Subgroups — #2355×C₂₂₃ #43251370

Order 28544.

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×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#391×C₂₂₃	C₂	1	2	yes	maximal
3	#387×C₂₂₃	C₂	1	4	yes	maximal
4	#384×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	#382×C₂₂₃	C₂	1	2	yes	maximal
6	C₄×C₄⋊C₄×C₂₂₃	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₂₂₃	C₂	1	2	yes	maximal
9	C₂²×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₂₂₃	C₄	1	8	yes	complemented
11	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₂₂₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₂₂₃	C₂²	1	4	yes	—
20	C₂×C₄²×C₂₂₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
22	#95×C₂₂₃	C₂²	1	4	yes	—
23	#95×C₂₂₃	C₂²	1	2	yes	—
24	#95×C₂₂₃	C₂²	1	4	yes	—
25	#95×C₂₂₃	C₂²	1	4	yes	—
26	C₂×Q₈×C₂₂₃	—	2	2	no	—
27	C₂×Q₈×C₂₂₃	—	2	2	no	—
28	C₂×Q₈×C₂₂₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₂₂₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
32	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₂₂₃	—	2	4	no	—
38	C₄⋊C₄×C₂₂₃	—	2	2	no	—
39	C₄⋊C₄×C₂₂₃	—	2	4	no	—
40	C₄⋊C₄×C₂₂₃	—	4	2	no	—
41	C₄⋊C₄×C₂₂₃	C₂×C₄	1	8	yes	—
42	C₄⋊C₄×C₂₂₃	C₂×C₄	1	4	yes	—
43	C₄⋊C₄×C₂₂₃	Q₈	1	4	yes	complemented
44	C₄²×C₂₂₃	—	2	4	no	—
45	C₄²×C₂₂₃	—	4	4	no	—
46	C₄²×C₂₂₃	C₂×C₄	1	8	yes	—
47	C₄²×C₂₂₃	C₂×C₄	1	4	yes	—
48	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
49	Q₈×C₂₂₃	—	2	8	no	—
50	C₂×C₄×C₂₂₃	—	2	2	no	—
51	C₂×C₄×C₂₂₃	—	2	2	no	—
52	C₂×C₄×C₂₂₃	—	2	2	no	—
53	C₂×C₄×C₂₂₃	—	2	2	no	—
54	C₂×C₄×C₂₂₃	—	2	2	no	—
55	C₂×C₄×C₂₂₃	—	2	1	no	—
56	C₂×C₄×C₂₂₃	—	2	2	no	—
57	C₂×C₄×C₂₂₃	—	2	1	no	—
58	C₂×C₄×C₂₂₃	—	2	2	no	—
59	C₂×C₄×C₂₂₃	—	2	2	no	—
60	C₂×C₄×C₂₂₃	—	2	4	no	—
61	C₂×C₄×C₂₂₃	—	2	4	no	—
62	C₂×C₄×C₂₂₃	—	2	4	no	—
63	C₂×C₄×C₂₂₃	—	2	2	no	—
64	C₂×C₄×C₂₂₃	C₂²×C₄	1	4	yes	—
65	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₂₂₃	C₂²×C₄	1	4	yes	—
67	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₂₂₃	C₂×Q₈	1	4	yes	—
70	C₂×C₄×C₂₂₃	C₂×Q₈	1	2	yes	—
71	C₂×C₄×C₂₂₃	#41	1	2	yes	—
72	C₂×C₄×C₂₂₃	#41	1	2	yes	—
73	C₂×C₄×C₂₂₃	#41	1	2	yes	—
74	C₂²×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₂₂₃	C₂²×Q₈	1	1	yes	characteristic
76	C₂²×C₂₂₃	C₂×#41	1	1	yes	characteristic
77	C₂²×C₂₂₃	C₂×#41	1	1	yes	characteristic
78	C₂²×C₂₂₃	C₂×#41	1	1	yes	characteristic
79	C₂²×C₂₂₃	H(5,𝔽₂)	1	1	yes	characteristic
80	C₂²×C₂₂₃	2₋¹⁺⁴	1	1	yes	characteristic
81	C₄×C₂₂₃	—	2	4	no	—
82	C₄×C₂₂₃	—	2	4	no	—
83	C₄×C₂₂₃	—	2	2	no	—
84	C₄×C₂₂₃	—	2	4	no	—
85	C₄×C₂₂₃	—	4	2	no	—
86	C₄×C₂₂₃	—	4	4	no	—
87	C₄×C₂₂₃	#117	1	4	yes	—
88	C₄×C₂₂₃	C₄×Q₈	1	4	yes	—
89	C₂×C₂₂₃	C₂×#117	1	1	yes	characteristic
90	C₂×C₂₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₂×C₂₂₃	#520	1	1	yes	characteristic
92	C₂×C₂₂₃	#554	1	1	yes	characteristic
93	C₂×C₂₂₃	#556	1	1	yes	characteristic
94	C₂×C₂₂₃	#557	1	1	yes	characteristic
95	C₂×C₂₂₃	#559	1	1	yes	characteristic
96	C₂₂₃	#2355	1	1	yes	characteristic, direct product factor, Sylow
97	#2355	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂×C₄×Q₈	C₂×C₂₂₃	1	1	yes	characteristic
99	#391	C₂×C₂₂₃	1	2	yes	—
100	#387	C₂×C₂₂₃	1	4	yes	—
101	#384	C₂×C₂₂₃	1	1	yes	characteristic
102	#382	C₂×C₂₂₃	1	2	yes	—
103	C₄×C₄⋊C₄	C₂×C₂₂₃	1	2	yes	—
104	C₄×C₄⋊C₄	C₂×C₂₂₃	1	1	yes	characteristic
105	C₄×C₄⋊C₄	C₂×C₂₂₃	1	2	yes	—
106	C₂²×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
107	C₄×Q₈	C₄×C₂₂₃	1	8	yes	complemented
108	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
109	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
110	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
111	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
112	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
113	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
114	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
115	C₂×C₄²	C₂²×C₂₂₃	1	2	yes	—
116	C₂×C₄²	C₂²×C₂₂₃	1	4	yes	—
117	C₂×C₄²	C₂²×C₂₂₃	1	2	yes	—
118	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
119	#95	C₂²×C₂₂₃	1	4	yes	—
120	#95	C₂²×C₂₂₃	1	2	yes	—
121	#95	C₂²×C₂₂₃	1	4	yes	—
122	#95	C₂²×C₂₂₃	1	4	yes	—
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	—	2	2	no	—
125	C₂×Q₈	C₂×C₄×C₂₂₃	1	4	yes	—
126	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
127	C₂²×C₄	C₂³×C₂₂₃	1	4	yes	—
128	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
129	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
130	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
131	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
132	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
133	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	2	4	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	C₂×C₄×C₂₂₃	1	8	yes	—
139	C₄⋊C₄	C₂×C₄×C₂₂₃	1	4	yes	—
140	C₄⋊C₄	Q₈×C₂₂₃	1	4	yes	complemented
141	C₄²	—	2	4	no	—
142	C₄²	—	4	4	no	—
143	C₄²	C₂×C₄×C₂₂₃	1	8	yes	—
144	C₄²	C₂×C₄×C₂₂₃	1	4	yes	—
145	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
146	Q₈	—	2	8	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	C₂²×C₄×C₂₂₃	1	4	yes	—
162	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
163	C₂×C₄	C₂²×C₄×C₂₂₃	1	4	yes	—
164	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
165	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
166	C₂×C₄	C₂×Q₈×C₂₂₃	1	4	yes	—
167	C₂×C₄	C₂×Q₈×C₂₂₃	1	2	yes	—
168	C₂×C₄	#41×C₂₂₃	1	2	yes	—
169	C₂×C₄	#41×C₂₂₃	1	2	yes	—
170	C₂×C₄	#41×C₂₂₃	1	2	yes	—
171	C₂²	C₂³×C₄×C₂₂₃	1	1	yes	characteristic
172	C₂²	C₂²×Q₈×C₂₂₃	1	1	yes	characteristic
173	C₂²	C₂×#41×C₂₂₃	1	1	yes	characteristic
174	C₂²	C₂×#41×C₂₂₃	1	1	yes	characteristic
175	C₂²	C₂×#41×C₂₂₃	1	1	yes	characteristic
176	C₂²	H(5,𝔽₂)×C₂₂₃	1	1	yes	characteristic
177	C₂²	2₋¹⁺⁴×C₂₂₃	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	2	2	no	—
180	C₄	—	2	4	no	—
181	C₄	—	2	4	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	4	no	—
184	C₄	#117×C₂₂₃	1	4	yes	—
185	C₄	C₄×Q₈×C₂₂₃	1	4	yes	—
186	C₂	C₂×#117×C₂₂₃	1	1	yes	characteristic
187	C₂	C₂×C₄×Q₈×C₂₂₃	1	1	yes	characteristic
188	C₂	#520×C₂₂₃	1	1	yes	characteristic
189	C₂	#554×C₂₂₃	1	1	yes	characteristic
190	C₂	#556×C₂₂₃	1	1	yes	characteristic
191	C₂	#557×C₂₂₃	1	1	yes	characteristic
192	C₂	#559×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₂₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₂₃
Frattini, cosocle	C₂³	C₂⁴×C₂₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₄×Q₈×C₂₂₃	C₄	semidirect
43. C₄⋊C₄×C₂₂₃	Q₈	semidirect
96. C₂₂₃	#2355	direct (N × H)
97. #2355	C₂₂₃	direct (N × H)
107. C₄×Q₈	C₄×C₂₂₃	semidirect
140. C₄⋊C₄	Q₈×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₂₂₃	2
#391×C₂₂₃	2
#387×C₂₂₃	2
#384×C₂₂₃	2
#382×C₂₂₃	2
C₄×C₄⋊C₄×C₂₂₃	2
#2355	223

Subgroups — #2355×C223 #43251370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2355×C₂₂₃ #43251370