Subgroups — C₂×C₂₃×#400 #15670013

Order 2944.

604 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#400	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×#95	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×#95	C₂	1	2	yes	maximal
6	C₂×C₂₃×#95	C₂	1	2	yes	maximal
7	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
8	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
10	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
13	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
16	C₂₃×#95	C₂²	1	8	yes	—
17	C₂₃×#95	C₂²	1	8	yes	—
18	C₂₃×#95	C₂²	1	4	yes	—
19	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	—	2	2	no	—
21	C₂²×C₄×C₂₃	—	2	2	no	—
22	C₂²×C₄×C₂₃	—	2	4	no	—
23	C₂²×C₄×C₂₃	—	2	2	no	—
24	C₂²×C₄×C₂₃	—	2	4	no	—
25	C₂²×C₄×C₂₃	—	2	4	no	—
26	C₂²×C₄×C₂₃	—	2	8	no	—
27	C₂²×C₄×C₂₃	—	2	4	no	—
28	C₂²×C₄×C₂₃	—	2	4	no	—
29	C₂²×C₄×C₂₃	—	2	4	no	—
30	C₂²×C₄×C₂₃	D₄	1	4	yes	—
31	C₂²×C₄×C₂₃	Q₈	1	4	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
34	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
35	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
36	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
37	C₄⋊C₄×C₂₃	—	2	8	no	—
38	C₄⋊C₄×C₂₃	—	2	8	no	—
39	C₂³×C₂₃	C₂×D₄	1	2	yes	—
40	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
41	C₂³×C₂₃	#41	1	2	yes	—
42	C₂³×C₂₃	#41	1	4	yes	—
43	C₂³×C₂₃	#41	1	4	yes	—
44	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
45	C₂×C₄×C₂₃	—	2	8	no	—
46	C₂×C₄×C₂₃	—	2	8	no	—
47	C₂×C₄×C₂₃	—	2	2	no	—
48	C₂×C₄×C₂₃	—	2	2	no	—
49	C₂×C₄×C₂₃	—	2	4	no	—
50	C₂×C₄×C₂₃	—	2	2	no	—
51	C₂×C₄×C₂₃	—	2	4	no	—
52	C₂×C₄×C₂₃	—	2	4	no	—
53	C₂×C₄×C₂₃	—	2	8	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	—	4	4	no	—
58	C₂×C₄×C₂₃	—	4	8	no	—
59	C₂×C₄×C₂₃	—	4	8	no	—
60	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
61	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
62	C₂×#400	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂²×C₂₃	C₂²⋊Q₈	1	4	yes	—
64	C₂²×C₂₃	#123	1	4	yes	—
65	C₂²×C₂₃	#124	1	4	yes	—
66	C₂²×C₂₃	#125	1	8	yes	—
67	C₂²×C₂₃	#126	1	8	yes	—
68	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
69	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
70	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
71	C₂²×C₂₃	C₂×#41	1	2	yes	—
72	C₂²×C₂₃	C₂×#41	1	2	yes	—
73	C₄×C₂₃	—	2	4	no	—
74	C₄×C₂₃	—	2	4	no	—
75	C₄×C₂₃	—	4	2	no	—
76	C₄×C₂₃	—	4	4	no	—
77	C₄×C₂₃	—	4	4	no	—
78	C₂²×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
79	C₂²×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
80	#400	C₂×C₂₃	1	8	yes	direct product factor
81	C₂×#95	C₂×C₂₃	1	2	yes	—
82	C₂×#95	C₂×C₂₃	1	2	yes	—
83	C₂×#95	C₂×C₂₃	1	1	yes	characteristic
84	C₂×C₂₃	#400	1	8	yes	direct product factor
85	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₂₃	C₂×#123	1	1	yes	characteristic
87	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
88	C₂×C₂₃	C₂×#125	1	2	yes	—
89	C₂×C₂₃	C₂×#126	1	2	yes	—
90	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
91	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
92	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
93	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
94	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
95	C₂×C₄⋊C₄	—	2	4	no	—
96	C₂×C₄⋊C₄	—	2	4	no	—
97	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
98	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
99	#95	C₂²×C₂₃	1	4	yes	—
100	#95	C₂²×C₂₃	1	8	yes	—
101	#95	C₂²×C₂₃	1	8	yes	—
102	C₂₃	C₂×#400	1	1	yes	characteristic, direct product factor, Sylow
103	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	—	2	4	no	—
109	C₂²×C₄	—	2	4	no	—
110	C₂²×C₄	—	2	8	no	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	—	2	4	no	—
114	C₂²×C₄	D₄×C₂₃	1	4	yes	—
115	C₂²×C₄	Q₈×C₂₃	1	4	yes	—
116	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
117	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
118	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
119	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
120	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
121	C₄⋊C₄	—	2	8	no	—
122	C₄⋊C₄	—	2	8	no	—
123	C₂³	C₂×D₄×C₂₃	1	2	yes	—
124	C₂³	C₂×Q₈×C₂₃	1	2	yes	—
125	C₂³	#41×C₂₃	1	2	yes	—
126	C₂³	#41×C₂₃	1	4	yes	—
127	C₂³	#41×C₂₃	1	4	yes	—
128	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
129	C₂×C₄	—	2	8	no	—
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	4	4	no	—
142	C₂×C₄	—	4	8	no	—
143	C₂×C₄	—	4	8	no	—
144	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
145	C₂×C₄	C₂×Q₈×C₂₃	1	4	yes	—
146	C₂²	C₂₃×C₂²⋊Q₈	1	4	yes	—
147	C₂²	C₂₃×#123	1	4	yes	—
148	C₂²	C₂₃×#124	1	4	yes	—
149	C₂²	C₂₃×#125	1	8	yes	—
150	C₂²	C₂₃×#126	1	8	yes	—
151	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
152	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
153	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
154	C₂²	C₂×#41×C₂₃	1	2	yes	—
155	C₂²	C₂×#41×C₂₃	1	2	yes	—
156	C₄	—	2	4	no	—
157	C₄	—	2	4	no	—
158	C₄	—	4	2	no	—
159	C₄	—	4	4	no	—
160	C₄	—	4	4	no	—
161	C₂	C₂₃×#400	1	8	yes	direct product factor
162	C₂	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
163	C₂	C₂×C₂₃×#123	1	1	yes	characteristic
164	C₂	C₂×C₂₃×#124	1	1	yes	characteristic
165	C₂	C₂×C₂₃×#125	1	2	yes	—
166	C₂	C₂×C₂₃×#126	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#400	C₂	direct (N × H)
62. C₂×#400	C₂₃	direct (N × H)
80. #400	C₂×C₂₃	direct (N × H)
84. C₂×C₂₃	#400	direct (N × H)
102. C₂₃	C₂×#400	direct (N × H)
161. C₂	C₂₃×#400	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₄×C₂₃	2
C₂₃×#400	2
C₂×C₂₃×#95	2
C₂×#400	23

Subgroups — C2×C23×#400 #15670013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#400 #15670013