Subgroups — C₂²×#14582652 #25065566

Order 11008.

942 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#14582652	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄₃⋊C₈	—	2	2	no	—
5	C₂×C₄⋊(C₄₃⋊C₈)	—	2	6	no	—
6	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	12	yes	—
7	#14582652	C₂²	1	16	yes	direct product factor
8	C₂³×C₈×C₄₃	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₄₃	C₄	1	1	yes	characteristic
10	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₄₃	C₄	1	6	yes	—
12	C₂×C₄×C₈×C₄₃	C₂²	1	6	yes	—
13	C₂²×C₄₃⋊C₈	—	2	12	no	—
14	C₂²×C₄₃⋊C₈	—	4	8	no	—
15	C₄⋊(C₄₃⋊C₈)	—	2	12	no	—
16	C₄⋊(C₄₃⋊C₈)	C₂³	1	8	yes	—
17	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₄₃	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₄₃	—	2	4	no	—
21	C₂²×C₈×C₄₃	C₈	1	8	yes	complemented
22	C₂²×C₈×C₄₃	C₂×C₄	1	6	yes	—
23	C₂²×C₈×C₄₃	C₂×C₄	1	6	yes	—
24	C₂×C₄²×C₄₃	C₂×C₄	1	3	yes	—
25	C₂×C₄²×C₄₃	D₄	1	3	yes	—
26	C₂×C₄²×C₄₃	Q₈	1	3	yes	—
27	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
28	C₄×C₈×C₄₃	C₂×C₄	1	12	yes	—
29	C₄×C₈×C₄₃	C₂³	1	4	yes	—
30	C₂⁴×C₄₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	—	2	4	no	—
32	C₂²×C₄×C₄₃	C₄⋊C₄	1	3	yes	—
33	C₂²×C₄×C₄₃	C₄⋊C₄	1	3	yes	—
34	C₂²×C₄×C₄₃	C₄⋊C₄	1	3	yes	—
35	C₂²×C₄×C₄₃	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₄₃	M₁₆	1	4	yes	—
37	C₂²×C₄×C₄₃	D₈	1	4	yes	—
38	C₂²×C₄×C₄₃	Q₁₆	1	4	yes	—
39	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
40	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
41	C₂²×C₄×C₄₃	C₂×Q₈	1	3	yes	—
42	C₂×C₈×C₄₃	—	2	12	no	—
43	C₂×C₈×C₄₃	C₂×C₈	1	24	yes	complemented
44	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
45	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
46	C₄²×C₄₃	C₄⋊C₄	1	6	yes	—
47	C₄²×C₄₃	C₂²×C₄	1	3	yes	—
48	C₄²×C₄₃	C₂×D₄	1	3	yes	—
49	C₄²×C₄₃	C₂×Q₈	1	3	yes	—
50	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
51	C₂×C₄₃⋊C₈	—	2	8	no	—
52	C₂×C₄₃⋊C₈	—	4	24	no	—
53	C₂³×C₄₃	C₄⋊C₈	1	4	yes	—
54	C₂³×C₄₃	#107	1	4	yes	—
55	C₂³×C₄₃	#108	1	4	yes	—
56	C₂³×C₄₃	C₂×C₄⋊C₄	1	3	yes	—
57	C₂×C₄×C₄₃	—	2	6	no	—
58	C₂×C₄×C₄₃	—	2	6	no	—
59	C₂×C₄×C₄₃	C₄⋊C₈	1	12	yes	—
60	C₂×C₄×C₄₃	#107	1	12	yes	—
61	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	3	yes	—
62	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	3	yes	—
63	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	3	yes	—
64	C₂×C₄×C₄₃	C₂²×C₈	1	6	yes	—
65	C₂×C₄×C₄₃	C₂×M₁₆	1	6	yes	—
66	C₂×C₄×C₄₃	C₂×D₈	1	6	yes	—
67	C₂×C₄×C₄₃	C₂×Q₁₆	1	6	yes	—
68	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
69	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
70	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
71	C₈×C₄₃	—	2	4	no	—
72	C₈×C₄₃	C₂²×C₈	1	8	yes	complemented
73	C₄₃⋊C₈	—	4	8	no	—
74	C₂²×#335	—	43	1	no	maximal, Sylow
75	C₂²×C₄₃	#335	1	16	yes	complemented
76	C₂²×C₄₃	C₂×C₄⋊C₈	1	6	yes	—
77	C₂²×C₄₃	C₂×#107	1	6	yes	—
78	C₂²×C₄₃	C₂×#108	1	6	yes	—
79	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₄₃	—	2	3	no	—
81	C₄×C₄₃	—	2	1	no	—
82	C₄×C₄₃	C₂×C₄⋊C₈	1	6	yes	—
83	C₄×C₄₃	C₂×#107	1	6	yes	—
84	C₄×C₄₃	C₂³×C₈	1	1	yes	characteristic
85	C₄×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
86	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
87	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
88	C₂²×C₄⋊C₈	—	43	2	no	—
89	C₂²×C₄×C₈	D₄₃	1	1	yes	characteristic
90	C₂×#335	—	43	12	no	—
91	C₂×C₄₃	C₂×#335	1	12	yes	complemented
92	C₂×C₄₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
93	C₂×C₄₃	C₂²×#107	1	1	yes	characteristic
94	C₂×C₄₃	C₂²×#108	1	1	yes	characteristic
95	C₂³×C₈	—	86	2	no	—
96	C₂³×C₈	Dic₄₃	1	1	yes	characteristic
97	C₂³×C₈	Dic₄₃	1	1	yes	characteristic
98	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
99	C₂×C₄⋊C₈	—	43	12	no	—
100	C₂×C₄⋊C₈	—	86	6	no	—
101	C₂×C₄×C₈	Dic₄₃	1	6	yes	—
102	C₂×C₄×C₈	C₂×D₄₃	1	6	yes	—
103	#335	—	43	16	no	—
104	C₄₃	C₂²×#335	1	1	yes	characteristic, complemented, Sylow
105	C₂³×C₄	Dic₈₆	1	1	yes	characteristic
106	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
107	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
108	C₂²×C₈	—	2	4	no	—
109	C₂²×C₈	—	86	12	no	—
110	C₂²×C₈	—	172	8	no	—
111	C₂²×C₈	C₄₃⋊C₈	1	8	yes	complemented
112	C₂²×C₈	C₂×Dic₄₃	1	6	yes	—
113	C₂²×C₈	C₂×Dic₄₃	1	6	yes	—
114	C₂×C₄²	Dic₈₆	1	3	yes	—
115	C₂×C₄²	D₁₇₂	1	3	yes	—
116	C₂×C₄²	C₂×Dic₄₃	1	3	yes	—
117	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
118	C₄⋊C₈	—	43	8	no	—
119	C₄⋊C₈	—	86	12	no	—
120	C₄×C₈	C₂×Dic₄₃	1	12	yes	—
121	C₄×C₈	C₂²×D₄₃	1	4	yes	—
122	C₂⁴	C₄⋊Dic₄₃	1	1	yes	characteristic
123	C₂²×C₄	—	2	4	no	—
124	C₂²×C₄	D₃₄₄	1	4	yes	—
125	C₂²×C₄	Dic₁₇₂	1	4	yes	—
126	C₂²×C₄	C₂×C₄₃⋊C₈	1	4	yes	—
127	C₂²×C₄	#10621877	1	4	yes	—
128	C₂²×C₄	C₄⋊Dic₄₃	1	3	yes	—
129	C₂²×C₄	C₄⋊Dic₄₃	1	3	yes	—
130	C₂²×C₄	C₄⋊Dic₄₃	1	3	yes	—
131	C₂²×C₄	C₂×Dic₈₆	1	3	yes	—
132	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
133	C₂²×C₄	C₂²×Dic₄₃	1	3	yes	—
134	C₂×C₈	—	2	12	no	—
135	C₂×C₈	—	86	8	no	—
136	C₂×C₈	—	172	24	no	—
137	C₂×C₈	C₂×C₄₃⋊C₈	1	24	yes	complemented
138	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
139	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
140	C₄²	C₄⋊Dic₄₃	1	6	yes	—
141	C₄²	C₂×Dic₈₆	1	3	yes	—
142	C₄²	C₂×D₁₇₂	1	3	yes	—
143	C₄²	C₂²×Dic₄₃	1	3	yes	—
144	C₄²	C₂³×D₄₃	1	1	yes	characteristic
145	C₂³	C₄⋊(C₄₃⋊C₈)	1	4	yes	—
146	C₂³	#13063259	1	4	yes	—
147	C₂³	#13063261	1	4	yes	—
148	C₂³	C₂×C₄⋊Dic₄₃	1	3	yes	—
149	C₂×C₄	—	2	6	no	—
150	C₂×C₄	—	2	6	no	—
151	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	12	yes	—
152	C₂×C₄	#13063259	1	12	yes	—
153	C₂×C₄	C₂×C₄⋊Dic₄₃	1	3	yes	—
154	C₂×C₄	C₂×C₄⋊Dic₄₃	1	3	yes	—
155	C₂×C₄	C₂×C₄⋊Dic₄₃	1	3	yes	—
156	C₂×C₄	C₂²×C₄₃⋊C₈	1	6	yes	—
157	C₂×C₄	C₂×#10621877	1	6	yes	—
158	C₂×C₄	C₂×D₃₄₄	1	6	yes	—
159	C₂×C₄	C₂×Dic₁₇₂	1	6	yes	—
160	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
161	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
162	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
163	C₈	—	2	4	no	—
164	C₈	—	172	8	no	—
165	C₈	C₂²×C₄₃⋊C₈	1	8	yes	complemented
166	C₂²	#14582652	1	16	yes	direct product factor
167	C₂²	C₂×C₄⋊(C₄₃⋊C₈)	1	6	yes	—
168	C₂²	C₂×#13063259	1	6	yes	—
169	C₂²	C₂×#13063261	1	6	yes	—
170	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
171	C₄	—	2	3	no	—
172	C₄	—	2	1	no	—
173	C₄	C₂×C₄⋊(C₄₃⋊C₈)	1	6	yes	—
174	C₄	C₂×#13063259	1	6	yes	—
175	C₄	C₂³×C₄₃⋊C₈	1	1	yes	characteristic
176	C₄	C₂²×#10621877	1	1	yes	characteristic
177	C₄	C₂²×D₃₄₄	1	1	yes	characteristic
178	C₄	C₂²×Dic₁₇₂	1	1	yes	characteristic
179	C₂	C₂×#14582652	1	12	yes	direct product factor
180	C₂	C₂²×C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
181	C₂	C₂²×#13063259	1	1	yes	characteristic
182	C₂	C₂²×#13063261	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₁₇₂
commutator	C₄×C₄₃	C₂³×C₈
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂²×C₄×C₈×C₄₃	C₂
socle, prime radical	C₂⁴×C₄₃	C₄⋊C₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₄₃
hypocenter	C₄₃	C₂²×#335
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#14582652	C₂	direct (N × H)
7. #14582652	C₂²	direct (N × H)
21. C₂²×C₈×C₄₃	C₈	semidirect
43. C₂×C₈×C₄₃	C₂×C₈	semidirect
72. C₈×C₄₃	C₂²×C₈	semidirect
75. C₂²×C₄₃	#335	semidirect
91. C₂×C₄₃	C₂×#335	semidirect
104. C₄₃	C₂²×#335	semidirect
111. C₂²×C₈	C₄₃⋊C₈	semidirect
137. C₂×C₈	C₂×C₄₃⋊C₈	semidirect
165. C₈	C₂²×C₄₃⋊C₈	semidirect
166. C₂²	#14582652	direct (N × H)
179. C₂	C₂×#14582652	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₄₃⋊C₈)	2
C₂×#14582652	2
C₂²×C₄×C₈×C₄₃	2
C₂²×#335	43

Subgroups — C22×#14582652 #25065566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14582652 #25065566