Subgroups — C₂²×C₂₃×#557 #19210335

Order 5888.

7414 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×#557	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₂₃×C₄⋊Q₈	C₂	1	3	yes	maximal
3	C₂²×C₂₃×#125	C₂	1	6	yes	maximal
4	C₂²×C₄×Q₈×C₂₃	C₂	1	3	yes	maximal
5	C₂²×C₄×Q₈×C₂₃	C₂	1	2	yes	maximal
6	C₂²×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₂₃	C₂²	1	3	yes	—
8	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#557	C₂²	1	256	yes	direct product factor
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	72	yes	—
11	C₂×C₂₃×#125	C₂²	1	144	yes	—
12	C₂×C₄×Q₈×C₂₃	C₂²	1	48	yes	—
13	C₂×C₄×Q₈×C₂₃	C₂²	1	72	yes	—
14	C₂×C₄×Q₈×C₂₃	C₂²	1	24	yes	—
15	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₂₃	C₂²	1	6	yes	—
17	C₂²×C₄⋊C₄×C₂₃	C₂²	1	3	yes	—
18	C₂²×C₄⋊C₄×C₂₃	C₂²	1	6	yes	—
19	C₂²×C₄⋊C₄×C₂₃	C₂²	1	6	yes	—
20	C₂²×C₄²×C₂₃	C₂²	1	6	yes	—
21	C₂²×C₄²×C₂₃	C₂²	1	3	yes	—
22	C₂²×Q₈×C₂₃	—	2	24	no	—
23	C₂²×Q₈×C₂₃	Q₈	1	16	yes	complemented
24	C₂²×Q₈×C₂₃	C₂³	1	36	yes	—
25	C₂²×Q₈×C₂₃	C₂³	1	12	yes	—
26	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
27	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
28	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
29	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
30	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
31	C₂₃×C₄⋊Q₈	C₂³	1	192	yes	—
32	C₂₃×#125	C₂³	1	384	yes	—
33	C₄×Q₈×C₂₃	C₂³	1	128	yes	—
34	C₄×Q₈×C₂₃	C₂³	1	192	yes	—
35	C₄×Q₈×C₂₃	C₂³	1	64	yes	—
36	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
37	C₂×C₄⋊C₄×C₂₃	C₂³	1	72	yes	—
38	C₂×C₄⋊C₄×C₂₃	C₂³	1	36	yes	—
39	C₂×C₄⋊C₄×C₂₃	C₂³	1	72	yes	—
40	C₂×C₄⋊C₄×C₂₃	C₂³	1	72	yes	—
41	C₂×C₄²×C₂₃	C₂³	1	72	yes	—
42	C₂×C₄²×C₂₃	C₂³	1	36	yes	—
43	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
44	C₂×Q₈×C₂₃	—	2	144	no	—
45	C₂×Q₈×C₂₃	C₂×Q₈	1	96	yes	complemented
46	C₂×Q₈×C₂₃	C₂⁴	1	48	yes	—
47	C₂×Q₈×C₂₃	C₂⁴	1	16	yes	—
48	C₂²×C₄×C₂₃	—	2	24	no	—
49	C₂²×C₄×C₂₃	—	2	8	no	—
50	C₂²×C₄×C₂₃	—	2	12	no	—
51	C₂²×C₄×C₂₃	C₂×Q₈	1	24	yes	—
52	C₂²×C₄×C₂₃	#41	1	8	yes	—
53	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
54	C₂²×C₄×C₂₃	C₂⁴	1	18	yes	—
55	C₂²×C₄×C₂₃	C₂⁴	1	36	yes	—
56	C₂²×C₄×C₂₃	C₂⁴	1	18	yes	—
57	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
58	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
59	C₄⋊C₄×C₂₃	C₂⁴	1	96	yes	—
60	C₄⋊C₄×C₂₃	C₂⁴	1	48	yes	—
61	C₄⋊C₄×C₂₃	C₂⁴	1	96	yes	—
62	C₄⋊C₄×C₂₃	C₂⁴	1	96	yes	—
63	C₄²×C₂₃	C₂⁴	1	96	yes	—
64	C₄²×C₂₃	C₂⁴	1	48	yes	—
65	C₂²×#557	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₂₃	C₂²×Q₈	1	4	yes	—
67	C₂³×C₂₃	C₂×#41	1	4	yes	—
68	C₂³×C₂₃	2₋¹⁺⁴	1	4	yes	—
69	C₂³×C₂₃	C₂⁵	1	3	yes	—
70	Q₈×C₂₃	—	2	96	no	—
71	Q₈×C₂₃	C₂²×Q₈	1	64	yes	complemented
72	C₂×C₄×C₂₃	—	2	72	no	—
73	C₂×C₄×C₂₃	—	2	24	no	—
74	C₂×C₄×C₂₃	—	2	36	no	—
75	C₂×C₄×C₂₃	C₂²×Q₈	1	72	yes	—
76	C₂×C₄×C₂₃	C₂×#41	1	24	yes	—
77	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
78	C₂×C₄×C₂₃	C₂⁵	1	12	yes	—
79	C₂×C₄×C₂₃	C₂⁵	1	24	yes	—
80	C₂×C₄×C₂₃	C₂⁵	1	12	yes	—
81	C₂×C₄×C₂₃	C₂⁵	1	4	yes	—
82	C₂×#557	C₂×C₂₃	1	48	yes	direct product factor
83	C₂²×C₄⋊Q₈	C₂×C₂₃	1	3	yes	—
84	C₂²×#125	C₂×C₂₃	1	6	yes	—
85	C₂²×C₄×Q₈	C₂×C₂₃	1	3	yes	—
86	C₂²×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic
87	C₂²×C₄×Q₈	C₂×C₂₃	1	2	yes	—
88	C₂²×C₂₃	#557	1	16	yes	direct product factor
89	C₂²×C₂₃	C₂³×Q₈	1	6	yes	—
90	C₂²×C₂₃	C₂²×#41	1	6	yes	—
91	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	6	yes	—
92	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
93	C₄×C₂₃	—	2	24	no	—
94	C₄×C₂₃	—	2	8	no	—
95	C₄×C₂₃	—	2	12	no	—
96	C₄×C₂₃	C₂³×Q₈	1	24	yes	—
97	C₄×C₂₃	C₂²×#41	1	8	yes	—
98	C₂³×Q₈	C₂²×C₂₃	1	3	yes	—
99	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
100	#557	C₂²×C₂₃	1	256	yes	direct product factor
101	C₂×C₄⋊Q₈	C₂²×C₂₃	1	72	yes	—
102	C₂×#125	C₂²×C₂₃	1	144	yes	—
103	C₂×C₄×Q₈	C₂²×C₂₃	1	48	yes	—
104	C₂×C₄×Q₈	C₂²×C₂₃	1	72	yes	—
105	C₂×C₄×Q₈	C₂²×C₂₃	1	24	yes	—
106	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
107	C₂²×C₄⋊C₄	C₂²×C₂₃	1	3	yes	—
108	C₂²×C₄⋊C₄	C₂²×C₂₃	1	6	yes	—
109	C₂²×C₄⋊C₄	C₂²×C₂₃	1	6	yes	—
110	C₂²×C₄⋊C₄	C₂²×C₂₃	1	6	yes	—
111	C₂²×C₄²	C₂²×C₂₃	1	6	yes	—
112	C₂²×C₄²	C₂²×C₂₃	1	3	yes	—
113	C₂×C₂₃	C₂×#557	1	12	yes	direct product factor
114	C₂×C₂₃	C₂⁴×Q₈	1	1	yes	characteristic
115	C₂×C₂₃	C₂³×#41	1	1	yes	characteristic
116	C₂×C₂₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
117	C₂²×Q₈	—	2	24	no	—
118	C₂²×Q₈	Q₈×C₂₃	1	16	yes	complemented
119	C₂²×Q₈	C₂³×C₂₃	1	36	yes	—
120	C₂²×Q₈	C₂³×C₂₃	1	12	yes	—
121	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
122	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
123	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
124	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
125	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
126	C₄⋊Q₈	C₂³×C₂₃	1	192	yes	—
127	#125	C₂³×C₂₃	1	384	yes	—
128	C₄×Q₈	C₂³×C₂₃	1	128	yes	—
129	C₄×Q₈	C₂³×C₂₃	1	192	yes	—
130	C₄×Q₈	C₂³×C₂₃	1	64	yes	—
131	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
132	C₂×C₄⋊C₄	C₂³×C₂₃	1	72	yes	—
133	C₂×C₄⋊C₄	C₂³×C₂₃	1	36	yes	—
134	C₂×C₄⋊C₄	C₂³×C₂₃	1	72	yes	—
135	C₂×C₄⋊C₄	C₂³×C₂₃	1	72	yes	—
136	C₂×C₄²	C₂³×C₂₃	1	72	yes	—
137	C₂×C₄²	C₂³×C₂₃	1	36	yes	—
138	C₂₃	C₂²×#557	1	1	yes	characteristic, direct product factor, Sylow
139	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
140	C₂×Q₈	—	2	144	no	—
141	C₂×Q₈	C₂×Q₈×C₂₃	1	96	yes	complemented
142	C₂×Q₈	C₂⁴×C₂₃	1	48	yes	—
143	C₂×Q₈	C₂⁴×C₂₃	1	16	yes	—
144	C₂²×C₄	—	2	24	no	—
145	C₂²×C₄	—	2	12	no	—
146	C₂²×C₄	—	2	8	no	—
147	C₂²×C₄	C₂×Q₈×C₂₃	1	24	yes	—
148	C₂²×C₄	#41×C₂₃	1	8	yes	—
149	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
150	C₂²×C₄	C₂⁴×C₂₃	1	18	yes	—
151	C₂²×C₄	C₂⁴×C₂₃	1	36	yes	—
152	C₂²×C₄	C₂⁴×C₂₃	1	18	yes	—
153	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
154	C₄⋊C₄	C₂⁴×C₂₃	1	16	yes	—
155	C₄⋊C₄	C₂⁴×C₂₃	1	96	yes	—
156	C₄⋊C₄	C₂⁴×C₂₃	1	48	yes	—
157	C₄⋊C₄	C₂⁴×C₂₃	1	96	yes	—
158	C₄⋊C₄	C₂⁴×C₂₃	1	96	yes	—
159	C₄²	C₂⁴×C₂₃	1	96	yes	—
160	C₄²	C₂⁴×C₂₃	1	48	yes	—
161	C₂³	C₂²×Q₈×C₂₃	1	4	yes	—
162	C₂³	C₂×#41×C₂₃	1	4	yes	—
163	C₂³	C₂₃×2₋¹⁺⁴	1	4	yes	—
164	C₂³	C₂⁵×C₂₃	1	3	yes	—
165	Q₈	—	2	96	no	—
166	Q₈	C₂²×Q₈×C₂₃	1	64	yes	complemented
167	C₂×C₄	—	2	72	no	—
168	C₂×C₄	—	2	24	no	—
169	C₂×C₄	—	2	36	no	—
170	C₂×C₄	C₂²×Q₈×C₂₃	1	72	yes	—
171	C₂×C₄	C₂×#41×C₂₃	1	24	yes	—
172	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
173	C₂×C₄	C₂⁵×C₂₃	1	12	yes	—
174	C₂×C₄	C₂⁵×C₂₃	1	24	yes	—
175	C₂×C₄	C₂⁵×C₂₃	1	12	yes	—
176	C₂×C₄	C₂⁵×C₂₃	1	4	yes	—
177	C₂²	C₂₃×#557	1	16	yes	direct product factor
178	C₂²	C₂³×Q₈×C₂₃	1	6	yes	—
179	C₂²	C₂²×#41×C₂₃	1	6	yes	—
180	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	6	yes	—
181	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
182	C₄	—	2	24	no	—
183	C₄	—	2	8	no	—
184	C₄	—	2	12	no	—
185	C₄	C₂³×Q₈×C₂₃	1	24	yes	—
186	C₄	C₂²×#41×C₂₃	1	8	yes	—
187	C₂	C₂×C₂₃×#557	1	12	yes	direct product factor
188	C₂	C₂⁴×Q₈×C₂₃	1	1	yes	characteristic
189	C₂	C₂³×#41×C₂₃	1	1	yes	characteristic
190	C₂	C₂²×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#557	C₂	direct (N × H)
9. C₂₃×#557	C₂²	direct (N × H)
23. C₂²×Q₈×C₂₃	Q₈	semidirect
45. C₂×Q₈×C₂₃	C₂×Q₈	semidirect
65. C₂²×#557	C₂₃	direct (N × H)
71. Q₈×C₂₃	C₂²×Q₈	semidirect
82. C₂×#557	C₂×C₂₃	direct (N × H)
88. C₂²×C₂₃	#557	direct (N × H)
100. #557	C₂²×C₂₃	direct (N × H)
113. C₂×C₂₃	C₂×#557	direct (N × H)
118. C₂²×Q₈	Q₈×C₂₃	semidirect
138. C₂₃	C₂²×#557	direct (N × H)
141. C₂×Q₈	C₂×Q₈×C₂₃	semidirect
166. Q₈	C₂²×Q₈×C₂₃	semidirect
177. C₂²	C₂₃×#557	direct (N × H)
187. C₂	C₂×C₂₃×#557	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×#557	2
C₂²×C₂₃×C₄⋊Q₈	2
C₂²×C₂₃×#125	2
C₂²×C₄×Q₈×C₂₃	2
C₂²×#557	23

Subgroups — C22×C23×#557 #19210335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#557 #19210335