Subgroups — C₂²×#14583331 #26003084

Order 11008.

2118 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063347	C₂	1	2	yes	maximal, complemented
2	C₂×#14583331	C₂	1	24	yes	maximal, direct product factor
3	C₂²×Dic₃₄₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#13063270	C₂	1	2	yes	maximal, complemented
5	C₂²×D₆₈₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#10621898	—	2	2	no	—
8	C₂×#13063347	—	2	12	no	—
9	C₂×#13063347	C₂²	1	24	yes	complemented
10	C₂²×Dic₁₇₂	C₂²	1	2	yes	—
11	C₂²×#10621873	—	2	2	no	—
12	C₂²×D₃₄₄	C₂²	1	2	yes	—
13	#14583331	C₂²	1	64	yes	direct product factor
14	C₂×Dic₃₄₄	C₂²	1	12	yes	complemented
15	C₂×#13063270	C₂²	1	24	yes	complemented
16	C₂×D₆₈₈	C₂²	1	12	yes	complemented
17	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₄₃	C₂²	1	12	yes	complemented
21	C₂×#10621898	—	2	24	no	—
22	C₂×#10621898	—	4	12	no	—
23	C₂²×Dic₈₆	—	2	2	no	—
24	C₂²×D₁₇₂	—	2	2	no	—
25	C₂²×C₂²⋊D₄₃	—	4	2	no	—
26	C₂²×C₄×D₄₃	—	4	2	no	—
27	#13063347	—	2	48	no	—
28	#13063347	C₂³	1	32	yes	complemented
29	C₂×Dic₁₇₂	—	2	6	no	—
30	C₂×Dic₁₇₂	C₂³	1	12	yes	—
31	C₂×#10621873	—	2	24	no	—
32	C₂×D₃₄₄	—	2	6	no	—
33	C₂×D₃₄₄	C₂³	1	12	yes	—
34	Dic₃₄₄	C₂³	1	16	yes	complemented
35	#13063270	C₂³	1	32	yes	complemented
36	D₆₈₈	C₂³	1	16	yes	complemented
37	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₃	D₄	1	6	yes	—
40	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
42	C₂×C₁₆×C₄₃	C₂³	1	6	yes	—
43	C₂×C₁₆×C₄₃	C₂³	1	6	yes	—
44	C₂×C₁₆×C₄₃	C₂³	1	16	yes	complemented
45	C₂⁴×C₄₃	D₈	1	1	yes	characteristic
46	C₂³×D₄₃	—	4	2	no	—
47	C₂²×C₄×C₄₃	D₈	1	6	yes	—
48	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
51	C₂×C₂²⋊D₄₃	—	4	24	no	—
52	C₂²×Dic₄₃	—	4	2	no	—
53	#10621898	—	2	32	no	—
54	#10621898	—	4	48	no	—
55	C₂×D₁₇₂	—	2	12	no	—
56	C₂×D₁₇₂	—	4	6	no	—
57	C₂×C₄×D₄₃	—	4	24	no	—
58	C₂×Dic₈₆	—	2	12	no	—
59	C₂×Dic₈₆	—	4	6	no	—
60	C₂×C₈×C₄₃	C₂×D₄	1	3	yes	—
61	C₂×C₈×C₄₃	C₂×D₄	1	6	yes	—
62	C₂×C₈×C₄₃	C₂×D₄	1	12	yes	—
63	C₂×C₈×C₄₃	C₂⁴	1	3	yes	—
64	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
65	Dic₁₇₂	—	2	12	no	—
66	Dic₁₇₂	C₂⁴	1	8	yes	—
67	D₃₄₄	—	2	12	no	—
68	D₃₄₄	C₂⁴	1	8	yes	—
69	#10621873	—	2	32	no	—
70	C₁₆×C₄₃	C₂⁴	1	4	yes	—
71	C₁₆×C₄₃	C₂⁴	1	4	yes	—
72	C₂³×C₄₃	—	2	4	no	—
73	C₂³×C₄₃	C₂×D₈	1	6	yes	—
74	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
75	C₂²×D₄₃	—	4	12	no	—
76	C₂²×D₄₃	—	8	8	no	—
77	C₂×C₄×C₄₃	C₂×D₈	1	3	yes	—
78	C₂×C₄×C₄₃	C₂×D₈	1	6	yes	—
79	C₂×C₄×C₄₃	C₂×D₈	1	12	yes	—
80	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
81	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
82	C₂²⋊D₄₃	—	4	32	no	—
83	C₂×Dic₄₃	—	4	12	no	—
84	D₁₇₂	—	2	8	no	—
85	D₁₇₂	—	4	12	no	—
86	C₄×D₄₃	—	4	32	no	—
87	Dic₈₆	—	2	8	no	—
88	Dic₈₆	—	4	12	no	—
89	C₈×C₄₃	C₂²×D₄	1	4	yes	—
90	C₈×C₄₃	C₂²×D₄	1	3	yes	—
91	C₈×C₄₃	C₂⁵	1	1	yes	characteristic
92	C₂²×#508	—	43	1	no	maximal, Sylow
93	C₂²×C₄₃	—	2	12	no	—
94	C₂²×C₄₃	#508	1	4	yes	complemented
95	C₂²×C₄₃	C₂²×D₈	1	4	yes	—
96	C₂²×C₄₃	C₂²×D₈	1	3	yes	—
97	C₂×D₄₃	—	4	8	no	—
98	C₂×D₄₃	—	8	24	no	—
99	C₄×C₄₃	C₂²×D₈	1	3	yes	—
100	C₄×C₄₃	C₂²×D₈	1	4	yes	—
101	C₄×C₄₃	C₂³×D₄	1	1	yes	characteristic
102	Dic₄₃	—	4	8	no	—
103	C₂²×#135	—	43	2	no	—
104	C₂×#508	—	43	24	no	—
105	C₂²×Q₃₂	—	43	1	no	—
106	C₂²×SD₃₂	—	43	2	no	—
107	C₂²×D₁₆	—	43	1	no	—
108	C₂³×C₁₆	D₄₃	1	1	yes	characteristic, complemented
109	C₂×C₄₃	—	2	4	no	—
110	C₂×C₄₃	C₂×#508	1	6	yes	complemented
111	C₂×C₄₃	C₂³×D₈	1	1	yes	characteristic
112	D₄₃	—	8	8	no	—
113	C₂²×#41	—	86	2	no	—
114	C₂×#135	—	43	24	no	—
115	C₂×#135	—	86	12	no	—
116	C₂²×Q₁₆	—	43	2	no	—
117	C₂²×SD₁₆	—	86	2	no	—
118	C₂²×D₈	—	43	2	no	—
119	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
120	#508	—	43	64	no	—
121	C₂×Q₃₂	—	43	12	no	—
122	C₂×SD₃₂	—	43	24	no	—
123	C₂×D₁₆	—	43	12	no	—
124	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
125	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
126	C₂²×C₁₆	C₂×D₄₃	1	12	yes	complemented
127	C₄₃	C₂²×#508	1	1	yes	characteristic, complemented, Sylow
128	C₂×#41	—	86	24	no	—
129	C₂×#41	—	172	12	no	—
130	C₂²×Q₈	—	86	2	no	—
131	C₂²×D₄	—	86	2	no	—
132	C₂²×D₄	—	172	2	no	—
133	C₂³×C₄	—	172	2	no	—
134	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
135	#135	—	43	32	no	—
136	#135	—	86	48	no	—
137	C₂×Q₁₆	—	43	12	no	—
138	C₂×Q₁₆	—	86	6	no	—
139	C₂×SD₁₆	—	86	24	no	—
140	C₂×D₈	—	43	12	no	—
141	C₂×D₈	—	86	6	no	—
142	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
143	C₂²×C₈	D₁₇₂	1	6	yes	—
144	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
145	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
146	Q₃₂	—	43	16	no	—
147	SD₃₂	—	43	32	no	—
148	D₁₆	—	43	16	no	—
149	C₂×C₁₆	C₂²×D₄₃	1	6	yes	—
150	C₂×C₁₆	C₂²×D₄₃	1	6	yes	—
151	C₂×C₁₆	C₂²×D₄₃	1	16	yes	complemented
152	C₂⁴	—	172	2	no	—
153	C₂⁴	D₃₄₄	1	1	yes	characteristic
154	#41	—	86	32	no	—
155	#41	—	172	48	no	—
156	C₂×Q₈	—	86	12	no	—
157	C₂×Q₈	—	172	6	no	—
158	C₂×D₄	—	86	12	no	—
159	C₂×D₄	—	172	24	no	—
160	C₂×D₄	—	172	6	no	—
161	C₂²×C₄	—	172	2	no	—
162	C₂²×C₄	—	172	24	no	—
163	C₂²×C₄	D₃₄₄	1	6	yes	—
164	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₁₇₂	1	6	yes	—
167	Q₁₆	—	43	8	no	—
168	Q₁₆	—	86	12	no	—
169	SD₁₆	—	86	32	no	—
170	D₈	—	43	8	no	—
171	D₈	—	86	12	no	—
172	C₂×C₈	C₂×D₁₇₂	1	3	yes	—
173	C₂×C₈	C₂×D₁₇₂	1	6	yes	—
174	C₂×C₈	C₂×D₁₇₂	1	12	yes	—
175	C₂×C₈	C₂³×D₄₃	1	3	yes	—
176	C₂×C₈	C₂³×D₄₃	1	4	yes	—
177	C₁₆	C₂³×D₄₃	1	4	yes	—
178	C₁₆	C₂³×D₄₃	1	4	yes	—
179	C₂³	—	2	4	no	—
180	C₂³	—	344	8	no	—
181	C₂³	—	172	12	no	—
182	C₂³	C₂×D₃₄₄	1	6	yes	—
183	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
184	Q₈	—	86	8	no	—
185	Q₈	—	172	12	no	—
186	D₄	—	86	8	no	—
187	D₄	—	172	32	no	—
188	D₄	—	172	12	no	—
189	C₂×C₄	—	172	12	no	—
190	C₂×C₄	—	172	32	no	—
191	C₂×C₄	C₂×D₃₄₄	1	6	yes	—
192	C₂×C₄	C₂×D₃₄₄	1	3	yes	—
193	C₂×C₄	C₂×D₃₄₄	1	12	yes	—
194	C₂×C₄	C₂²×D₁₇₂	1	3	yes	—
195	C₂×C₄	C₂²×D₁₇₂	1	4	yes	—
196	C₈	C₂²×D₁₇₂	1	3	yes	—
197	C₈	C₂²×D₁₇₂	1	4	yes	—
198	C₈	C₂⁴×D₄₃	1	1	yes	characteristic
199	C₂²	—	2	12	no	—
200	C₂²	—	344	24	no	—
201	C₂²	—	172	8	no	—
202	C₂²	#14583331	1	4	yes	direct product factor
203	C₂²	C₂²×D₃₄₄	1	4	yes	—
204	C₂²	C₂²×D₃₄₄	1	3	yes	—
205	C₄	—	172	8	no	—
206	C₄	C₂²×D₃₄₄	1	4	yes	—
207	C₄	C₂²×D₃₄₄	1	3	yes	—
208	C₄	C₂³×D₁₇₂	1	1	yes	characteristic
209	C₂	—	2	4	no	—
210	C₂	—	344	8	no	—
211	C₂	C₂×#14583331	1	6	yes	direct product factor
212	C₂	C₂³×D₃₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₃₄₄
commutator, cosocle	C₈×C₄₃	C₂⁵
Frattini	C₈	C₂⁴×D₄₃
Fitting	C₂³×C₁₆×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₈
hypercenter	C₂³×C₁₆	D₄₃
hypocenter	C₄₃	C₂²×#508
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#13063347	C₂	semidirect
2. C₂×#14583331	C₂	direct (N × H)
3. C₂²×Dic₃₄₄	C₂	semidirect
4. C₂²×#13063270	C₂	semidirect
5. C₂²×D₆₈₈	C₂	semidirect
6. C₂³×C₁₆×C₄₃	C₂	semidirect
9. C₂×#13063347	C₂²	semidirect
13. #14583331	C₂²	direct (N × H)
14. C₂×Dic₃₄₄	C₂²	semidirect
15. C₂×#13063270	C₂²	semidirect
16. C₂×D₆₈₈	C₂²	semidirect
20. C₂²×C₁₆×C₄₃	C₂²	semidirect
28. #13063347	C₂³	semidirect
34. Dic₃₄₄	C₂³	semidirect
35. #13063270	C₂³	semidirect
36. D₆₈₈	C₂³	semidirect
44. C₂×C₁₆×C₄₃	C₂³	semidirect
94. C₂²×C₄₃	#508	semidirect
108. C₂³×C₁₆	D₄₃	semidirect
110. C₂×C₄₃	C₂×#508	semidirect
126. C₂²×C₁₆	C₂×D₄₃	semidirect
127. C₄₃	C₂²×#508	semidirect
151. C₂×C₁₆	C₂²×D₄₃	semidirect
202. C₂²	#14583331	direct (N × H)
211. C₂	C₂×#14583331	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×#13063347	2
C₂×#14583331	2
C₂²×Dic₃₄₄	2
C₂²×#13063270	2
C₂²×D₆₈₈	2
C₂³×C₁₆×C₄₃	2
C₂²×#508	43

Subgroups — C22×#14583331 #26003084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14583331 #26003084