Subgroups — #15676352

Order 2944.

480 conjugacy classes of subgroups.

The subgroup lattice has 194 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	#13092323	C₂	1	2	yes	maximal
3	#13092304	C₂	1	4	yes	maximal
4	#13092287	C₂	1	1	yes	maximal, characteristic
5	#13092282	C₂	1	2	yes	maximal
6	C₄⋊C₄×Dic₂₃	C₂	1	2	yes	maximal
7	C₄×C₄⋊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	1	yes	characteristic
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
19	C₂×C₄×Dic₂₃	C₂²	1	4	yes	—
20	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
21	C₄×Dic₄₆	C₄	1	8	yes	complemented
22	C₂₃×#95	C₂²	1	2	yes	—
23	C₂₃⋊#95	C₂²	1	4	yes	—
24	C₂₃⋊#95	C₂²	1	4	yes	—
25	C₂₃⋊#95	C₂²	1	4	yes	—
26	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₂₃	C₂³	1	4	yes	—
32	C₂²×Dic₂₃	C₂³	1	2	yes	—
33	C₂²×Dic₂₃	C₂³	1	2	yes	—
34	C₂×Dic₄₆	—	2	2	no	—
35	C₂×Dic₄₆	—	2	2	no	—
36	C₂×Dic₄₆	C₂×C₄	1	4	yes	—
37	C₄⋊C₄×C₂₃	—	2	2	no	—
38	C₄⋊C₄×C₂₃	Q₈	1	4	yes	complemented
39	C₄²×C₂₃	—	2	4	no	—
40	C₄²×C₂₃	C₂×C₄	1	4	yes	—
41	C₄⋊Dic₂₃	—	2	4	no	—
42	C₄⋊Dic₂₃	—	2	4	no	—
43	C₄⋊Dic₂₃	—	4	2	no	—
44	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
45	Dic₂₃⋊C₄	C₂×C₄	1	8	yes	—
46	C₄×Dic₂₃	—	4	4	no	—
47	C₄×Dic₂₃	C₂×C₄	1	8	yes	—
48	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
49	C₂×C₄×C₂₃	—	2	2	no	—
50	C₂×C₄×C₂₃	—	2	2	no	—
51	C₂×C₄×C₂₃	—	2	1	no	—
52	C₂×C₄×C₂₃	—	2	1	no	—
53	C₂×C₄×C₂₃	—	2	2	no	—
54	C₂×C₄×C₂₃	—	2	2	no	—
55	C₂×C₄×C₂₃	—	2	2	no	—
56	C₂×C₄×C₂₃	C₂²×C₄	1	2	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	4	yes	—
60	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₂₃	#41	1	2	yes	—
62	C₂×C₄×C₂₃	#41	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	4	no	—
69	C₂×Dic₂₃	—	2	4	no	—
70	C₂×Dic₂₃	—	2	4	no	—
71	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
72	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
73	Dic₄₆	—	2	8	no	—
74	#2355	—	23	1	no	maximal, Sylow
75	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
76	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
77	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
78	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
79	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
80	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
81	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
82	C₄×C₂₃	—	2	2	no	—
83	C₄×C₂₃	—	2	4	no	—
84	C₄×C₂₃	—	4	2	no	—
85	C₄×C₂₃	#117	1	4	yes	—
86	C₄×C₂₃	C₄×Q₈	1	4	yes	—
87	Dic₂₃	—	2	4	no	—
88	Dic₂₃	—	2	4	no	—
89	Dic₂₃	—	4	4	no	—
90	C₂×C₄×Q₈	—	23	1	no	—
91	#391	—	23	2	no	—
92	#387	—	23	4	no	—
93	#384	—	23	1	no	—
94	#382	—	23	2	no	—
95	C₄×C₄⋊C₄	—	23	2	no	—
96	C₄×C₄⋊C₄	—	23	2	no	—
97	C₄×C₄⋊C₄	D₂₃	1	1	yes	characteristic
98	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
99	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
100	C₂×C₂₃	#520	1	1	yes	characteristic
101	C₂×C₂₃	#554	1	1	yes	characteristic
102	C₂×C₂₃	#556	1	1	yes	characteristic
103	C₂×C₂₃	#557	1	1	yes	characteristic
104	C₂×C₂₃	#559	1	1	yes	characteristic
105	C₂²×Q₈	—	23	1	no	—
106	C₄×Q₈	—	23	8	no	—
107	C₂×C₄⋊C₄	—	23	2	no	—
108	C₂×C₄⋊C₄	—	23	1	no	—
109	C₂×C₄⋊C₄	—	23	2	no	—
110	C₂×C₄⋊C₄	—	23	2	no	—
111	C₂×C₄⋊C₄	—	23	2	no	—
112	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
113	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
114	C₂×C₄²	—	23	2	no	—
115	C₂×C₄²	—	23	4	no	—
116	C₂×C₄²	C₂×D₂₃	1	2	yes	—
117	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
118	#95	—	23	4	no	—
119	#95	—	23	4	no	—
120	#95	—	23	4	no	—
121	#95	C₂×D₂₃	1	2	yes	—
122	C₂₃	#2355	1	1	yes	characteristic, complemented, Sylow
123	C₂×Q₈	—	23	4	no	—
124	C₂×Q₈	—	46	2	no	—
125	C₂×Q₈	—	46	2	no	—
126	C₂²×C₄	—	23	2	no	—
127	C₂²×C₄	—	23	2	no	—
128	C₂²×C₄	—	23	4	no	—
129	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
130	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
131	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
132	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	23	8	no	—
136	C₄⋊C₄	—	23	4	no	—
137	C₄⋊C₄	—	46	4	no	—
138	C₄⋊C₄	—	46	4	no	—
139	C₄⋊C₄	—	92	2	no	—
140	C₄⋊C₄	Dic₄₆	1	4	yes	complemented
141	C₄²	—	2	4	no	—
142	C₄²	—	23	8	no	—
143	C₄²	—	92	4	no	—
144	C₄²	C₄×D₂₃	1	4	yes	—
145	C₂³	C₂³×D₂₃	1	1	yes	characteristic
146	Q₈	—	46	8	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	23	4	no	—
155	C₂×C₄	—	23	4	no	—
156	C₂×C₄	—	46	2	no	—
157	C₂×C₄	—	46	2	no	—
158	C₂×C₄	—	46	2	no	—
159	C₂×C₄	—	46	2	no	—
160	C₂×C₄	—	46	4	no	—
161	C₂×C₄	—	46	4	no	—
162	C₂×C₄	—	46	4	no	—
163	C₂×C₄	C₂×Dic₄₆	1	4	yes	—
164	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
165	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
166	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
167	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
168	C₂×C₄	#67288	1	2	yes	—
169	C₂×C₄	#67290	1	2	yes	—
170	C₂×C₄	#67292	1	2	yes	—
171	C₂²	C₂²×Dic₄₆	1	1	yes	characteristic
172	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
173	C₂²	C₂×#67288	1	1	yes	characteristic
174	C₂²	C₂×#67290	1	1	yes	characteristic
175	C₂²	C₂×#67292	1	1	yes	characteristic
176	C₂²	#10625160	1	1	yes	characteristic
177	C₂²	#10625161	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	2	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	46	4	no	—
182	C₄	—	46	4	no	—
183	C₄	—	92	4	no	—
184	C₄	C₄×Dic₄₆	1	4	yes	—
185	C₄	#10625063	1	4	yes	—
186	C₂	C₂×#10625063	1	1	yes	characteristic
187	C₂	C₂×C₄×Dic₄₆	1	1	yes	characteristic
188	C₂	#13092897	1	1	yes	characteristic
189	C₂	#13093124	1	1	yes	characteristic
190	C₂	#13093137	1	1	yes	characteristic
191	C₂	#13093146	1	1	yes	characteristic
192	C₂	#13093152	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₂₃	C₂
socle, prime radical, cosocle	C₂³×C₂₃	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₂₃
hypocenter	C₂₃	#2355
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
21. C₄×Dic₄₆	C₄	semidirect
38. C₄⋊C₄×C₂₃	Q₈	semidirect
122. C₂₃	#2355	semidirect
140. C₄⋊C₄	Dic₄₆	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
#13092323	2
#13092304	2
#13092287	2
#13092282	2
C₄⋊C₄×Dic₂₃	2
C₄×C₄⋊Dic₂₃	2
C₄×C₄⋊C₄×C₂₃	2
#2355	23