Subgroups — C₂³×C₂₃×#103 #19207704

Order 5888.

5502 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	C₂⁴×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₂₃×#103	C₂	1	28	yes	maximal, direct product factor
5	C₂³×Q₈×C₂₃	—	2	8	no	—
6	C₂³×Q₈×C₂₃	—	2	7	no	—
7	C₂³×Q₈×C₂₃	C₄	1	16	yes	complemented
8	C₂³×Q₈×C₂₃	C₂²	1	14	yes	—
9	C₂⁴×C₄×C₂₃	—	2	1	no	—
10	C₂⁴×C₄×C₂₃	—	2	1	no	—
11	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₂₃	—	2	8	no	—
13	C₂³×C₈×C₂₃	C₂²	1	14	yes	—
14	C₂²×C₄⋊C₄×C₂₃	—	2	7	no	—
15	C₂²×C₄⋊C₄×C₂₃	C₂²	1	14	yes	—
16	C₂×C₂₃×#103	C₂²	1	112	yes	direct product factor
17	C₂⁵×C₂₃	D₄	1	1	yes	characteristic
18	C₂²×Q₈×C₂₃	—	2	112	no	—
19	C₂²×Q₈×C₂₃	—	2	56	no	—
20	C₂²×Q₈×C₂₃	—	2	42	no	—
21	C₂²×Q₈×C₂₃	C₂×C₄	1	112	yes	complemented
22	C₂²×Q₈×C₂₃	C₂³	1	28	yes	—
23	C₂³×C₄×C₂₃	—	2	16	no	—
24	C₂³×C₄×C₂₃	—	2	14	no	—
25	C₂³×C₄×C₂₃	—	2	14	no	—
26	C₂³×C₄×C₂₃	—	4	8	no	—
27	C₂³×C₄×C₂₃	C₂×C₄	1	8	yes	—
28	C₂³×C₄×C₂₃	D₄	1	7	yes	—
29	C₂³×C₄×C₂₃	D₄	1	8	yes	—
30	C₂³×C₄×C₂₃	C₂³	1	7	yes	—
31	C₂²×C₈×C₂₃	—	2	56	no	—
32	C₂²×C₈×C₂₃	C₂³	1	28	yes	—
33	C₂×C₄⋊C₄×C₂₃	—	2	42	no	—
34	C₂×C₄⋊C₄×C₂₃	C₂³	1	28	yes	—
35	C₂₃×#103	C₂³	1	64	yes	direct product factor
36	C₂⁴×C₂₃	C₂²⋊C₄	1	8	yes	—
37	C₂⁴×C₂₃	SD₁₆	1	8	yes	—
38	C₂⁴×C₂₃	Q₁₆	1	8	yes	—
39	C₂⁴×C₂₃	C₂×D₄	1	7	yes	—
40	C₂×Q₈×C₂₃	—	2	336	no	—
41	C₂×Q₈×C₂₃	—	2	56	no	—
42	C₂×Q₈×C₂₃	—	2	28	no	—
43	C₂×Q₈×C₂₃	C₂²×C₄	1	112	yes	complemented
44	C₂×Q₈×C₂₃	C₂⁴	1	8	yes	—
45	C₂²×C₄×C₂₃	—	2	112	no	—
46	C₂²×C₄×C₂₃	—	2	28	no	—
47	C₂²×C₄×C₂₃	—	2	28	no	—
48	C₂²×C₄×C₂₃	—	4	56	no	—
49	C₂²×C₄×C₂₃	C₂²⋊C₄	1	56	yes	—
50	C₂²×C₄×C₂₃	C₂²×C₄	1	28	yes	—
51	C₂²×C₄×C₂₃	C₂×D₄	1	21	yes	—
52	C₂²×C₄×C₂₃	C₂×D₄	1	28	yes	—
53	C₂²×C₄×C₂₃	C₂⁴	1	7	yes	—
54	C₂×C₈×C₂₃	—	2	56	no	—
55	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
56	C₄⋊C₄×C₂₃	—	2	28	no	—
57	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
58	C₂³×#103	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₂₃	#103	1	64	yes	direct product factor
60	C₂³×C₂₃	C₂×C₂²⋊C₄	1	28	yes	—
61	C₂³×C₂₃	C₂×SD₁₆	1	28	yes	—
62	C₂³×C₂₃	C₂×Q₁₆	1	28	yes	—
63	C₂³×C₂₃	C₂²×D₄	1	7	yes	—
64	Q₈×C₂₃	—	2	112	no	—
65	Q₈×C₂₃	—	2	8	no	—
66	Q₈×C₂₃	C₂³×C₄	1	16	yes	complemented
67	C₂×C₄×C₂₃	—	2	112	no	—
68	C₂×C₄×C₂₃	—	2	8	no	—
69	C₂×C₄×C₂₃	—	2	8	no	—
70	C₂×C₄×C₂₃	—	4	56	no	—
71	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	84	yes	—
72	C₂×C₄×C₂₃	C₂³×C₄	1	14	yes	—
73	C₂×C₄×C₂₃	C₂²×D₄	1	7	yes	—
74	C₂×C₄×C₂₃	C₂²×D₄	1	14	yes	—
75	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
76	C₈×C₂₃	—	2	8	no	—
77	C₂⁴×Q₈	C₂×C₂₃	1	1	yes	characteristic
78	C₂⁴×C₈	C₂×C₂₃	1	1	yes	characteristic
79	C₂³×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
80	C₂²×#103	C₂×C₂₃	1	28	yes	direct product factor
81	C₂²×C₂₃	C₂×#103	1	112	yes	direct product factor
82	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	14	yes	—
83	C₂²×C₂₃	C₂²×SD₁₆	1	14	yes	—
84	C₂²×C₂₃	C₂²×Q₁₆	1	14	yes	—
85	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
86	C₄×C₂₃	—	2	16	no	—
87	C₄×C₂₃	—	4	8	no	—
88	C₄×C₂₃	C₂²×C₂²⋊C₄	1	14	yes	—
89	C₄×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
90	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
91	C₂³×Q₈	—	2	8	no	—
92	C₂³×Q₈	—	2	7	no	—
93	C₂³×Q₈	C₄×C₂₃	1	16	yes	complemented
94	C₂³×Q₈	C₂²×C₂₃	1	14	yes	—
95	C₂⁴×C₄	—	2	1	no	—
96	C₂⁴×C₄	—	2	1	no	—
97	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
98	C₂³×C₈	—	2	8	no	—
99	C₂³×C₈	C₂²×C₂₃	1	14	yes	—
100	C₂²×C₄⋊C₄	—	2	7	no	—
101	C₂²×C₄⋊C₄	C₂²×C₂₃	1	14	yes	—
102	C₂×#103	C₂²×C₂₃	1	112	yes	direct product factor
103	C₂×C₂₃	C₂²×#103	1	28	yes	direct product factor
104	C₂×C₂₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₂₃	C₂³×SD₁₆	1	1	yes	characteristic
106	C₂×C₂₃	C₂³×Q₁₆	1	1	yes	characteristic
107	C₂⁵	D₄×C₂₃	1	1	yes	characteristic
108	C₂²×Q₈	—	2	112	no	—
109	C₂²×Q₈	—	2	56	no	—
110	C₂²×Q₈	—	2	42	no	—
111	C₂²×Q₈	C₂×C₄×C₂₃	1	112	yes	complemented
112	C₂²×Q₈	C₂³×C₂₃	1	28	yes	—
113	C₂³×C₄	—	2	16	no	—
114	C₂³×C₄	—	2	14	no	—
115	C₂³×C₄	—	2	14	no	—
116	C₂³×C₄	—	4	8	no	—
117	C₂³×C₄	C₂×C₄×C₂₃	1	8	yes	—
118	C₂³×C₄	D₄×C₂₃	1	7	yes	—
119	C₂³×C₄	D₄×C₂₃	1	8	yes	—
120	C₂³×C₄	C₂³×C₂₃	1	7	yes	—
121	C₂²×C₈	—	2	56	no	—
122	C₂²×C₈	C₂³×C₂₃	1	28	yes	—
123	C₂×C₄⋊C₄	—	2	42	no	—
124	C₂×C₄⋊C₄	C₂³×C₂₃	1	28	yes	—
125	#103	C₂³×C₂₃	1	64	yes	direct product factor
126	C₂₃	C₂³×#103	1	1	yes	characteristic, direct product factor, Sylow
127	C₂⁴	C₂²⋊C₄×C₂₃	1	8	yes	—
128	C₂⁴	SD₁₆×C₂₃	1	8	yes	—
129	C₂⁴	Q₁₆×C₂₃	1	8	yes	—
130	C₂⁴	C₂×D₄×C₂₃	1	7	yes	—
131	C₂×Q₈	—	2	336	no	—
132	C₂×Q₈	—	2	28	no	—
133	C₂×Q₈	—	2	56	no	—
134	C₂×Q₈	C₂²×C₄×C₂₃	1	112	yes	complemented
135	C₂×Q₈	C₂⁴×C₂₃	1	8	yes	—
136	C₂²×C₄	—	2	112	no	—
137	C₂²×C₄	—	2	28	no	—
138	C₂²×C₄	—	2	28	no	—
139	C₂²×C₄	—	4	56	no	—
140	C₂²×C₄	C₂²⋊C₄×C₂₃	1	56	yes	—
141	C₂²×C₄	C₂²×C₄×C₂₃	1	28	yes	—
142	C₂²×C₄	C₂×D₄×C₂₃	1	21	yes	—
143	C₂²×C₄	C₂×D₄×C₂₃	1	28	yes	—
144	C₂²×C₄	C₂⁴×C₂₃	1	7	yes	—
145	C₂×C₈	—	2	56	no	—
146	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
147	C₄⋊C₄	—	2	28	no	—
148	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
149	C₂³	C₂₃×#103	1	64	yes	direct product factor
150	C₂³	C₂×C₂²⋊C₄×C₂₃	1	28	yes	—
151	C₂³	C₂×SD₁₆×C₂₃	1	28	yes	—
152	C₂³	C₂×Q₁₆×C₂₃	1	28	yes	—
153	C₂³	C₂²×D₄×C₂₃	1	7	yes	—
154	Q₈	—	2	8	no	—
155	Q₈	—	2	112	no	—
156	Q₈	C₂³×C₄×C₂₃	1	16	yes	complemented
157	C₂×C₄	—	2	8	no	—
158	C₂×C₄	—	2	112	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	4	56	no	—
161	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	84	yes	—
162	C₂×C₄	C₂³×C₄×C₂₃	1	14	yes	—
163	C₂×C₄	C₂²×D₄×C₂₃	1	14	yes	—
164	C₂×C₄	C₂²×D₄×C₂₃	1	7	yes	—
165	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
166	C₈	—	2	8	no	—
167	C₂²	C₂×C₂₃×#103	1	112	yes	direct product factor
168	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	14	yes	—
169	C₂²	C₂²×SD₁₆×C₂₃	1	14	yes	—
170	C₂²	C₂²×Q₁₆×C₂₃	1	14	yes	—
171	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
172	C₄	—	2	16	no	—
173	C₄	—	4	8	no	—
174	C₄	C₂²×C₂²⋊C₄×C₂₃	1	14	yes	—
175	C₄	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
176	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
177	C₂	C₂²×C₂₃×#103	1	28	yes	direct product factor
178	C₂	C₂³×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
179	C₂	C₂³×SD₁₆×C₂₃	1	1	yes	characteristic
180	C₂	C₂³×Q₁₆×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₂₃	D₄
commutator	C₄	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₂₃ ◁ C₂⁴×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂²×C₂₃×#103	C₂	direct (N × H)
7. C₂³×Q₈×C₂₃	C₄	semidirect
16. C₂×C₂₃×#103	C₂²	direct (N × H)
21. C₂²×Q₈×C₂₃	C₂×C₄	semidirect
35. C₂₃×#103	C₂³	direct (N × H)
43. C₂×Q₈×C₂₃	C₂²×C₄	semidirect
58. C₂³×#103	C₂₃	direct (N × H)
59. C₂³×C₂₃	#103	direct (N × H)
66. Q₈×C₂₃	C₂³×C₄	semidirect
80. C₂²×#103	C₂×C₂₃	direct (N × H)
81. C₂²×C₂₃	C₂×#103	direct (N × H)
93. C₂³×Q₈	C₄×C₂₃	semidirect
102. C₂×#103	C₂²×C₂₃	direct (N × H)
103. C₂×C₂₃	C₂²×#103	direct (N × H)
111. C₂²×Q₈	C₂×C₄×C₂₃	semidirect
125. #103	C₂³×C₂₃	direct (N × H)
126. C₂₃	C₂³×#103	direct (N × H)
134. C₂×Q₈	C₂²×C₄×C₂₃	semidirect
149. C₂³	C₂₃×#103	direct (N × H)
156. Q₈	C₂³×C₄×C₂₃	semidirect
167. C₂²	C₂×C₂₃×#103	direct (N × H)
177. C₂	C₂²×C₂₃×#103	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×Q₈×C₂₃	2
C₂⁴×C₈×C₂₃	2
C₂³×C₄⋊C₄×C₂₃	2
C₂²×C₂₃×#103	2
C₂³×#103	23

Subgroups — C23×C23×#103 #19207704

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂₃×#103 #19207704