Subgroups — C₂×C₂₃×#479 #15670694

Order 2944.

384 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#479	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×#107	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×#106	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₂×C₂₃×#103	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₃×#103	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#125	—	2	2	no	—
13	C₂₃×#125	C₂²	1	4	yes	—
14	C₄×Q₈×C₂₃	—	2	2	no	—
15	C₄×Q₈×C₂₃	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
17	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
18	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₂₃	—	2	1	no	—
23	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#107	C₂²	1	4	yes	—
25	C₂₃×#106	C₂²	1	4	yes	—
26	C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
27	C₂₃×#103	C₂²	1	4	yes	—
28	C₂₃×#103	C₂²	1	4	yes	—
29	C₂×Q₈×C₂₃	—	2	1	no	—
30	C₂×Q₈×C₂₃	—	2	2	no	—
31	C₂×Q₈×C₂₃	Q₈	1	4	yes	—
32	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₃	—	2	1	no	—
34	C₂²×C₄×C₂₃	—	2	1	no	—
35	C₂²×C₄×C₂₃	—	2	1	no	—
36	C₂²×C₄×C₂₃	—	2	1	no	—
37	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₂₃	—	2	2	no	—
41	C₂×C₈×C₂₃	—	2	2	no	—
42	C₂×C₈×C₂₃	C₂³	1	2	yes	—
43	C₂×C₈×C₂₃	C₂³	1	2	yes	—
44	C₄⋊C₄×C₂₃	—	2	4	no	—
45	C₄⋊C₄×C₂₃	—	2	4	no	—
46	C₄⋊C₄×C₂₃	—	2	4	no	—
47	C₄⋊C₄×C₂₃	—	2	1	no	—
48	C₄⋊C₄×C₂₃	—	2	1	no	—
49	C₄⋊C₄×C₂₃	—	2	1	no	—
50	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
51	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
52	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
53	C₄²×C₂₃	—	2	4	no	—
54	C₄²×C₂₃	D₄	1	2	yes	—
55	C₄²×C₂₃	C₂³	1	2	yes	—
56	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
57	Q₈×C₂₃	—	2	4	no	—
58	Q₈×C₂₃	—	2	2	no	—
59	Q₈×C₂₃	C₂×Q₈	1	4	yes	—
60	C₂×C₄×C₂₃	—	2	4	no	—
61	C₂×C₄×C₂₃	—	2	2	no	—
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	4	2	no	—
68	C₂×C₄×C₂₃	—	4	2	no	—
69	C₂×C₄×C₂₃	—	4	2	no	—
70	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
74	C₂×C₄×C₂₃	#41	1	2	yes	—
75	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
76	C₈×C₂₃	—	2	2	no	—
77	C₈×C₂₃	—	2	2	no	—
78	C₂×#479	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
80	C₂²×C₂₃	#135	1	2	yes	—
81	C₂²×C₂₃	#137	1	2	yes	—
82	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
83	C₄×C₂₃	—	2	4	no	—
84	C₄×C₂₃	—	2	2	no	—
85	C₄×C₂₃	—	2	2	no	—
86	C₄×C₂₃	—	4	2	no	—
87	C₄×C₂₃	—	4	2	no	—
88	C₄×C₂₃	—	4	2	no	—
89	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
90	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
91	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
92	C₂×#125	C₂×C₂₃	1	1	yes	characteristic
93	C₂×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic
94	#479	C₂×C₂₃	1	8	yes	direct product factor
95	C₂×#107	C₂×C₂₃	1	1	yes	characteristic
96	C₂×#106	C₂×C₂₃	1	1	yes	characteristic
97	C₂×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
98	C₂×#103	C₂×C₂₃	1	1	yes	characteristic
99	C₂×#103	C₂×C₂₃	1	1	yes	characteristic
100	C₂×C₂₃	#479	1	4	yes	direct product factor
101	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
103	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
104	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
105	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
106	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
107	#125	—	2	2	no	—
108	#125	C₂²×C₂₃	1	4	yes	—
109	C₄×Q₈	—	2	2	no	—
110	C₄×Q₈	C₂²×C₂₃	1	4	yes	—
111	C₂×C₄⋊C₄	—	2	1	no	—
112	C₂×C₄⋊C₄	—	2	1	no	—
113	C₂×C₄⋊C₄	—	2	1	no	—
114	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
116	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
117	C₂×C₄²	—	2	1	no	—
118	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
119	#107	C₂²×C₂₃	1	4	yes	—
120	#106	C₂²×C₂₃	1	4	yes	—
121	C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
122	#103	C₂²×C₂₃	1	4	yes	—
123	#103	C₂²×C₂₃	1	4	yes	—
124	C₂₃	C₂×#479	1	1	yes	characteristic, direct product factor, Sylow
125	C₂×Q₈	—	2	1	no	—
126	C₂×Q₈	—	2	2	no	—
127	C₂×Q₈	Q₈×C₂₃	1	4	yes	—
128	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
134	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	C₂³×C₂₃	1	2	yes	—
139	C₂×C₈	C₂³×C₂₃	1	2	yes	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	1	no	—
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	—	2	1	no	—
146	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
147	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
148	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
149	C₄²	—	2	4	no	—
150	C₄²	D₄×C₂₃	1	2	yes	—
151	C₄²	C₂³×C₂₃	1	2	yes	—
152	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
153	Q₈	—	2	4	no	—
154	Q₈	—	2	2	no	—
155	Q₈	C₂×Q₈×C₂₃	1	4	yes	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
167	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
168	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
169	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
170	C₂×C₄	#41×C₂₃	1	2	yes	—
171	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
172	C₈	—	2	2	no	—
173	C₈	—	2	2	no	—
174	C₂²	C₂₃×C₂²⋊Q₈	1	2	yes	—
175	C₂²	C₂₃×#135	1	2	yes	—
176	C₂²	C₂₃×#137	1	2	yes	—
177	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	2	2	no	—
180	C₄	—	2	4	no	—
181	C₄	—	4	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
185	C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
186	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
187	C₂	C₂₃×#479	1	4	yes	direct product factor
188	C₂	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
189	C₂	C₂×C₂₃×#135	1	1	yes	characteristic
190	C₂	C₂×C₂₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#479	C₂	direct (N × H)
78. C₂×#479	C₂₃	direct (N × H)
94. #479	C₂×C₂₃	direct (N × H)
100. C₂×C₂₃	#479	direct (N × H)
124. C₂₃	C₂×#479	direct (N × H)
187. C₂	C₂₃×#479	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×C₂₃×#125	2
C₂×C₄×Q₈×C₂₃	2
C₂₃×#479	2
C₂×C₂₃×#107	2
C₂×C₂₃×#106	2
C₂×C₂₃×C₄⋊C₈	2
C₂×C₂₃×#103	2
C₂×#479	23

Subgroups — C2×C23×#479 #15670694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#479 #15670694