Subgroups — C₄×#10625058 #15675746

Order 2944.

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×Dic₄₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625058	C₂	1	2	yes	maximal
3	C₄⋊C₄×Dic₂₃	C₂	1	2	yes	maximal
4	C₄×Dic₂₃⋊C₄	C₂	1	2	yes	maximal
5	#13092268	C₂	1	4	yes	maximal
6	C₄²×Dic₂₃	C₂	1	1	yes	maximal, characteristic
7	C₄²×Dic₂₃	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₃⋊C₄	C₂²	1	4	yes	—
15	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
16	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
17	C₂×C₄×Dic₂₃	C₂²	1	4	yes	—
18	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₂₃	C₂²	1	4	yes	—
20	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
21	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
22	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
23	#10625058	C₄	1	16	yes	direct product factor
24	C₄×Dic₄₆	C₄	1	8	yes	complemented
25	C₂₃×#95	C₂²	1	2	yes	—
26	C₂₃⋊#95	C₂²	1	4	yes	—
27	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
29	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₂₃	C₂³	1	2	yes	—
32	C₂²×Dic₂₃	C₂³	1	2	yes	—
33	C₂²×Dic₂₃	C₂³	1	2	yes	—
34	C₂²×Dic₂₃	C₂³	1	2	yes	—
35	C₂×Dic₄₆	C₂×C₄	1	4	yes	—
36	C₂×Dic₄₆	C₂×C₄	1	8	yes	—
37	C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
38	C₄²×C₂₃	—	2	4	no	—
39	C₄²×C₂₃	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
41	Dic₂₃⋊C₄	C₂×C₄	1	8	yes	—
42	Dic₂₃⋊C₄	C₂×C₄	1	16	yes	—
43	C₄×Dic₂₃	—	2	2	no	—
44	C₄×Dic₂₃	—	2	4	no	—
45	C₄×Dic₂₃	—	2	8	no	—
46	C₄×Dic₂₃	C₂×C₄	1	8	yes	—
47	C₄×Dic₂₃	C₂×C₄	1	8	yes	—
48	C₄×Dic₂₃	C₂×C₄	1	16	yes	—
49	C₄×Dic₂₃	Q₈	1	4	yes	complemented
50	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
51	C₂×C₄×C₂₃	—	2	2	no	—
52	C₂×C₄×C₂₃	—	2	4	no	—
53	C₂×C₄×C₂₃	—	2	4	no	—
54	C₂×C₄×C₂₃	C₂²×C₄	1	8	yes	—
55	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₂₃	#41	1	2	yes	—
61	C₂×Dic₂₃	—	2	2	no	—
62	C₂×Dic₂₃	—	2	2	no	—
63	C₂×Dic₂₃	—	2	4	no	—
64	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
65	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
66	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
67	C₂×Dic₂₃	C₂²×C₄	1	8	yes	—
68	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
69	C₂×Dic₂₃	C₂×Q₈	1	4	yes	—
70	C₂×Dic₂₃	#41	1	4	yes	—
71	Dic₄₆	C₄²	1	16	yes	complemented
72	C₄²×Q₈	—	23	1	no	maximal, Sylow
73	C₂²×C₂₃	C₂³×C₄	1	2	yes	—
74	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
76	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
77	C₂²×C₂₃	C₂×#41	1	2	yes	—
78	C₄×C₂₃	—	2	8	no	—
79	C₄×C₂₃	—	2	2	no	—
80	C₄×C₂₃	C₂×C₄²	1	4	yes	—
81	C₄×C₂₃	C₄×Q₈	1	4	yes	complemented
82	Dic₂₃	—	2	4	no	—
83	Dic₂₃	—	2	4	no	—
84	Dic₂₃	C₂×C₄²	1	8	yes	—
85	Dic₂₃	C₄×Q₈	1	8	yes	complemented
86	C₂×C₄×Q₈	—	23	2	no	—
87	C₂×C₄×Q₈	—	23	1	no	—
88	C₄×C₄⋊C₄	—	23	2	no	—
89	C₄×C₄⋊C₄	—	23	2	no	—
90	C₄×C₄⋊C₄	—	23	4	no	—
91	C₄×C₄⋊C₄	D₂₃	1	1	yes	characteristic
92	C₄³	—	23	2	no	—
93	C₄³	—	23	1	no	—
94	C₂×C₂₃	C₂²×C₄²	1	1	yes	characteristic
95	C₂×C₂₃	C₂×C₄×Q₈	1	2	yes	—
96	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
97	C₂×C₂₃	C₄×#41	1	2	yes	—
98	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
99	C₂²×Q₈	—	23	1	no	—
100	C₄×Q₈	—	23	16	no	—
101	C₄×Q₈	—	23	8	no	—
102	C₂×C₄⋊C₄	—	23	4	no	—
103	C₂×C₄⋊C₄	—	23	2	no	—
104	C₂×C₄⋊C₄	—	23	1	no	—
105	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
106	C₂×C₄²	—	23	4	no	—
107	C₂×C₄²	—	23	2	no	—
108	C₂×C₄²	—	23	2	no	—
109	C₂×C₄²	—	23	1	no	—
110	C₂×C₄²	—	23	4	no	—
111	C₂×C₄²	—	23	2	no	—
112	C₂×C₄²	—	23	1	no	—
113	C₂×C₄²	C₂×D₂₃	1	2	yes	—
114	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
115	#95	—	23	4	no	—
116	#95	C₂×D₂₃	1	2	yes	—
117	C₂₃	C₄²×Q₈	1	1	yes	characteristic, complemented, Sylow
118	C₂×Q₈	—	23	8	no	—
119	C₂×Q₈	—	23	4	no	—
120	C₂²×C₄	—	23	2	no	—
121	C₂²×C₄	—	23	2	no	—
122	C₂²×C₄	—	23	2	no	—
123	C₂²×C₄	—	23	2	no	—
124	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
125	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
128	C₄⋊C₄	—	23	16	no	—
129	C₄⋊C₄	—	23	8	no	—
130	C₄⋊C₄	—	23	4	no	—
131	C₄⋊C₄	C₄×D₂₃	1	8	yes	—
132	C₄²	—	2	4	no	—
133	C₄²	—	23	16	no	—
134	C₄²	—	23	8	no	—
135	C₄²	—	23	8	no	—
136	C₄²	—	23	4	no	—
137	C₄²	—	46	8	no	—
138	C₄²	—	46	4	no	—
139	C₄²	—	46	2	no	—
140	C₄²	C₄×D₂₃	1	4	yes	—
141	C₂³	C₂³×D₂₃	1	1	yes	characteristic
142	Q₈	—	23	16	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	23	4	no	—
147	C₂×C₄	—	23	8	no	—
148	C₂×C₄	—	23	4	no	—
149	C₂×C₄	—	23	4	no	—
150	C₂×C₄	—	23	4	no	—
151	C₂×C₄	—	23	4	no	—
152	C₂×C₄	—	23	4	no	—
153	C₂×C₄	—	46	2	no	—
154	C₂×C₄	—	46	2	no	—
155	C₂×C₄	—	46	4	no	—
156	C₂×C₄	C₂×C₄×D₂₃	1	8	yes	—
157	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
158	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
159	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
160	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
161	C₂×C₄	#67290	1	2	yes	—
162	C₂×C₄	Q₈×D₂₃	1	2	yes	—
163	C₂²	C₂²×C₄×D₂₃	1	2	yes	—
164	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
165	C₂²	C₂×#67290	1	1	yes	characteristic
166	C₂²	C₂×Q₈×D₂₃	1	1	yes	characteristic
167	C₂²	#41×D₂₃	1	2	yes	—
168	C₄	—	2	8	no	—
169	C₄	—	2	2	no	—
170	C₄	—	23	8	no	—
171	C₄	—	23	8	no	—
172	C₄	—	46	4	no	—
173	C₄	—	46	4	no	—
174	C₄	C₄²×D₂₃	1	4	yes	—
175	C₄	#10625058	1	4	yes	direct product factor
176	C₂	C₂×C₄²×D₂₃	1	1	yes	characteristic
177	C₂	C₄×Q₈×D₂₃	1	2	yes	—
178	C₂	C₂×#10625058	1	1	yes	characteristic
179	C₂	C₄×#67290	1	2	yes	—
180	C₂	#13092885	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₃
commutator	C₂×C₂₃	C₂²×C₄²
Frattini	C₂³	C₂³×D₂₃
Fitting	C₄×C₄⋊C₄×C₂₃	C₂
socle, prime radical, cosocle	C₂³×C₂₃	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₂₃
hypocenter	C₂₃	C₄²×Q₈
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
23. #10625058	C₄	direct (N × H)
24. C₄×Dic₄₆	C₄	semidirect
49. C₄×Dic₂₃	Q₈	semidirect
71. Dic₄₆	C₄²	semidirect
81. C₄×C₂₃	C₄×Q₈	semidirect
85. Dic₂₃	C₄×Q₈	semidirect
117. C₂₃	C₄²×Q₈	semidirect
175. C₄	#10625058	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×Dic₄₆	2
C₂×#10625058	2
C₄⋊C₄×Dic₂₃	2
C₄×Dic₂₃⋊C₄	2
#13092268	2
C₄²×Dic₂₃	2
C₄×C₄⋊C₄×C₂₃	2
C₄²×Q₈	23

Subgroups — C4×#10625058 #15675746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#10625058 #15675746