Subgroups — C₄×#15688874 #26107459

Order 12032.

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⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#15688874	C₂	1	2	yes	maximal
3	#20241223	C₂	1	2	yes	maximal
4	C₄²×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
6	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
7	#15688878	—	2	2	no	—
8	#15688874	C₄	1	8	yes	direct product factor
9	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄³×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄₇⋊C₈	—	2	4	no	—
15	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
16	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
17	C₄×C₄₇⋊C₈	—	4	4	no	—
18	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
25	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
26	C₄×C₈×C₄₇	—	2	4	no	—
27	C₄×C₈×C₄₇	—	2	2	no	—
28	C₄×C₈×C₄₇	C₈	1	8	yes	complemented
29	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₄₇	—	2	2	no	—
36	C₂×C₈×C₄₇	—	2	2	no	—
37	C₂×C₈×C₄₇	—	2	2	no	—
38	C₂×C₈×C₄₇	C₄²	1	2	yes	—
39	C₂×C₈×C₄₇	C₄²	1	2	yes	—
40	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
42	C₄²×C₄₇	—	2	4	no	—
43	C₄²×C₄₇	—	2	2	no	—
44	C₄²×C₄₇	C₂×C₈	1	4	yes	—
45	C₄²×C₄₇	M₁₆	1	4	yes	—
46	C₄²×C₄₇	SD₁₆	1	4	yes	—
47	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
48	C₄²×C₄₇	#41	1	2	yes	—
49	C₂×C₄₇⋊C₈	—	2	8	no	—
50	C₂×C₄₇⋊C₈	—	4	4	no	—
51	C₂×C₄₇⋊C₈	—	4	4	no	—
52	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	—	2	2	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	—	2	2	no	—
56	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
57	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
58	C₂×C₄×C₄₇	#106	1	4	yes	—
59	C₂×C₄×C₄₇	#108	1	4	yes	—
60	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
65	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
67	C₂×C₄×C₄₇	#135	1	2	yes	—
68	C₈×C₄₇	—	2	2	no	—
69	C₈×C₄₇	—	2	4	no	—
70	C₈×C₄₇	C₄×C₈	1	4	yes	complemented
71	C₄₇⋊C₈	—	4	8	no	—
72	C₄×#334	—	47	1	no	maximal, Sylow
73	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
75	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
76	C₂²×C₄₇	#427	1	1	yes	characteristic
77	C₂²×C₄₇	C₂×#108	1	2	yes	—
78	C₄×C₄₇	—	2	4	no	—
79	C₄×C₄₇	—	2	2	no	—
80	C₄×C₄₇	#334	1	8	yes	complemented
81	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
82	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
83	C₄×C₄₇	#431	1	2	yes	—
84	C₄×C₄₇	C₄×SD₁₆	1	4	yes	—
85	C₄×C₄⋊C₈	—	47	2	no	—
86	C₄²×C₈	D₄₇	1	1	yes	characteristic
87	C₂×#334	—	47	2	no	—
88	#1277	—	47	2	no	—
89	C₂×C₄₇	C₂×#334	1	2	yes	—
90	C₂×C₄₇	#1564	1	2	yes	—
91	C₂×C₄₇	C₄×C₄⋊C₈	1	1	yes	characteristic
92	C₂×C₄₇	C₄×#106	1	1	yes	characteristic
93	C₂×C₄₇	C₄×#108	1	1	yes	characteristic
94	C₂×C₄⋊C₈	—	47	4	no	—
95	C₂×C₄×C₈	—	94	2	no	—
96	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
97	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
98	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
100	C₄³	C₂×D₄₇	1	1	yes	characteristic
101	#336	—	94	2	no	—
102	#334	—	47	8	no	—
103	C₄₇	C₄×#334	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₈	—	94	4	no	—
105	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
106	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
108	C₂×C₄²	Dic₉₄	1	2	yes	—
109	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
110	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
111	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
112	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
113	C₄⋊C₈	—	47	8	no	—
114	C₄⋊C₈	—	94	4	no	—
115	C₄×C₈	—	2	4	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	—	188	4	no	—
118	C₄×C₈	C₄₇⋊C₈	1	8	yes	complemented
119	C₄×C₈	C₄×D₄₇	1	4	yes	—
120	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
121	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
122	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
123	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
124	C₂²×C₄	C₂²×Dic₄₇	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	—	94	8	no	—
129	C₂×C₈	—	188	4	no	—
130	C₂×C₈	—	188	4	no	—
131	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
132	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
133	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
134	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
135	C₄²	—	2	4	no	—
136	C₄²	—	2	2	no	—
137	C₄²	#10625325	1	4	yes	—
138	C₄²	C₂×C₄₇⋊C₈	1	4	yes	—
139	C₄²	#10625329	1	4	yes	—
140	C₄²	C₂×C₄×D₄₇	1	2	yes	—
141	C₄²	#10625350	1	2	yes	—
142	C₂³	C₂×C₄⋊Dic₄₇	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⋊(C₄₇⋊C₈)	1	4	yes	—
147	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
148	C₂×C₄	#13093956	1	4	yes	—
149	C₂×C₄	#13093960	1	4	yes	—
150	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
151	C₂×C₄	#13093983	1	1	yes	characteristic
152	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
153	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
154	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
155	C₂×C₄	C₂×#10625329	1	2	yes	—
156	C₂×C₄	C₂×#10625325	1	2	yes	—
157	C₂×C₄	#13094046	1	2	yes	—
158	C₈	—	2	2	no	—
159	C₈	—	2	4	no	—
160	C₈	—	188	8	no	—
161	C₈	C₄×C₄₇⋊C₈	1	4	yes	complemented
162	C₂²	C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
163	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
164	C₂²	C₂×#13093956	1	1	yes	characteristic
165	C₂²	#15689161	1	1	yes	characteristic
166	C₂²	C₂×#13093960	1	2	yes	—
167	C₄	—	2	2	no	—
168	C₄	—	2	4	no	—
169	C₄	#15688874	1	8	yes	direct product factor
170	C₄	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
171	C₄	C₄×#10625329	1	1	yes	characteristic
172	C₄	#15689176	1	2	yes	—
173	C₄	C₄×#10625325	1	4	yes	—
174	C₂	C₂×#15688874	1	2	yes	—
175	C₂	#20242136	1	2	yes	—
176	C₂	C₄×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
177	C₂	C₄×#13093956	1	1	yes	characteristic
178	C₂	C₄×#13093960	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	D₁₈₈
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄²×C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄²×C₈	D₄₇
hypocenter	C₄₇	C₄×#334
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
8. #15688874	C₄	direct (N × H)
28. C₄×C₈×C₄₇	C₈	semidirect
70. C₈×C₄₇	C₄×C₈	semidirect
80. C₄×C₄₇	#334	semidirect
103. C₄₇	C₄×#334	semidirect
118. C₄×C₈	C₄₇⋊C₈	semidirect
161. C₈	C₄×C₄₇⋊C₈	semidirect
169. C₄	#15688874	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⋊(C₄₇⋊C₈)	2
C₂×#15688874	2
#20241223	2
C₄²×C₈×C₄₇	2
C₄×#334	47

Subgroups — C4×#15688874 #26107459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15688874 #26107459