Subgroups — #15676309

Order 2944.

568 conjugacy classes of subgroups.

The subgroup lattice has 200 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	#13092313	C₂	1	2	yes	maximal
4	#13092312	C₂	1	4	yes	maximal
5	#13092286	C₂	1	2	yes	maximal
6	C₄⋊C₄×Dic₂₃	C₂	1	2	yes	maximal
7	C₄×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#384	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×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₃⋊C₄	C₂²	1	4	yes	—
18	C₂×Dic₂₃⋊C₄	C₂²	1	4	yes	—
19	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
20	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
21	C₂×C₄×Dic₂₃	C₂²	1	4	yes	—
22	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
23	#10625058	—	2	8	no	—
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	4	yes	—
28	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
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	D₄	1	4	yes	—
37	C₂×Dic₄₆	Q₈	1	4	yes	—
38	C₄⋊C₄×C₂₃	—	2	4	no	—
39	C₄⋊C₄×C₂₃	—	2	4	no	—
40	C₄⋊C₄×C₂₃	—	4	2	no	—
41	C₄²×C₂₃	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₂₃	—	2	2	no	—
43	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₂₃	Q₈	1	4	yes	complemented
45	Dic₂₃⋊C₄	—	2	8	no	—
46	Dic₂₃⋊C₄	—	4	4	no	—
47	Dic₂₃⋊C₄	C₂×C₄	1	8	yes	—
48	C₄×Dic₂₃	—	2	4	no	—
49	C₄×Dic₂₃	—	2	8	no	—
50	C₄×Dic₂₃	C₂×C₄	1	8	yes	—
51	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₂₃	—	2	1	no	—
53	C₂×C₄×C₂₃	—	2	1	no	—
54	C₂×C₄×C₂₃	—	2	4	no	—
55	C₂×C₄×C₂₃	—	2	4	no	—
56	C₂×C₄×C₂₃	—	2	4	no	—
57	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₂₃	#41	1	2	yes	—
64	C₂×Dic₂₃	—	2	2	no	—
65	C₂×Dic₂₃	—	2	2	no	—
66	C₂×Dic₂₃	—	2	2	no	—
67	C₂×Dic₂₃	—	2	2	no	—
68	C₂×Dic₂₃	—	2	2	no	—
69	C₂×Dic₂₃	—	2	2	no	—
70	C₂×Dic₂₃	—	2	2	no	—
71	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
72	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
73	C₂×Dic₂₃	C₂×D₄	1	4	yes	—
74	C₂×Dic₂₃	C₂×Q₈	1	4	yes	—
75	C₂×Dic₂₃	C₂×Q₈	1	4	yes	—
76	Dic₄₆	C₄⋊C₄	1	16	yes	complemented
77	Q₈×C₄⋊C₄	—	23	1	no	maximal, Sylow
78	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
79	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
82	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
83	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
84	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
85	C₄×C₂₃	—	2	2	no	—
86	C₄×C₂₃	—	4	4	no	—
87	C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
88	C₄×C₂₃	C₄×Q₈	1	4	yes	complemented
89	Dic₂₃	—	2	4	no	—
90	Dic₂₃	—	2	4	no	—
91	Dic₂₃	—	4	2	no	—
92	Dic₂₃	C₂×C₄⋊C₄	1	8	yes	—
93	C₂×C₄×Q₈	—	23	2	no	—
94	C₂×C₄×Q₈	—	23	1	no	—
95	#389	—	23	4	no	—
96	#389	—	23	2	no	—
97	#384	—	23	2	no	—
98	#384	D₂₃	1	1	yes	characteristic
99	C₄×C₄⋊C₄	—	23	2	no	—
100	C₄×C₄⋊C₄	—	23	1	no	—
101	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
103	C₂×C₂₃	#520	1	1	yes	characteristic
104	C₂×C₂₃	D₄×Q₈	1	1	yes	characteristic
105	C₂×C₂₃	#550	1	1	yes	characteristic
106	C₂×C₂₃	#557	1	1	yes	characteristic
107	C₂×C₂₃	Q₈²	1	1	yes	characteristic
108	C₂²×Q₈	—	23	1	no	—
109	C₄×Q₈	—	23	8	no	—
110	C₄×Q₈	—	46	8	no	—
111	C₂×C₄⋊C₄	—	23	2	no	—
112	C₂×C₄⋊C₄	—	23	1	no	—
113	C₂×C₄⋊C₄	—	23	1	no	—
114	C₂×C₄⋊C₄	—	23	4	no	—
115	C₂×C₄⋊C₄	—	23	1	no	—
116	C₂×C₄⋊C₄	—	23	4	no	—
117	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
118	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
119	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
120	C₂×C₄²	—	23	2	no	—
121	C₂×C₄²	—	23	4	no	—
122	C₂×C₄²	—	23	2	no	—
123	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
124	#95	—	23	4	no	—
125	#95	—	23	2	no	—
126	C₂₃	Q₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
127	C₂×Q₈	—	23	4	no	—
128	C₂×Q₈	—	23	4	no	—
129	C₂×Q₈	—	23	4	no	—
130	C₂²×C₄	—	23	2	no	—
131	C₂²×C₄	—	23	2	no	—
132	C₂²×C₄	—	23	2	no	—
133	C₂²×C₄	—	23	2	no	—
134	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
135	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	23	8	no	—
142	C₄⋊C₄	—	23	4	no	—
143	C₄⋊C₄	—	23	4	no	—
144	C₄⋊C₄	—	46	8	no	—
145	C₄⋊C₄	—	46	2	no	—
146	C₄⋊C₄	—	92	4	no	—
147	C₄²	—	23	8	no	—
148	C₄²	—	46	8	no	—
149	C₄²	—	46	4	no	—
150	C₄²	C₄×D₂₃	1	4	yes	—
151	C₂³	C₂³×D₂₃	1	1	yes	characteristic
152	Q₈	—	23	16	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	23	4	no	—
159	C₂×C₄	—	23	4	no	—
160	C₂×C₄	—	23	4	no	—
161	C₂×C₄	—	23	4	no	—
162	C₂×C₄	—	23	4	no	—
163	C₂×C₄	—	46	2	no	—
164	C₂×C₄	—	46	2	no	—
165	C₂×C₄	—	46	2	no	—
166	C₂×C₄	—	46	2	no	—
167	C₂×C₄	—	46	2	no	—
168	C₂×C₄	—	46	2	no	—
169	C₂×C₄	—	46	2	no	—
170	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
171	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
172	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
173	C₂×C₄	D₄×D₂₃	1	2	yes	—
174	C₂×C₄	#67290	1	2	yes	—
175	C₂×C₄	Q₈×D₂₃	1	2	yes	—
176	C₂×C₄	Q₈×D₂₃	1	2	yes	—
177	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
178	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
179	C₂²	C₂×#67290	1	1	yes	characteristic
180	C₂²	#10625155	1	1	yes	characteristic
181	C₂²	C₂×Q₈×D₂₃	1	1	yes	characteristic
182	C₂²	C₂×Q₈×D₂₃	1	1	yes	characteristic
183	C₂²	#10625158	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	4	no	—
186	C₄	—	23	8	no	—
187	C₄	—	46	4	no	—
188	C₄	—	46	4	no	—
189	C₄	—	92	2	no	—
190	C₄	#10625058	1	4	yes	complemented
191	C₄	C₄⋊C₄×D₂₃	1	4	yes	—
192	C₂	C₂×C₄⋊C₄×D₂₃	1	1	yes	characteristic
193	C₂	C₂×#10625058	1	1	yes	characteristic
194	C₂	#13092898	1	1	yes	characteristic
195	C₂	#13093082	1	1	yes	characteristic
196	C₂	#13093087	1	1	yes	characteristic
197	C₂	#13093142	1	1	yes	characteristic
198	C₂	#13093147	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#384	C₂
socle, prime radical, cosocle	C₂³×C₂₃	C₂⁴
hypercenter	#384	D₂₃
hypocenter	C₂₃	Q₈×C₄⋊C₄
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
24. C₄×Dic₄₆	C₄	semidirect
44. C₄⋊Dic₂₃	Q₈	semidirect
76. Dic₄₆	C₄⋊C₄	semidirect
88. C₄×C₂₃	C₄×Q₈	semidirect
126. C₂₃	Q₈×C₄⋊C₄	semidirect
190. C₄	#10625058	semidirect

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
#13092313	2
#13092312	2
#13092286	2
C₄⋊C₄×Dic₂₃	2
C₄×C₄⋊Dic₂₃	2
C₂₃×#384	2
Q₈×C₄⋊C₄	23