46	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
49	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₇₁	#41	1	2	yes	—
51	C₁₆×C₇₁	—	2	2	no	—
52	C₁₆×C₇₁	C₂²⋊C₄	1	4	yes	complemented
53	C₂³×C₇₁	—	2	2	no	—
54	C₂³×C₇₁	—	2	2	no	—
55	C₂³×C₇₁	#107	1	2	yes	—
56	C₂³×C₇₁	D₁₆	1	2	yes	—
57	C₂³×C₇₁	Q₃₂	1	2	yes	—
58	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₇₁	—	2	2	no	—
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	—	2	2	no	—
62	C₂×C₄×C₇₁	—	2	4	no	—
63	C₂×C₄×C₇₁	#107	1	2	yes	—
64	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#121	1	2	yes	—
66	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#135	1	2	yes	—
68	C₂×Dic₇₁	—	8	4	no	—
69	C₂×Dic₇₁	—	8	4	no	—
70	C₂×Dic₇₁	—	8	4	no	—
71	C₈×C₇₁	—	2	2	no	—
72	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₈×C₇₁	#117	1	1	yes	characteristic
74	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
75	C₂²×C₇₁	—	2	2	no	—
76	C₂²×C₇₁	—	2	2	no	—
77	C₂²×C₇₁	—	2	4	no	—
78	C₂²×C₇₁	—	2	4	no	—
79	C₂²×C₇₁	#366	1	4	yes	complemented
80	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
81	C₂²×C₇₁	#466	1	2	yes	—
82	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
83	C₂²×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
84	C₂²×C₇₁	#508	1	2	yes	—
85	C₄×C₇₁	—	2	2	no	—
86	C₄×C₇₁	#380	1	1	yes	characteristic
87	C₄×C₇₁	#427	1	1	yes	characteristic
88	C₄×C₇₁	#467	1	2	yes	—
89	Dic₇₁	—	16	4	no	—
90	#12834	—	71	1	no	maximal, Sylow
91	C₂×C₇₁	—	2	4	no	—
92	C₂×C₇₁	#1808	1	1	yes	characteristic
93	C₂×C₇₁	C₂×#366	1	1	yes	characteristic
94	C₂×C₇₁	#2158	1	1	yes	characteristic
95	C₂×C₇₁	#2215	1	2	yes	—
96	C₂×C₇₁	#2216	1	2	yes	—
97	C₂³×C₁₆	D₇₁	1	1	yes	characteristic
98	C₂×#366	—	71	2	no	—
99	#1808	—	71	2	no	—
100	#1380	—	71	2	no	—
101	C₇₁	#12834	1	1	yes	characteristic, complemented, Sylow
102	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
103	C₂²×C₁₆	—	2	4	no	—
104	C₂²×C₁₆	Dic₇₁	1	4	yes	complemented
105	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
106	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
107	C₂×#107	—	71	4	no	—
108	#380	—	142	2	no	—
109	#366	—	142	4	no	—
110	#340	—	142	2	no	—
111	C₂³×C₄	D₂₈₄	1	1	yes	characteristic
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	Dic₁₄₂	1	2	yes	—
115	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
116	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
117	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	142	4	no	—
119	C₂×C₂²⋊C₄	—	284	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	2	8	no	—
122	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
123	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
124	C₂×C₁₆	C₂²⋊D₇₁	1	4	yes	—
125	#107	—	142	4	no	—
126	#107	—	142	4	no	—
127	#95	—	284	2	no	—
128	C₂⁴	D₅₆₈	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	284	4	no	—
132	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
133	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
134	C₂²×C₄	Dic₂₈₄	1	2	yes	—
135	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
141	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
142	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
143	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
144	C₂×C₈	#11765865	1	2	yes	—
145	C₄⋊C₄	—	284	4	no	—
146	C₄⋊C₄	—	284	4	no	—
147	C₂²⋊C₄	—	568	4	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	C₂²⋊Dic₇₁	1	4	yes	complemented
150	C₂³	—	2	2	no	—
151	C₂³	—	2	2	no	—
152	C₂³	#14556062	1	2	yes	—
153	C₂³	D₁₁₃₆	1	2	yes	—
154	C₂³	Dic₅₆₈	1	2	yes	—
155	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	568	4	no	—
161	C₂×C₄	—	568	4	no	—
162	C₂×C₄	—	568	4	no	—
163	C₂×C₄	#14556062	1	2	yes	—
164	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
165	C₂×C₄	#14556102	1	2	yes	—
166	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
167	C₂×C₄	#14556150	1	2	yes	—
168	C₈	—	2	2	no	—
169	C₈	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
170	C₈	#14556087	1	1	yes	characteristic
171	C₈	#14556107	1	2	yes	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	4	no	—
175	C₂²	—	2	4	no	—
176	C₂²	#17985825	1	4	yes	complemented
177	C₂²	C₂×#14556062	1	1	yes	characteristic
178	C₂²	#17986235	1	2	yes	—
179	C₂²	C₂×D₁₁₃₆	1	1	yes	characteristic
180	C₂²	C₂×Dic₅₆₈	1	1	yes	characteristic
181	C₂²	#17986431	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	1136	4	no	—
184	C₄	#17985859	1	1	yes	characteristic
185	C₄	#17986037	1	1	yes	characteristic
186	C₄	#17986240	1	2	yes	—
187	C₂	—	2	4	no	—
188	C₂	#22588369	1	1	yes	characteristic
189	C₂	C₂×#17985825	1	1	yes	characteristic
190	C₂	#22590124	1	1	yes	characteristic
191	C₂	#22590426	1	2	yes	—
192	C₂	#22590431	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₆₈
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂³×C₁₆×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₈
prime radical	C₂⁴×C₇₁	D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂³×C₁₆	D₇₁
hypocenter	C₇₁	#12834
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#17985825	C₂	semidirect
3. #22586432	C₂	semidirect
11. C₂²×C₁₆×C₇₁	C₄	semidirect
52. C₁₆×C₇₁	C₂²⋊C₄	semidirect
79. C₂²×C₇₁	#366	semidirect
101. C₇₁	#12834	semidirect
104. C₂²×C₁₆	Dic₇₁	semidirect
149. C₁₆	C₂²⋊Dic₇₁	semidirect
176. C₂²	#17985825	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985825	2
#22588369	2
#22586432	2
C₂³×C₁₆×C₇₁	2
#12834	71