Subgroups — #3127×C₂₂₃ #43252142

Order 28544.

358 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	#541×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#519×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#485×C₂₂₃	C₂	1	2	yes	maximal
4	#479×C₂₂₃	C₂	1	2	yes	maximal, complemented
5	#462×C₂₂₃	C₂	1	2	yes	maximal, complemented
6	#451×C₂₂₃	C₂	1	2	yes	maximal
7	#441×C₂₂₃	C₂	1	2	yes	maximal, complemented
8	C₄×Q₁₆×C₂₂₃	C₂	1	2	yes	maximal, complemented
9	#433×C₂₂₃	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₂₂₃	C₂²	1	2	yes	—
12	C₂×Q₁₆×C₂₂₃	C₂²	1	2	yes	—
13	C₄⋊Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
14	#126×C₂₂₃	—	2	2	no	—
15	#125×C₂₂₃	—	2	2	no	—
16	#125×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂²⋊Q₈×C₂₂₃	C₂²	1	2	yes	—
18	C₄×Q₈×C₂₂₃	—	2	2	no	—
19	C₄×Q₈×C₂₂₃	C₂²	1	2	yes	—
20	C₄×Q₈×C₂₂₃	C₂²	1	2	yes	—
21	C₄×Q₈×C₂₂₃	C₂²	1	2	yes	—
22	#117×C₂₂₃	—	2	2	no	—
23	#117×C₂₂₃	C₂²	1	1	yes	characteristic
24	#117×C₂₂₃	C₂²	1	1	yes	characteristic
25	#107×C₂₂₃	C₂²	1	2	yes	—
26	#106×C₂₂₃	C₂²	1	2	yes	—
27	C₄⋊C₈×C₂₂₃	C₂²	1	2	yes	—
28	#103×C₂₂₃	C₂²	1	2	yes	—
29	#103×C₂₂₃	C₂²	1	2	yes	—
30	#103×C₂₂₃	C₂²	1	2	yes	—
31	#103×C₂₂₃	C₂²	1	2	yes	—
32	C₂²⋊C₈×C₂₂₃	C₂²	1	2	yes	—
33	#97×C₂₂₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₂₂₃	—	2	4	no	—
36	C₂×Q₈×C₂₂₃	—	2	1	no	—
37	C₂×Q₈×C₂₂₃	C₂³	1	2	yes	—
38	C₂×Q₈×C₂₂₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₂₃	—	2	1	no	—
40	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
41	Q₁₆×C₂₂₃	—	2	4	no	—
42	Q₁₆×C₂₂₃	—	2	4	no	—
43	C₂×C₈×C₂₂₃	C₂³	1	2	yes	—
44	C₂×C₈×C₂₂₃	C₂³	1	2	yes	—
45	C₄⋊C₄×C₂₂₃	—	2	2	no	—
46	C₄⋊C₄×C₂₂₃	—	2	2	no	—
47	C₄⋊C₄×C₂₂₃	—	2	2	no	—
48	C₄⋊C₄×C₂₂₃	—	2	2	no	—
49	C₄⋊C₄×C₂₂₃	—	2	2	no	—
50	C₄⋊C₄×C₂₂₃	—	2	2	no	—
51	C₄⋊C₄×C₂₂₃	D₄	1	2	yes	—
52	C₄⋊C₄×C₂₂₃	C₂³	1	2	yes	—
53	C₄⋊C₄×C₂₂₃	C₂³	1	2	yes	—
54	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
55	C₂²⋊C₄×C₂₂₃	—	2	2	no	—
56	C₂²⋊C₄×C₂₂₃	D₄	1	2	yes	—
57	C₄²×C₂₂₃	—	2	2	no	—
58	C₄²×C₂₂₃	—	2	2	no	—
59	C₄²×C₂₂₃	—	2	1	no	—
60	C₄²×C₂₂₃	—	2	2	no	—
61	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
62	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
63	C₂³×C₂₂₃	C₂×D₄	1	1	yes	characteristic
64	Q₈×C₂₂₃	—	2	4	no	—
65	Q₈×C₂₂₃	—	2	2	no	—
66	Q₈×C₂₂₃	—	2	2	no	—
67	Q₈×C₂₂₃	—	2	2	no	—
68	Q₈×C₂₂₃	#41	1	4	yes	—
69	C₂×C₄×C₂₂₃	—	2	4	no	—
70	C₂×C₄×C₂₂₃	—	2	2	no	—
71	C₂×C₄×C₂₂₃	—	2	2	no	—
72	C₂×C₄×C₂₂₃	—	2	2	no	—
73	C₂×C₄×C₂₂₃	—	2	2	no	—
74	C₂×C₄×C₂₂₃	—	2	2	no	—
75	C₂×C₄×C₂₂₃	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₂₂₃	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
79	C₈×C₂₂₃	—	2	2	no	—
80	C₈×C₂₂₃	—	2	2	no	—
81	C₂²×C₂₂₃	—	2	1	no	—
82	C₂²×C₂₂₃	—	2	2	no	—
83	C₂²×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
84	C₄×C₂₂₃	—	2	4	no	—
85	C₄×C₂₂₃	—	2	1	no	—
86	C₄×C₂₂₃	—	2	2	no	—
87	C₄×C₂₂₃	—	2	2	no	—
88	C₄×C₂₂₃	—	4	2	no	—
89	C₄×C₂₂₃	—	4	2	no	—
90	C₄×C₂₂₃	—	4	2	no	—
91	C₄×C₂₂₃	C₂×#41	1	2	yes	—
92	C₂×C₂₂₃	—	4	1	no	—
93	C₂×C₂₂₃	#525	1	1	yes	characteristic
94	C₂×C₂₂₃	#578	1	2	yes	—
95	C₂₂₃	#3127	1	1	yes	characteristic, direct product factor, Sylow
96	#3127	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	#541	C₂×C₂₂₃	1	1	yes	characteristic
98	#519	C₂×C₂₂₃	1	1	yes	characteristic
99	#485	C₂×C₂₂₃	1	2	yes	—
100	#479	C₂×C₂₂₃	1	2	yes	complemented
101	#462	C₂×C₂₂₃	1	2	yes	complemented
102	#451	C₂×C₂₂₃	1	2	yes	—
103	#441	C₂×C₂₂₃	1	2	yes	complemented
104	C₄×Q₁₆	C₂×C₂₂₃	1	2	yes	complemented
105	#433	C₂×C₂₂₃	1	1	yes	characteristic
106	C₂²×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
107	C₂×Q₁₆	C₂²×C₂₂₃	1	2	yes	—
108	C₂×Q₁₆	C₂²×C₂₂₃	1	2	yes	—
109	C₄⋊Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
110	#126	—	2	2	no	—
111	#125	—	2	2	no	—
112	#125	C₂²×C₂₂₃	1	1	yes	characteristic
113	C₂²⋊Q₈	C₂²×C₂₂₃	1	2	yes	—
114	C₄×Q₈	—	2	2	no	—
115	C₄×Q₈	C₂²×C₂₂₃	1	2	yes	—
116	C₄×Q₈	C₂²×C₂₂₃	1	2	yes	—
117	C₄×Q₈	C₂²×C₂₂₃	1	2	yes	—
118	#117	—	2	2	no	—
119	#117	C₂²×C₂₂₃	1	1	yes	characteristic
120	#117	C₂²×C₂₂₃	1	1	yes	characteristic
121	#107	C₂²×C₂₂₃	1	2	yes	—
122	#106	C₂²×C₂₂₃	1	2	yes	—
123	C₄⋊C₈	C₂²×C₂₂₃	1	2	yes	—
124	#103	C₂²×C₂₂₃	1	2	yes	—
125	#103	C₂²×C₂₂₃	1	2	yes	—
126	#103	C₂²×C₂₂₃	1	2	yes	—
127	#103	C₂²×C₂₂₃	1	2	yes	—
128	C₂²⋊C₈	C₂²×C₂₂₃	1	2	yes	—
129	#97	C₂²×C₂₂₃	1	1	yes	characteristic
130	C₄×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	—	2	1	no	—
133	C₂×Q₈	C₂³×C₂₂₃	1	2	yes	—
134	C₂×Q₈	C₂³×C₂₂₃	1	2	yes	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
137	Q₁₆	—	2	4	no	—
138	Q₁₆	—	2	4	no	—
139	C₂×C₈	C₂³×C₂₂₃	1	2	yes	—
140	C₂×C₈	C₂³×C₂₂₃	1	2	yes	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	D₄×C₂₂₃	1	2	yes	—
148	C₄⋊C₄	C₂³×C₂₂₃	1	2	yes	—
149	C₄⋊C₄	C₂³×C₂₂₃	1	2	yes	—
150	C₂²⋊C₄	—	2	1	no	—
151	C₂²⋊C₄	—	2	2	no	—
152	C₂²⋊C₄	D₄×C₂₂₃	1	2	yes	—
153	C₄²	—	2	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	2	2	no	—
157	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
158	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
159	C₂³	C₂×D₄×C₂₂₃	1	1	yes	characteristic
160	Q₈	—	2	4	no	—
161	Q₈	—	2	2	no	—
162	Q₈	—	2	2	no	—
163	Q₈	—	2	2	no	—
164	Q₈	#41×C₂₂₃	1	4	yes	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	C₂×D₄×C₂₂₃	1	2	yes	—
172	C₂×C₄	C₂×D₄×C₂₂₃	1	2	yes	—
173	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
174	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	2	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	2	no	—
179	C₂²	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
180	C₄	—	2	4	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	2	no	—
186	C₄	—	4	2	no	—
187	C₄	C₂×#41×C₂₂₃	1	2	yes	—
188	C₂	—	4	1	no	—
189	C₂	#525×C₂₂₃	1	1	yes	characteristic
190	C₂	#578×C₂₂₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₃
prime radical	C₂³×C₂₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ #117×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. #479×C₂₂₃	C₂	semidirect
5. #462×C₂₂₃	C₂	semidirect
7. #441×C₂₂₃	C₂	semidirect
8. C₄×Q₁₆×C₂₂₃	C₂	semidirect
95. C₂₂₃	#3127	direct (N × H)
96. #3127	C₂₂₃	direct (N × H)
100. #479	C₂×C₂₂₃	semidirect
101. #462	C₂×C₂₂₃	semidirect
103. #441	C₂×C₂₂₃	semidirect
104. C₄×Q₁₆	C₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#541×C₂₂₃	2
#519×C₂₂₃	2
#485×C₂₂₃	2
#479×C₂₂₃	2
#462×C₂₂₃	2
#451×C₂₂₃	2
#441×C₂₂₃	2
C₄×Q₁₆×C₂₂₃	2
#433×C₂₂₃	2
#3127	223

Subgroups — #3127×C223 #43252142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3127×C₂₂₃ #43252142