Subgroups — #3487×C₂₂₃ #43252502

Order 28544.

1536 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×2₋¹⁺⁴×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#41×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#550×C₂₂₃	C₂	1	8	yes	maximal, complemented
4	#548×C₂₂₃	C₂	1	8	yes	maximal, complemented
5	#535×C₂₂₃	C₂	1	6	yes	maximal, complemented
6	#525×C₂₂₃	C₂	1	6	yes	maximal, complemented
7	#519×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	2₋¹⁺⁴×C₂₂₃	—	2	8	no	—
9	C₂×#41×C₂₂₃	—	2	8	no	—
10	C₂×#41×C₂₂₃	C₂²	1	6	yes	—
11	C₂×#41×C₂₂₃	C₂²	1	8	yes	—
12	C₂×#41×C₂₂₃	C₂²	1	4	yes	complemented
13	C₂²×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₂₂₃	C₂²	1	4	yes	complemented
15	C₂²×D₄×C₂₂₃	C₂²	1	3	yes	—
16	C₂³×C₄×C₂₂₃	C₂²	1	3	yes	—
17	#127×C₂₂₃	C₂²	1	12	yes	complemented
18	#124×C₂₂₃	C₂²	1	12	yes	complemented
19	#123×C₂₂₃	C₂²	1	4	yes	—
20	C₂²⋊Q₈×C₂₂₃	C₂²	1	12	yes	complemented
21	#121×C₂₂₃	C₂²	1	12	yes	complemented
22	#121×C₂₂₃	C₂²	1	24	yes	complemented
23	#120×C₂₂₃	C₂²	1	12	yes	—
24	C₄×Q₈×C₂₂₃	C₂²	1	8	yes	complemented
25	C₄×D₄×C₂₂₃	C₂²	1	24	yes	complemented
26	#117×C₂₂₃	C₂²	1	6	yes	—
27	C₂⁴×C₂₂₃	C₂³	1	3	yes	—
28	#41×C₂₂₃	—	2	32	no	—
29	#41×C₂₂₃	—	2	16	no	—
30	#41×C₂₂₃	—	2	24	no	—
31	C₂×Q₈×C₂₂₃	—	2	16	no	—
32	C₂×Q₈×C₂₂₃	D₄	1	8	yes	complemented
33	C₂×Q₈×C₂₂₃	C₂³	1	6	yes	—
34	C₂×Q₈×C₂₂₃	C₂³	1	4	yes	—
35	C₂×D₄×C₂₂₃	—	2	12	no	—
36	C₂×D₄×C₂₂₃	C₂³	1	4	yes	—
37	C₂×D₄×C₂₂₃	C₂³	1	12	yes	—
38	C₂×D₄×C₂₂₃	C₂³	1	6	yes	—
39	C₂×D₄×C₂₂₃	C₂³	1	24	yes	—
40	C₂²×C₄×C₂₂₃	—	2	12	no	—
41	C₂²×C₄×C₂₂₃	C₂³	1	12	yes	—
42	C₂²×C₄×C₂₂₃	C₂³	1	12	yes	—
43	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₂₃	C₂³	1	3	yes	—
45	C₄⋊C₄×C₂₂₃	C₂³	1	12	yes	—
46	C₄⋊C₄×C₂₂₃	C₂³	1	8	yes	—
47	C₂²⋊C₄×C₂₂₃	C₂³	1	4	yes	—
48	C₂²⋊C₄×C₂₂₃	C₂³	1	8	yes	—
49	C₂²⋊C₄×C₂₂₃	C₂³	1	24	yes	—
50	C₄²×C₂₂₃	C₂³	1	12	yes	—
51	C₂³×C₂₂₃	—	2	6	no	—
52	C₂³×C₂₂₃	—	4	6	no	—
53	C₂³×C₂₂₃	C₂⁴	1	6	yes	—
54	C₂³×C₂₂₃	C₂⁴	1	4	yes	—
55	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
56	Q₈×C₂₂₃	—	2	12	no	—
57	Q₈×C₂₂₃	C₂×D₄	1	16	yes	—
58	D₄×C₂₂₃	—	2	8	no	—
59	D₄×C₂₂₃	—	2	24	no	—
60	D₄×C₂₂₃	—	2	12	no	—
61	D₄×C₂₂₃	—	4	24	no	—
62	C₂×C₄×C₂₂₃	—	2	2	no	—
63	C₂×C₄×C₂₂₃	—	2	24	no	—
64	C₂×C₄×C₂₂₃	—	2	24	no	—
65	C₂×C₄×C₂₂₃	C₂×D₄	1	12	yes	—
66	C₂×C₄×C₂₂₃	C₂⁴	1	8	yes	—
67	C₂×C₄×C₂₂₃	C₂⁴	1	4	yes	—
68	C₂×C₄×C₂₂₃	C₂⁴	1	2	yes	—
69	C₂×C₄×C₂₂₃	C₂⁴	1	6	yes	—
70	C₂²×C₂₂₃	—	2	2	no	—
71	C₂²×C₂₂₃	—	2	4	no	—
72	C₂²×C₂₂₃	—	2	6	no	—
73	C₂²×C₂₂₃	—	2	8	no	—
74	C₂²×C₂₂₃	—	2	12	no	—
75	C₂²×C₂₂₃	—	4	12	no	—
76	C₂²×C₂₂₃	C₂²×D₄	1	2	yes	—
77	C₂²×C₂₂₃	C₂⁵	1	1	yes	characteristic
78	C₄×C₂₂₃	—	2	2	no	—
79	C₄×C₂₂₃	—	2	4	no	—
80	C₄×C₂₂₃	—	2	8	no	—
81	C₄×C₂₂₃	C₂²×D₄	1	12	yes	—
82	C₂×C₂₂₃	—	2	2	no	—
83	C₂×C₂₂₃	—	4	4	no	—
84	C₂×C₂₂₃	—	4	6	no	—
85	C₂×C₂₂₃	C₂³×D₄	1	1	yes	characteristic
86	C₂×C₂₂₃	#585	1	2	yes	—
87	C₂₂₃	#3487	1	1	yes	characteristic, direct product factor, Sylow
88	#3487	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂×2₋¹⁺⁴	C₂×C₂₂₃	1	1	yes	characteristic, complemented
90	C₂²×#41	C₂×C₂₂₃	1	1	yes	characteristic, complemented
91	#550	C₂×C₂₂₃	1	8	yes	complemented
92	#548	C₂×C₂₂₃	1	8	yes	complemented
93	#535	C₂×C₂₂₃	1	6	yes	complemented
94	#525	C₂×C₂₂₃	1	6	yes	complemented
95	#519	C₂×C₂₂₃	1	1	yes	characteristic, complemented
96	2₋¹⁺⁴	—	2	8	no	—
97	C₂×#41	—	2	8	no	—
98	C₂×#41	C₂²×C₂₂₃	1	6	yes	—
99	C₂×#41	C₂²×C₂₂₃	1	8	yes	—
100	C₂×#41	C₂²×C₂₂₃	1	4	yes	complemented
101	C₂²×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
102	C₂²×Q₈	C₂²×C₂₂₃	1	4	yes	complemented
103	C₂²×D₄	C₂²×C₂₂₃	1	3	yes	—
104	C₂³×C₄	C₂²×C₂₂₃	1	3	yes	—
105	#127	C₂²×C₂₂₃	1	12	yes	complemented
106	#124	C₂²×C₂₂₃	1	12	yes	complemented
107	#123	C₂²×C₂₂₃	1	4	yes	—
108	C₂²⋊Q₈	C₂²×C₂₂₃	1	12	yes	complemented
109	#121	C₂²×C₂₂₃	1	12	yes	complemented
110	#121	C₂²×C₂₂₃	1	24	yes	complemented
111	#120	C₂²×C₂₂₃	1	12	yes	—
112	C₄×Q₈	C₂²×C₂₂₃	1	8	yes	complemented
113	C₄×D₄	C₂²×C₂₂₃	1	24	yes	complemented
114	#117	C₂²×C₂₂₃	1	6	yes	—
115	C₂⁴	C₂³×C₂₂₃	1	3	yes	—
116	#41	—	2	32	no	—
117	#41	—	2	16	no	—
118	#41	—	2	24	no	—
119	C₂×Q₈	—	2	16	no	—
120	C₂×Q₈	D₄×C₂₂₃	1	8	yes	complemented
121	C₂×Q₈	C₂³×C₂₂₃	1	6	yes	—
122	C₂×Q₈	C₂³×C₂₂₃	1	4	yes	—
123	C₂×D₄	—	2	12	no	—
124	C₂×D₄	C₂³×C₂₂₃	1	4	yes	—
125	C₂×D₄	C₂³×C₂₂₃	1	12	yes	—
126	C₂×D₄	C₂³×C₂₂₃	1	6	yes	—
127	C₂×D₄	C₂³×C₂₂₃	1	24	yes	—
128	C₂²×C₄	—	2	12	no	—
129	C₂²×C₄	C₂³×C₂₂₃	1	12	yes	—
130	C₂²×C₄	C₂³×C₂₂₃	1	12	yes	—
131	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
132	C₂²×C₄	C₂³×C₂₂₃	1	3	yes	—
133	C₄⋊C₄	C₂³×C₂₂₃	1	12	yes	—
134	C₄⋊C₄	C₂³×C₂₂₃	1	8	yes	—
135	C₂²⋊C₄	C₂³×C₂₂₃	1	4	yes	—
136	C₂²⋊C₄	C₂³×C₂₂₃	1	8	yes	—
137	C₂²⋊C₄	C₂³×C₂₂₃	1	24	yes	—
138	C₄²	C₂³×C₂₂₃	1	12	yes	—
139	C₂³	—	2	6	no	—
140	C₂³	—	4	6	no	—
141	C₂³	C₂⁴×C₂₂₃	1	6	yes	—
142	C₂³	C₂⁴×C₂₂₃	1	4	yes	—
143	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
144	Q₈	—	2	12	no	—
145	Q₈	C₂×D₄×C₂₂₃	1	16	yes	—
146	D₄	—	2	8	no	—
147	D₄	—	2	24	no	—
148	D₄	—	2	12	no	—
149	D₄	—	4	24	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	24	no	—
152	C₂×C₄	—	2	24	no	—
153	C₂×C₄	C₂×D₄×C₂₂₃	1	12	yes	—
154	C₂×C₄	C₂⁴×C₂₂₃	1	8	yes	—
155	C₂×C₄	C₂⁴×C₂₂₃	1	4	yes	—
156	C₂×C₄	C₂⁴×C₂₂₃	1	2	yes	—
157	C₂×C₄	C₂⁴×C₂₂₃	1	6	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	—	2	4	no	—
160	C₂²	—	2	6	no	—
161	C₂²	—	2	8	no	—
162	C₂²	—	2	12	no	—
163	C₂²	—	4	12	no	—
164	C₂²	C₂²×D₄×C₂₂₃	1	2	yes	—
165	C₂²	C₂⁵×C₂₂₃	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	2	4	no	—
168	C₄	—	2	8	no	—
169	C₄	C₂²×D₄×C₂₂₃	1	12	yes	—
170	C₂	—	2	2	no	—
171	C₂	—	4	6	no	—
172	C₂	—	4	4	no	—
173	C₂	C₂³×D₄×C₂₂₃	1	1	yes	characteristic
174	C₂	#585×C₂₂₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×2₋¹⁺⁴×C₂₂₃	C₂	semidirect
2. C₂²×#41×C₂₂₃	C₂	semidirect
3. #550×C₂₂₃	C₂	semidirect
4. #548×C₂₂₃	C₂	semidirect
5. #535×C₂₂₃	C₂	semidirect
6. #525×C₂₂₃	C₂	semidirect
7. #519×C₂₂₃	C₂	semidirect
12. C₂×#41×C₂₂₃	C₂²	semidirect
14. C₂²×Q₈×C₂₂₃	C₂²	semidirect
17. #127×C₂₂₃	C₂²	semidirect
18. #124×C₂₂₃	C₂²	semidirect
20. C₂²⋊Q₈×C₂₂₃	C₂²	semidirect
21. #121×C₂₂₃	C₂²	semidirect
22. #121×C₂₂₃	C₂²	semidirect
24. C₄×Q₈×C₂₂₃	C₂²	semidirect
25. C₄×D₄×C₂₂₃	C₂²	semidirect
32. C₂×Q₈×C₂₂₃	D₄	semidirect
87. C₂₂₃	#3487	direct (N × H)
88. #3487	C₂₂₃	direct (N × H)
89. C₂×2₋¹⁺⁴	C₂×C₂₂₃	semidirect
90. C₂²×#41	C₂×C₂₂₃	semidirect
91. #550	C₂×C₂₂₃	semidirect
92. #548	C₂×C₂₂₃	semidirect
93. #535	C₂×C₂₂₃	semidirect
94. #525	C₂×C₂₂₃	semidirect
95. #519	C₂×C₂₂₃	semidirect
100. C₂×#41	C₂²×C₂₂₃	semidirect
102. C₂²×Q₈	C₂²×C₂₂₃	semidirect
105. #127	C₂²×C₂₂₃	semidirect
106. #124	C₂²×C₂₂₃	semidirect
108. C₂²⋊Q₈	C₂²×C₂₂₃	semidirect
109. #121	C₂²×C₂₂₃	semidirect
110. #121	C₂²×C₂₂₃	semidirect
112. C₄×Q₈	C₂²×C₂₂₃	semidirect
113. C₄×D₄	C₂²×C₂₂₃	semidirect
120. C₂×Q₈	D₄×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×2₋¹⁺⁴×C₂₂₃	2
C₂²×#41×C₂₂₃	2
#550×C₂₂₃	2
#548×C₂₂₃	2
#535×C₂₂₃	2
#525×C₂₂₃	2
#519×C₂₂₃	2
#3487	223

Subgroups — #3487×C223 #43252502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3487×C₂₂₃ #43252502