Subgroups — C₄×#16813587 #28392923

Order 14464.

684 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₂₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16813587	C₂	1	2	yes	maximal
3	C₄⋊C₄×Dic₁₁₃	C₂	1	2	yes	maximal
4	C₄×#14268626	C₂	1	2	yes	maximal
5	#20298449	C₂	1	4	yes	maximal
6	C₄²×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic
7	C₄²×Dic₁₁₃	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₂₆	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₂₆	C₄	1	8	yes	complemented
11	#16813587	C₄	1	16	yes	direct product factor
12	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
13	C₂×#14268626	C₂²	1	4	yes	—
14	C₂×#14268626	C₂²	1	2	yes	—
15	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₁₃	C₂²	1	4	yes	—
17	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₁₁₃	C₂²	1	4	yes	—
19	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
22	#16813529	C₂²	1	4	yes	—
23	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₁₃	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
26	#95×C₁₁₃	C₂²	1	2	yes	—
27	C₂×Dic₂₂₆	C₂×C₄	1	8	yes	—
28	C₂×Dic₂₂₆	C₂×C₄	1	4	yes	—
29	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
30	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
31	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
32	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
33	C₄⋊Dic₁₁₃	C₂×C₄	1	4	yes	—
34	#14268626	C₂×C₄	1	16	yes	—
35	#14268626	C₂×C₄	1	8	yes	—
36	C₄×Dic₁₁₃	—	2	2	no	—
37	C₄×Dic₁₁₃	—	2	4	no	—
38	C₄×Dic₁₁₃	—	2	8	no	—
39	C₄×Dic₁₁₃	C₂×C₄	1	8	yes	—
40	C₄×Dic₁₁₃	C₂×C₄	1	16	yes	—
41	C₄×Dic₁₁₃	C₂×C₄	1	8	yes	—
42	C₄×Dic₁₁₃	Q₈	1	4	yes	complemented
43	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
45	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₁₁₃	C₂×C₄	1	8	yes	—
48	C₄²×C₁₁₃	—	2	4	no	—
49	C₄²×C₁₁₃	C₂×C₄	1	4	yes	—
50	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
51	C₂×C₄×C₁₁₃	—	2	4	no	—
52	C₂×C₄×C₁₁₃	—	2	2	no	—
53	C₂×C₄×C₁₁₃	—	2	4	no	—
54	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₁₁₃	C₂²×C₄	1	8	yes	—
56	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
58	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₁₁₃	#41	1	2	yes	—
61	C₂×Dic₁₁₃	—	2	2	no	—
62	C₂×Dic₁₁₃	—	2	4	no	—
63	C₂×Dic₁₁₃	—	2	2	no	—
64	C₂×Dic₁₁₃	C₂²×C₄	1	4	yes	—
65	C₂×Dic₁₁₃	C₂²×C₄	1	4	yes	—
66	C₂×Dic₁₁₃	C₂²×C₄	1	8	yes	—
67	C₂×Dic₁₁₃	C₂²×C₄	1	4	yes	—
68	C₂×Dic₁₁₃	C₂²×C₄	1	4	yes	—
69	C₂×Dic₁₁₃	C₂×Q₈	1	4	yes	—
70	C₂×Dic₁₁₃	#41	1	4	yes	—
71	Dic₂₂₆	C₄²	1	16	yes	complemented
72	C₂²×C₁₁₃	C₂³×C₄	1	2	yes	—
73	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
74	C₂²×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
76	C₂²×C₁₁₃	C₂×#41	1	1	yes	characteristic
77	C₄×C₁₁₃	—	2	8	no	—
78	C₄×C₁₁₃	—	2	2	no	—
79	C₄×C₁₁₃	C₂×C₄²	1	4	yes	—
80	C₄×C₁₁₃	C₄×Q₈	1	4	yes	complemented
81	Dic₁₁₃	—	2	4	no	—
82	Dic₁₁₃	—	2	4	no	—
83	Dic₁₁₃	C₂×C₄²	1	8	yes	—
84	Dic₁₁₃	C₄×Q₈	1	8	yes	complemented
85	C₂×C₁₁₃	C₂²×C₄²	1	1	yes	characteristic
86	C₂×C₁₁₃	C₂×C₄×Q₈	1	2	yes	—
87	C₂×C₁₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₂×C₁₁₃	C₄×#41	1	2	yes	—
89	C₂×C₁₁₃	C₄×#41	1	1	yes	characteristic
90	C₄²×Q₈	—	113	1	no	maximal, Sylow
91	C₁₁₃	C₄²×Q₈	1	1	yes	characteristic, complemented, Sylow
92	C₂×C₄×Q₈	—	113	2	no	—
93	C₂×C₄×Q₈	—	113	1	no	—
94	C₄×C₄⋊C₄	—	113	2	no	—
95	C₄×C₄⋊C₄	—	113	2	no	—
96	C₄×C₄⋊C₄	—	113	4	no	—
97	C₄×C₄⋊C₄	D₁₁₃	1	1	yes	characteristic
98	C₄³	—	113	2	no	—
99	C₄³	—	113	1	no	—
100	C₂²×Q₈	—	113	1	no	—
101	C₄×Q₈	—	113	16	no	—
102	C₄×Q₈	—	113	8	no	—
103	C₂×C₄⋊C₄	—	113	4	no	—
104	C₂×C₄⋊C₄	—	113	2	no	—
105	C₂×C₄⋊C₄	—	113	1	no	—
106	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
107	C₂×C₄²	—	113	4	no	—
108	C₂×C₄²	—	113	2	no	—
109	C₂×C₄²	—	113	2	no	—
110	C₂×C₄²	—	113	1	no	—
111	C₂×C₄²	—	113	4	no	—
112	C₂×C₄²	—	113	2	no	—
113	C₂×C₄²	—	113	1	no	—
114	C₂×C₄²	C₂×D₁₁₃	1	2	yes	—
115	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
116	#95	—	113	4	no	—
117	#95	C₂×D₁₁₃	1	2	yes	—
118	C₂×Q₈	—	113	8	no	—
119	C₂×Q₈	—	113	4	no	—
120	C₂²×C₄	—	113	2	no	—
121	C₂²×C₄	—	113	2	no	—
122	C₂²×C₄	—	113	2	no	—
123	C₂²×C₄	—	113	2	no	—
124	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
125	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
128	C₄⋊C₄	—	113	16	no	—
129	C₄⋊C₄	—	113	8	no	—
130	C₄⋊C₄	—	113	4	no	—
131	C₄⋊C₄	C₄×D₁₁₃	1	8	yes	—
132	C₄²	—	2	4	no	—
133	C₄²	—	113	16	no	—
134	C₄²	—	113	8	no	—
135	C₄²	—	113	8	no	—
136	C₄²	—	113	4	no	—
137	C₄²	—	226	8	no	—
138	C₄²	—	226	4	no	—
139	C₄²	—	226	2	no	—
140	C₄²	C₄×D₁₁₃	1	4	yes	—
141	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
142	Q₈	—	113	16	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	113	4	no	—
147	C₂×C₄	—	113	8	no	—
148	C₂×C₄	—	113	4	no	—
149	C₂×C₄	—	113	4	no	—
150	C₂×C₄	—	113	4	no	—
151	C₂×C₄	—	113	4	no	—
152	C₂×C₄	—	113	4	no	—
153	C₂×C₄	—	226	2	no	—
154	C₂×C₄	—	226	4	no	—
155	C₂×C₄	—	226	2	no	—
156	C₂×C₄	C₂×C₄×D₁₁₃	1	8	yes	—
157	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
158	C₂×C₄	C₂×C₄×D₁₁₃	1	4	yes	—
159	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
160	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
161	C₂×C₄	Q₈×D₁₁₃	1	2	yes	—
162	C₂×C₄	#14268646	1	2	yes	—
163	C₂²	C₂²×C₄×D₁₁₃	1	2	yes	—
164	C₂²	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
165	C₂²	C₂×Q₈×D₁₁₃	1	1	yes	characteristic
166	C₂²	#41×D₁₁₃	1	2	yes	—
167	C₂²	C₂×#14268646	1	1	yes	characteristic
168	C₄	—	2	8	no	—
169	C₄	—	2	2	no	—
170	C₄	—	113	8	no	—
171	C₄	—	113	8	no	—
172	C₄	—	226	4	no	—
173	C₄	—	226	4	no	—
174	C₄	C₄²×D₁₁₃	1	4	yes	—
175	C₄	#16813587	1	4	yes	direct product factor
176	C₂	C₂×C₄²×D₁₁₃	1	1	yes	characteristic
177	C₂	C₄×Q₈×D₁₁₃	1	2	yes	—
178	C₂	C₂×#16813587	1	1	yes	characteristic
179	C₂	C₄×#14268646	1	2	yes	—
180	C₂	#20299066	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₁₃
commutator	C₂×C₁₁₃	C₂²×C₄²
Frattini	C₂³	C₂³×D₁₁₃
Fitting	C₄×C₄⋊C₄×C₁₁₃	C₂
socle, prime radical, cosocle	C₂³×C₁₁₃	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₁₁₃
hypocenter	C₁₁₃	C₄²×Q₈
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₂₂₆	C₄	semidirect
11. #16813587	C₄	direct (N × H)
42. C₄×Dic₁₁₃	Q₈	semidirect
71. Dic₂₂₆	C₄²	semidirect
80. C₄×C₁₁₃	C₄×Q₈	semidirect
84. Dic₁₁₃	C₄×Q₈	semidirect
91. C₁₁₃	C₄²×Q₈	semidirect
175. C₄	#16813587	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×C₄×Dic₂₂₆	2
C₂×#16813587	2
C₄⋊C₄×Dic₁₁₃	2
C₄×#14268626	2
#20298449	2
C₄²×Dic₁₁₃	2
C₄×C₄⋊C₄×C₁₁₃	2
C₄²×Q₈	113

Subgroups — C4×#16813587 #28392923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16813587 #28392923