Subgroups — C₄×#17985750 #31888633

Order 18176.

470 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₇₁⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#17985750	C₂	1	2	yes	maximal
3	#22586175	C₂	1	2	yes	maximal
4	C₄²×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	4	yes	—
6	C₂×C₄×C₇₁⋊C₈	—	2	2	no	—
7	#17985754	—	2	2	no	—
8	#17985750	C₄	1	8	yes	direct product factor
9	C₂×C₄×C₈×C₇₁	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₇₁	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₄³×C₇₁	C₂²	1	1	yes	characteristic
14	C₂²×C₇₁⋊C₈	—	2	4	no	—
15	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
16	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	8	yes	—
17	C₄×C₇₁⋊C₈	—	4	4	no	—
18	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₇₁	Q₈	1	2	yes	—
25	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
26	C₄×C₈×C₇₁	—	2	4	no	—
27	C₄×C₈×C₇₁	—	2	2	no	—
28	C₄×C₈×C₇₁	C₈	1	8	yes	complemented
29	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
30	C₂×C₇₁⋊C₈	—	2	8	no	—
31	C₂×C₇₁⋊C₈	—	4	4	no	—
32	C₂×C₇₁⋊C₈	—	4	4	no	—
33	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₇₁	—	2	2	no	—
39	C₂×C₈×C₇₁	—	2	2	no	—
40	C₂×C₈×C₇₁	—	2	2	no	—
41	C₂×C₈×C₇₁	C₄²	1	2	yes	—
42	C₂×C₈×C₇₁	C₄²	1	2	yes	—
43	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
44	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
45	C₄²×C₇₁	—	2	4	no	—
46	C₄²×C₇₁	—	2	2	no	—
47	C₄²×C₇₁	C₂×C₈	1	4	yes	—
48	C₄²×C₇₁	M₁₆	1	4	yes	—
49	C₄²×C₇₁	SD₁₆	1	4	yes	—
50	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
51	C₄²×C₇₁	#41	1	2	yes	—
52	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₁	—	2	2	no	—
54	C₂×C₄×C₇₁	—	2	2	no	—
55	C₂×C₄×C₇₁	—	2	2	no	—
56	C₂×C₄×C₇₁	C₄⋊C₈	1	4	yes	—
57	C₂×C₄×C₇₁	C₄⋊C₈	1	4	yes	—
58	C₂×C₄×C₇₁	#106	1	4	yes	—
59	C₂×C₄×C₇₁	#108	1	4	yes	—
60	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
65	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
67	C₂×C₄×C₇₁	#135	1	2	yes	—
68	C₈×C₇₁	—	2	2	no	—
69	C₈×C₇₁	—	2	4	no	—
70	C₈×C₇₁	C₄×C₈	1	4	yes	complemented
71	C₇₁⋊C₈	—	4	8	no	—
72	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₇₁	C₂×C₄⋊C₈	1	2	yes	—
74	C₂²×C₇₁	C₂×#106	1	1	yes	characteristic
75	C₂²×C₇₁	#427	1	1	yes	characteristic
76	C₂²×C₇₁	C₂×#108	1	2	yes	—
77	C₄×C₇₁	—	2	4	no	—
78	C₄×C₇₁	—	2	2	no	—
79	C₄×C₇₁	#334	1	8	yes	complemented
80	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
81	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
82	C₄×C₇₁	#431	1	2	yes	—
83	C₄×C₇₁	C₄×SD₁₆	1	4	yes	—
84	C₄×#334	—	71	1	no	maximal, Sylow
85	C₂×C₇₁	C₂×#334	1	2	yes	—
86	C₂×C₇₁	#1564	1	2	yes	—
87	C₂×C₇₁	C₄×C₄⋊C₈	1	1	yes	characteristic
88	C₂×C₇₁	C₄×#106	1	1	yes	characteristic
89	C₂×C₇₁	C₄×#108	1	1	yes	characteristic
90	C₄×C₄⋊C₈	—	71	2	no	—
91	C₄²×C₈	D₇₁	1	1	yes	characteristic
92	C₂×#334	—	71	2	no	—
93	#1277	—	71	2	no	—
94	C₇₁	C₄×#334	1	1	yes	characteristic, complemented, Sylow
95	C₂×C₄⋊C₈	—	71	4	no	—
96	C₂×C₄×C₈	—	142	2	no	—
97	C₂×C₄×C₈	Dic₇₁	1	2	yes	—
98	C₂×C₄×C₈	Dic₇₁	1	2	yes	—
99	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
101	C₄³	C₂×D₇₁	1	1	yes	characteristic
102	#336	—	142	2	no	—
103	#334	—	71	8	no	—
104	C₂²×C₈	—	142	4	no	—
105	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
106	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
107	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
108	C₂×C₄²	Dic₁₄₂	1	2	yes	—
109	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
110	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
111	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
112	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
113	C₄⋊C₈	—	71	8	no	—
114	C₄⋊C₈	—	142	4	no	—
115	C₄×C₈	—	2	4	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	—	284	4	no	—
118	C₄×C₈	C₇₁⋊C₈	1	8	yes	complemented
119	C₄×C₈	C₄×D₇₁	1	4	yes	—
120	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
121	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
122	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
123	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
124	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	284	4	no	—
129	C₂×C₈	—	284	4	no	—
130	C₂×C₈	—	142	8	no	—
131	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
132	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
133	C₂×C₈	C₂×C₇₁⋊C₈	1	4	yes	—
134	C₂×C₈	C₂×C₇₁⋊C₈	1	4	yes	—
135	C₄²	—	2	4	no	—
136	C₄²	—	2	2	no	—
137	C₄²	C₂×C₇₁⋊C₈	1	4	yes	—
138	C₄²	#11765849	1	4	yes	—
139	C₄²	#11765854	1	4	yes	—
140	C₄²	C₂×C₄×D₇₁	1	2	yes	—
141	C₄²	#11765865	1	2	yes	—
142	C₂³	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
147	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
148	C₂×C₄	#14556060	1	4	yes	—
149	C₂×C₄	#14556064	1	4	yes	—
150	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
151	C₂×C₄	#14556087	1	1	yes	characteristic
152	C₂×C₄	C₄×D₂₈₄	1	2	yes	—
153	C₂×C₄	C₄×Dic₁₄₂	1	2	yes	—
154	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
155	C₂×C₄	C₂×#11765849	1	2	yes	—
156	C₂×C₄	C₂×#11765854	1	2	yes	—
157	C₂×C₄	#14556150	1	2	yes	—
158	C₈	—	2	4	no	—
159	C₈	—	2	2	no	—
160	C₈	—	284	8	no	—
161	C₈	C₄×C₇₁⋊C₈	1	4	yes	complemented
162	C₂²	C₄×C₄⋊Dic₇₁	1	1	yes	characteristic
163	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	2	yes	—
164	C₂²	C₂×#14556060	1	1	yes	characteristic
165	C₂²	#17986037	1	1	yes	characteristic
166	C₂²	C₂×#14556064	1	2	yes	—
167	C₄	—	2	2	no	—
168	C₄	—	2	4	no	—
169	C₄	#17985750	1	8	yes	direct product factor
170	C₄	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
171	C₄	C₄×#11765849	1	1	yes	characteristic
172	C₄	#17986052	1	2	yes	—
173	C₄	C₄×#11765854	1	4	yes	—
174	C₂	C₂×#17985750	1	2	yes	—
175	C₂	#22587088	1	2	yes	—
176	C₂	C₄×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
177	C₂	C₄×#14556060	1	1	yes	characteristic
178	C₂	C₄×#14556064	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	D₂₈₄
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄²×C₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄²×C₈	D₇₁
hypocenter	C₇₁	C₄×#334
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
8. #17985750	C₄	direct (N × H)
28. C₄×C₈×C₇₁	C₈	semidirect
70. C₈×C₇₁	C₄×C₈	semidirect
79. C₄×C₇₁	#334	semidirect
94. C₇₁	C₄×#334	semidirect
118. C₄×C₈	C₇₁⋊C₈	semidirect
161. C₈	C₄×C₇₁⋊C₈	semidirect
169. C₄	#17985750	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₇₁⋊C₈)	2
C₂×#17985750	2
#22586175	2
C₄²×C₈×C₇₁	2
C₄×#334	71

Subgroups — C4×#17985750 #31888633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17985750 #31888633