Subgroups — #368×Dic₇₁ #31912866

Order 18176.

314 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985829	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#108×Dic₇₁	C₂	1	2	yes	maximal
4	#22586433	C₂	1	2	yes	maximal
5	C₂×#368×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14556064	C₂²	1	2	yes	—
8	M₁₆×Dic₇₁	—	2	2	no	—
9	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
10	#17985829	C₄	1	4	yes	complemented
11	C₁₆×Dic₇₁	C₄	1	4	yes	—
12	#17985763	—	2	2	no	—
13	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×#108×C₇₁	C₂²	1	2	yes	—
15	#368×C₇₁	C₄	1	4	yes	complemented
16	C₂×#11765849	—	2	2	no	—
17	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
19	C₂×C₇₁⋊C₁₆	—	2	1	no	—
20	C₂×C₇₁⋊C₁₆	—	2	1	no	—
21	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
22	#14556064	—	2	2	no	—
23	#14556064	C₂×C₄	1	4	yes	—
24	#14556038	—	4	2	no	—
25	C₈×Dic₇₁	—	4	2	no	—
26	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
27	C₈×Dic₇₁	Q₈	1	2	yes	—
28	C₂×M₁₆×C₇₁	—	2	2	no	—
29	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
33	#108×C₇₁	—	2	2	no	—
34	#108×C₇₁	C₂×C₄	1	4	yes	—
35	C₂²×Dic₇₁	Q₁₆	1	2	yes	—
36	#11765849	—	2	4	no	—
37	#11765849	—	4	2	no	—
38	C₂×C₇₁⋊C₈	—	2	1	no	—
39	C₂×C₇₁⋊C₈	—	2	1	no	—
40	C₂×C₇₁⋊C₈	—	4	2	no	—
41	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
42	C₄×Dic₇₁	C₄⋊C₄	1	2	yes	—
43	C₄×Dic₇₁	D₈	1	1	yes	characteristic
44	C₄×Dic₇₁	D₈	1	1	yes	characteristic
45	C₇₁⋊C₁₆	—	2	2	no	—
46	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	M₁₆×C₇₁	—	2	4	no	—
48	M₁₆×C₇₁	—	4	2	no	—
49	C₂×C₈×C₇₁	—	4	2	no	—
50	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
55	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
56	C₁₆×C₇₁	C₄²	1	4	yes	—
57	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
63	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
64	C₂×Dic₇₁	—	2	4	no	—
65	C₂×Dic₇₁	#107	1	2	yes	—
66	C₂×Dic₇₁	#107	1	2	yes	—
67	C₈×C₇₁	—	4	4	no	—
68	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₇₁	#117	1	1	yes	characteristic
70	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
71	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
72	C₇₁⋊C₈	—	2	2	no	—
73	C₇₁⋊C₈	—	4	4	no	—
74	C₂²×C₇₁	—	2	2	no	—
75	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
76	C₂²×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
77	C₂²×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
78	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
79	C₄×C₇₁	#427	1	1	yes	characteristic
80	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
81	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
82	Dic₇₁	—	2	2	no	—
83	Dic₇₁	#368	1	4	yes	direct product factor
84	C₄×#368	—	71	1	no	maximal, Sylow
85	C₂×C₇₁	—	2	1	no	—
86	C₂×C₇₁	—	2	1	no	—
87	C₂×C₇₁	C₄×#107	1	1	yes	characteristic
88	C₂×C₇₁	C₂×#368	1	2	yes	—
89	C₂×#368	—	71	1	no	—
90	C₂×#368	D₇₁	1	1	yes	characteristic
91	C₂×C₄×C₁₆	—	71	1	no	—
92	C₄×#108	—	71	2	no	—
93	#1381	—	71	2	no	—
94	C₇₁	C₄×#368	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₁₆	—	71	1	no	—
96	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
97	C₂×#108	—	71	2	no	—
98	C₂×#108	C₂×D₇₁	1	2	yes	—
99	C₄×M₁₆	—	142	2	no	—
100	C₂×C₄×C₈	—	71	1	no	—
101	#368	—	71	4	no	—
102	#368	Dic₇₁	1	4	yes	direct product factor
103	C₄×C₁₆	—	71	4	no	—
104	#341	—	142	2	no	—
105	C₂×M₁₆	—	2	2	no	—
106	C₂×M₁₆	—	142	2	no	—
107	C₂²×C₈	—	71	1	no	—
108	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
109	C₂×C₄²	—	71	1	no	—
110	C₂×C₁₆	—	71	2	no	—
111	C₂×C₁₆	—	142	1	no	—
112	C₂×C₁₆	—	142	1	no	—
113	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
114	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
115	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
116	#108	—	2	2	no	—
117	#108	—	71	4	no	—
118	#108	—	142	2	no	—
119	#108	C₂×Dic₇₁	1	4	yes	—
120	#97	—	284	2	no	—
121	C₄×C₈	—	284	2	no	—
122	C₄×C₈	—	71	2	no	—
123	C₄×C₈	—	71	2	no	—
124	C₂²×C₄	—	71	2	no	—
125	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
126	M₁₆	—	2	4	no	—
127	M₁₆	—	4	2	no	—
128	M₁₆	—	284	2	no	—
129	M₁₆	—	142	4	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	284	2	no	—
132	C₂×C₈	—	71	2	no	—
133	C₂×C₈	—	142	1	no	—
134	C₂×C₈	—	142	1	no	—
135	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
137	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
139	C₂×C₈	#11765864	1	1	yes	characteristic
140	C₂×C₈	#11765866	1	1	yes	characteristic
141	C₄²	—	71	2	no	—
142	C₄²	—	71	1	no	—
143	C₄²	—	71	1	no	—
144	C₁₆	—	142	2	no	—
145	C₁₆	C₄×Dic₇₁	1	4	yes	—
146	C₂³	Q₁₆×D₇₁	1	1	yes	characteristic
147	C₂×C₄	—	71	2	no	—
148	C₂×C₄	—	71	2	no	—
149	C₂×C₄	—	142	4	no	—
150	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
151	C₂×C₄	#14556090	1	1	yes	characteristic
152	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
153	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
154	C₂×C₄	#14556147	1	1	yes	characteristic
155	C₂×C₄	#14556151	1	1	yes	characteristic
156	C₈	—	4	4	no	—
157	C₈	—	284	4	no	—
158	C₈	—	142	2	no	—
159	C₈	C₂×C₄×Dic₇₁	1	1	yes	characteristic
160	C₈	#14556086	1	1	yes	characteristic
161	C₈	#14556093	1	1	yes	characteristic
162	C₈	Q₈×Dic₇₁	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	#107×D₇₁	1	1	yes	characteristic
165	C₂²	#17986093	1	1	yes	characteristic
166	C₂²	Q₁₆×Dic₇₁	1	1	yes	characteristic
167	C₄	—	71	4	no	—
168	C₄	—	142	2	no	—
169	C₄	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
170	C₄	#17986036	1	1	yes	characteristic
171	C₄	#17986081	1	1	yes	characteristic
172	C₄	D₈×Dic₇₁	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#107×Dic₇₁	1	1	yes	characteristic
176	C₂	#368×D₇₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₇₁
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#368×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#107
prime radical	C₂³×C₇₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#368	D₇₁
hypocenter	C₇₁	C₄×#368
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
10. #17985829	C₄	semidirect
15. #368×C₇₁	C₄	semidirect
83. Dic₇₁	#368	direct (N × H)
94. C₇₁	C₄×#368	semidirect
102. #368	Dic₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17985829	2
C₂×C₁₆×Dic₇₁	2
#108×Dic₇₁	2
#22586433	2
C₂×#368×C₇₁	2
C₄×#368	71

Subgroups — #368×Dic71 #31912866

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #368×Dic₇₁ #31912866