Subgroups — #3334×C₂₂₃ #43252349

Order 28544.

420 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#547×C₂₂₃	C₂	1	2	yes	maximal, complemented
2	#500×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#480×C₂₂₃	C₂	1	4	yes	maximal, complemented
4	#474×C₂₂₃	C₂	1	2	yes	maximal, complemented
5	#466×C₂₂₃	C₂	1	2	yes	maximal, complemented
6	C₄×D₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#107×C₂₂₃	C₂	1	2	yes	maximal, complemented
9	C₂×#41×C₂₂₃	—	2	2	no	—
10	C₂×D₈×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₂₂₃	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₂₂₃	C₂²	1	2	yes	—
13	#123×C₂₂₃	—	2	4	no	—
14	C₂²⋊Q₈×C₂₂₃	—	2	2	no	—
15	#121×C₂₂₃	C₂²	1	4	yes	—
16	C₄×D₄×C₂₂₃	C₂²	1	2	yes	—
17	C₄×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	4	yes	—
19	#107×C₂₂₃	C₂²	1	1	yes	characteristic
20	#107×C₂₂₃	C₂²	1	2	yes	—
21	#107×C₂₂₃	C₂²	1	4	yes	—
22	#107×C₂₂₃	C₂²	1	2	yes	—
23	C₄⋊C₈×C₂₂₃	C₂²	1	1	yes	characteristic
24	#102×C₂₂₃	C₂²	1	2	yes	—
25	#102×C₂₂₃	C₂²	1	4	yes	—
26	C₂²⋊C₈×C₂₂₃	C₂²	1	2	yes	—
27	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
28	#41×C₂₂₃	—	4	8	no	—
29	C₂×Q₈×C₂₂₃	—	2	2	no	—
30	C₂×D₄×C₂₂₃	—	2	4	no	—
31	C₂×D₄×C₂₂₃	D₄	1	1	yes	characteristic
32	C₂×D₄×C₂₂₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₂₃	—	2	4	no	—
34	C₂²×C₄×C₂₂₃	—	2	2	no	—
35	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
36	D₈×C₂₂₃	—	2	2	no	—
37	C₂×C₈×C₂₂₃	—	2	4	no	—
38	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₂₂₃	C₂³	1	2	yes	—
40	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₂₂₃	—	2	4	no	—
42	C₄⋊C₄×C₂₂₃	—	2	2	no	—
43	C₄⋊C₄×C₂₂₃	D₄	1	1	yes	characteristic
44	C₄⋊C₄×C₂₂₃	C₂³	1	2	yes	—
45	C₄⋊C₄×C₂₂₃	C₂³	1	4	yes	—
46	C₂²⋊C₄×C₂₂₃	—	2	2	no	—
47	C₂²⋊C₄×C₂₂₃	—	2	4	no	—
48	C₂²⋊C₄×C₂₂₃	D₄	1	2	yes	—
49	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
50	C₂³×C₂₂₃	—	2	2	no	—
51	C₂³×C₂₂₃	C₂×D₄	1	2	yes	—
52	Q₈×C₂₂₃	—	4	4	no	—
53	D₄×C₂₂₃	—	2	2	no	—
54	D₄×C₂₂₃	—	2	2	no	—
55	D₄×C₂₂₃	—	4	8	no	—
56	D₄×C₂₂₃	D₈	1	4	yes	complemented
57	C₂×C₄×C₂₂₃	—	2	2	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	—	4	8	no	—
62	C₂×C₄×C₂₂₃	—	4	4	no	—
63	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₂₂₃	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
67	C₈×C₂₂₃	—	2	2	no	—
68	C₈×C₂₂₃	—	2	1	no	—
69	C₈×C₂₂₃	#41	1	2	yes	complemented
70	C₂²×C₂₂₃	—	2	2	no	—
71	C₂²×C₂₂₃	—	2	2	no	—
72	C₂²×C₂₂₃	—	4	2	no	—
73	C₂²×C₂₂₃	—	4	2	no	—
74	C₂²×C₂₂₃	—	4	2	no	—
75	C₂²×C₂₂₃	C₂×D₈	1	4	yes	—
76	C₂²×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
77	C₄×C₂₂₃	—	2	2	no	—
78	C₄×C₂₂₃	—	2	1	no	—
79	C₄×C₂₂₃	—	4	2	no	—
80	C₄×C₂₂₃	—	4	4	no	—
81	C₄×C₂₂₃	C₂×D₈	1	2	yes	—
82	C₄×C₂₂₃	C₂×#41	1	1	yes	characteristic
83	C₄×C₂₂₃	2₋¹⁺⁴	1	1	yes	characteristic
84	C₂×C₂₂₃	—	2	4	no	—
85	C₂×C₂₂₃	—	8	2	no	—
86	C₂×C₂₂₃	#547	1	1	yes	characteristic
87	C₂×C₂₂₃	C₂²×D₈	1	1	yes	characteristic
88	C₂×C₂₂₃	#578	1	1	yes	characteristic
89	C₂₂₃	#3334	1	1	yes	characteristic, direct product factor, Sylow
90	#3334	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	#547	C₂×C₂₂₃	1	2	yes	complemented
92	#500	C₂×C₂₂₃	1	1	yes	characteristic, complemented
93	#480	C₂×C₂₂₃	1	4	yes	complemented
94	#474	C₂×C₂₂₃	1	2	yes	complemented
95	#466	C₂×C₂₂₃	1	2	yes	complemented
96	C₄×D₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
97	C₈×D₄	C₂×C₂₂₃	1	1	yes	characteristic, complemented
98	C₂×#107	C₂×C₂₂₃	1	2	yes	complemented
99	C₂×#41	—	2	2	no	—
100	C₂×D₈	C₂²×C₂₂₃	1	1	yes	characteristic
101	C₂²×C₈	C₂²×C₂₂₃	1	2	yes	—
102	C₄⋊Q₈	C₂²×C₂₂₃	1	2	yes	—
103	#123	—	2	4	no	—
104	C₂²⋊Q₈	—	2	2	no	—
105	#121	C₂²×C₂₂₃	1	4	yes	—
106	C₄×D₄	C₂²×C₂₂₃	1	2	yes	—
107	C₄×D₄	C₂²×C₂₂₃	1	1	yes	characteristic
108	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	4	yes	—
109	#107	C₂²×C₂₂₃	1	1	yes	characteristic
110	#107	C₂²×C₂₂₃	1	2	yes	—
111	#107	C₂²×C₂₂₃	1	4	yes	—
112	#107	C₂²×C₂₂₃	1	2	yes	—
113	C₄⋊C₈	C₂²×C₂₂₃	1	1	yes	characteristic
114	#102	C₂²×C₂₂₃	1	2	yes	—
115	#102	C₂²×C₂₂₃	1	4	yes	—
116	C₂²⋊C₈	C₂²×C₂₂₃	1	2	yes	—
117	C₄×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
118	#41	—	4	8	no	—
119	C₂×Q₈	—	2	2	no	—
120	C₂×D₄	—	2	4	no	—
121	C₂×D₄	D₄×C₂₂₃	1	1	yes	characteristic
122	C₂×D₄	C₂³×C₂₂₃	1	2	yes	—
123	C₂²×C₄	—	2	4	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
126	D₈	—	2	2	no	—
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
129	C₂×C₈	C₂³×C₂₂₃	1	2	yes	—
130	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	D₄×C₂₂₃	1	1	yes	characteristic
134	C₄⋊C₄	C₂³×C₂₂₃	1	2	yes	—
135	C₄⋊C₄	C₂³×C₂₂₃	1	4	yes	—
136	C₂²⋊C₄	—	2	2	no	—
137	C₂²⋊C₄	—	2	4	no	—
138	C₂²⋊C₄	D₄×C₂₂₃	1	2	yes	—
139	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
140	C₂³	—	2	2	no	—
141	C₂³	C₂×D₄×C₂₂₃	1	2	yes	—
142	Q₈	—	4	4	no	—
143	D₄	—	2	2	no	—
144	D₄	—	2	2	no	—
145	D₄	—	4	8	no	—
146	D₄	D₈×C₂₂₃	1	4	yes	complemented
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	4	8	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
154	C₂×C₄	C₂×D₄×C₂₂₃	1	2	yes	—
155	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
156	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
157	C₈	—	2	2	no	—
158	C₈	—	2	1	no	—
159	C₈	#41×C₂₂₃	1	2	yes	complemented
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	4	2	no	—
163	C₂²	—	4	2	no	—
164	C₂²	—	4	2	no	—
165	C₂²	C₂×D₈×C₂₂₃	1	4	yes	—
166	C₂²	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	1	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	4	no	—
171	C₄	C₂×D₈×C₂₂₃	1	2	yes	—
172	C₄	C₂×#41×C₂₂₃	1	1	yes	characteristic
173	C₄	2₋¹⁺⁴×C₂₂₃	1	1	yes	characteristic
174	C₂	—	2	4	no	—
175	C₂	—	8	2	no	—
176	C₂	#547×C₂₂₃	1	1	yes	characteristic
177	C₂	C₂²×D₈×C₂₂₃	1	1	yes	characteristic
178	C₂	#578×C₂₂₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #547×C₂₂₃	C₂	semidirect
2. #500×C₂₂₃	C₂	semidirect
3. #480×C₂₂₃	C₂	semidirect
4. #474×C₂₂₃	C₂	semidirect
5. #466×C₂₂₃	C₂	semidirect
6. C₄×D₈×C₂₂₃	C₂	semidirect
7. C₈×D₄×C₂₂₃	C₂	semidirect
8. C₂×#107×C₂₂₃	C₂	semidirect
56. D₄×C₂₂₃	D₈	semidirect
69. C₈×C₂₂₃	#41	semidirect
89. C₂₂₃	#3334	direct (N × H)
90. #3334	C₂₂₃	direct (N × H)
91. #547	C₂×C₂₂₃	semidirect
92. #500	C₂×C₂₂₃	semidirect
93. #480	C₂×C₂₂₃	semidirect
94. #474	C₂×C₂₂₃	semidirect
95. #466	C₂×C₂₂₃	semidirect
96. C₄×D₈	C₂×C₂₂₃	semidirect
97. C₈×D₄	C₂×C₂₂₃	semidirect
98. C₂×#107	C₂×C₂₂₃	semidirect
146. D₄	D₈×C₂₂₃	semidirect
159. C₈	#41×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#547×C₂₂₃	2
#500×C₂₂₃	2
#480×C₂₂₃	2
#474×C₂₂₃	2
#466×C₂₂₃	2
C₄×D₈×C₂₂₃	2
C₈×D₄×C₂₂₃	2
C₂×#107×C₂₂₃	2
#3334	223

Subgroups — #3334×C223 #43252349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3334×C₂₂₃ #43252349