Subgroups — C₂²×#14582701 #25249470

Order 11008.

1470 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	C₂³×D₃₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13063259	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582701	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₁₇₂	—	2	1	no	—
6	C₂²×D₃₄₄	—	2	3	no	—
7	C₂²×D₃₄₄	—	2	4	no	—
8	C₂²×D₃₄₄	C₄	1	8	yes	complemented
9	C₂²×D₃₄₄	C₂²	1	6	yes	—
10	C₂²×C₄⋊Dic₄₃	—	2	1	no	—
11	C₂×#13063259	—	2	3	no	—
12	C₂×#13063259	C₂²	1	6	yes	complemented
13	#14582701	C₂²	1	16	yes	direct product factor
14	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₄₃	—	2	4	no	—
16	C₂²×C₁₆×C₄₃	C₂²	1	6	yes	complemented
17	C₂⁴×D₄₃	—	4	1	no	—
18	C₂²×D₁₇₂	—	2	8	no	—
19	C₂²×D₁₇₂	—	2	6	no	—
20	C₂²×D₁₇₂	—	4	3	no	—
21	C₂²×D₁₇₂	—	4	4	no	—
22	C₂³×Dic₄₃	—	4	1	no	—
23	C₂×D₃₄₄	—	2	24	no	—
24	C₂×D₃₄₄	—	2	6	no	—
25	C₂×D₃₄₄	—	2	12	no	—
26	C₂×D₃₄₄	C₂×C₄	1	24	yes	complemented
27	C₂×D₃₄₄	C₂³	1	4	yes	—
28	C₂×C₄⋊Dic₄₃	—	2	6	no	—
29	C₂×C₄⋊Dic₄₃	—	4	3	no	—
30	#13063259	—	2	6	no	—
31	#13063259	C₂³	1	4	yes	complemented
32	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₄₃	D₄	1	3	yes	—
35	C₂²×C₈×C₄₃	D₄	1	4	yes	—
36	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
37	C₂×C₁₆×C₄₃	—	2	12	no	—
38	C₂×C₁₆×C₄₃	C₂³	1	4	yes	complemented
39	C₂⁴×C₄₃	D₈	1	1	yes	characteristic
40	C₂³×D₄₃	—	4	8	no	—
41	C₂³×D₄₃	—	4	6	no	—
42	C₂³×D₄₃	—	8	4	no	—
43	C₂³×D₄₃	—	8	4	no	—
44	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
45	C₂²×C₄×C₄₃	D₈	1	3	yes	—
46	C₂²×C₄×C₄₃	SD₁₆	1	4	yes	—
47	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
48	C₂²×Dic₄₃	—	4	6	no	—
49	C₂²×Dic₄₃	—	8	4	no	—
50	C₂×D₁₇₂	—	2	24	no	—
51	C₂×D₁₇₂	—	2	4	no	—
52	C₂×D₁₇₂	—	4	24	no	—
53	C₂×D₁₇₂	—	4	6	no	—
54	C₂×D₁₇₂	—	4	12	no	—
55	C₂×C₈×C₄₃	C₂²⋊C₄	1	12	yes	—
56	C₂×C₈×C₄₃	C₂²×C₄	1	6	yes	—
57	C₂×C₈×C₄₃	C₂×D₄	1	3	yes	—
58	C₂×C₈×C₄₃	C₂×D₄	1	6	yes	—
59	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
60	C₄⋊Dic₄₃	—	2	4	no	—
61	C₄⋊Dic₄₃	—	4	6	no	—
62	D₃₄₄	—	2	24	no	—
63	D₃₄₄	—	2	4	no	—
64	D₃₄₄	C₂²×C₄	1	8	yes	complemented
65	C₁₆×C₄₃	—	2	4	no	—
66	C₂³×C₄₃	#102	1	4	yes	—
67	C₂³×C₄₃	D₁₆	1	4	yes	—
68	C₂³×C₄₃	SD₃₂	1	4	yes	—
69	C₂³×C₄₃	C₂×D₈	1	3	yes	—
70	C₂²×D₄₃	—	4	4	no	—
71	C₂²×D₄₃	—	4	24	no	—
72	C₂²×D₄₃	—	8	32	no	—
73	C₂²×D₄₃	—	8	12	no	—
74	C₂²×D₄₃	—	8	12	no	—
75	C₂×C₄×C₄₃	#102	1	12	yes	—
76	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	6	yes	—
77	C₂×C₄×C₄₃	C₂×D₈	1	3	yes	—
78	C₂×C₄×C₄₃	C₂×SD₁₆	1	6	yes	—
79	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
80	C₂×Dic₄₃	—	4	4	no	—
81	C₂×Dic₄₃	—	8	12	no	—
82	D₁₇₂	—	2	8	no	—
83	D₁₇₂	—	4	4	no	—
84	D₁₇₂	—	4	24	no	—
85	C₈×C₄₃	C₂×C₂²⋊C₄	1	6	yes	—
86	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
87	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
88	C₂²×#357	—	43	1	no	maximal, Sylow
89	C₂²×C₄₃	#357	1	16	yes	complemented
90	C₂²×C₄₃	C₂×#102	1	6	yes	—
91	C₂²×C₄₃	C₂×D₁₆	1	6	yes	—
92	C₂²×C₄₃	C₂×SD₃₂	1	6	yes	—
93	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
94	C₂×D₄₃	—	4	8	no	—
95	C₂×D₄₃	—	8	4	no	—
96	C₂×D₄₃	—	8	4	no	—
97	C₂×D₄₃	—	8	48	no	—
98	C₄×C₄₃	C₂×#102	1	6	yes	—
99	C₄×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
100	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
101	Dic₄₃	—	8	4	no	—
102	C₂³×D₈	—	43	1	no	—
103	C₂³×C₁₆	D₄₃	1	1	yes	characteristic, complemented
104	C₂²×#107	—	43	1	no	—
105	C₂×#357	—	43	12	no	—
106	C₂×C₄₃	C₂×#357	1	12	yes	complemented
107	C₂×C₄₃	C₂²×#102	1	1	yes	characteristic
108	C₂×C₄₃	C₂²×D₁₆	1	1	yes	characteristic
109	C₂×C₄₃	C₂²×SD₃₂	1	1	yes	characteristic
110	D₄₃	—	8	8	no	—
111	C₂³×D₄	—	86	1	no	—
112	C₂²×D₈	—	43	6	no	—
113	C₂²×D₈	—	43	8	no	—
114	C₂²×D₈	—	86	3	no	—
115	C₂²×D₈	—	86	4	no	—
116	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
117	C₂²×C₄⋊C₄	—	86	1	no	—
118	C₂²×C₁₆	—	2	4	no	—
119	C₂²×C₁₆	C₂×D₄₃	1	6	yes	complemented
120	C₂×#107	—	43	6	no	—
121	C₂×#107	—	86	3	no	—
122	#357	—	43	16	no	—
123	C₄₃	C₂²×#357	1	1	yes	characteristic, complemented, Sylow
124	C₂⁵	—	172	1	no	—
125	C₂²×D₄	—	86	8	no	—
126	C₂²×D₄	—	86	6	no	—
127	C₂²×D₄	—	172	4	no	—
128	C₂²×D₄	—	172	3	no	—
129	C₂³×C₄	—	172	1	no	—
130	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
131	C₂×D₈	—	43	24	no	—
132	C₂×D₈	—	43	4	no	—
133	C₂×D₈	—	86	24	no	—
134	C₂×D₈	—	86	6	no	—
135	C₂×D₈	—	86	12	no	—
136	C₂²×C₈	C₄×D₄₃	1	4	yes	—
137	C₂²×C₈	D₁₇₂	1	3	yes	—
138	C₂²×C₈	C₂²⋊D₄₃	1	4	yes	—
139	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
140	C₂×C₄⋊C₄	—	86	6	no	—
141	C₂×C₄⋊C₄	—	172	3	no	—
142	C₂×C₁₆	—	2	12	no	—
143	C₂×C₁₆	C₂²×D₄₃	1	4	yes	complemented
144	#107	—	43	4	no	—
145	#107	—	86	6	no	—
146	C₂⁴	—	344	4	no	—
147	C₂⁴	—	344	4	no	—
148	C₂⁴	—	172	8	no	—
149	C₂⁴	—	172	6	no	—
150	C₂⁴	D₃₄₄	1	1	yes	characteristic
151	C₂×D₄	—	86	24	no	—
152	C₂×D₄	—	86	4	no	—
153	C₂×D₄	—	172	24	no	—
154	C₂×D₄	—	172	6	no	—
155	C₂×D₄	—	172	12	no	—
156	C₂²×C₄	—	344	4	no	—
157	C₂²×C₄	—	172	6	no	—
158	C₂²×C₄	#10621873	1	4	yes	—
159	C₂²×C₄	D₃₄₄	1	3	yes	—
160	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	4	yes	—
161	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
162	D₈	—	43	8	no	—
163	D₈	—	86	24	no	—
164	D₈	—	86	4	no	—
165	C₂×C₈	(C₂×D₄₃)⋊C₄	1	12	yes	—
166	C₂×C₈	C₂×C₄×D₄₃	1	6	yes	—
167	C₂×C₈	C₂×D₁₇₂	1	3	yes	—
168	C₂×C₈	C₂×C₂²⋊D₄₃	1	6	yes	—
169	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
170	C₄⋊C₄	—	86	4	no	—
171	C₄⋊C₄	—	172	6	no	—
172	C₁₆	—	2	4	no	—
173	C₂³	—	344	12	no	—
174	C₂³	—	344	12	no	—
175	C₂³	—	344	32	no	—
176	C₂³	—	172	24	no	—
177	C₂³	—	172	4	no	—
178	C₂³	#13063247	1	4	yes	—
179	C₂³	D₆₈₈	1	4	yes	—
180	C₂³	#13063270	1	4	yes	—
181	C₂³	C₂×D₃₄₄	1	3	yes	—
182	D₄	—	86	8	no	—
183	D₄	—	172	24	no	—
184	D₄	—	172	4	no	—
185	C₂×C₄	—	344	12	no	—
186	C₂×C₄	—	172	4	no	—
187	C₂×C₄	#13063247	1	12	yes	—
188	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	6	yes	—
189	C₂×C₄	C₂×D₃₄₄	1	3	yes	—
190	C₂×C₄	C₂×#10621873	1	6	yes	—
191	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
192	C₈	C₂×(C₂×D₄₃)⋊C₄	1	6	yes	—
193	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
194	C₈	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
195	C₂²	—	344	48	no	—
196	C₂²	—	344	4	no	—
197	C₂²	—	344	4	no	—
198	C₂²	—	172	8	no	—
199	C₂²	#14582701	1	16	yes	direct product factor
200	C₂²	C₂×#13063247	1	6	yes	—
201	C₂²	C₂×D₆₈₈	1	6	yes	—
202	C₂²	C₂×#13063270	1	6	yes	—
203	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
204	C₄	—	344	4	no	—
205	C₄	C₂×#13063247	1	6	yes	—
206	C₄	C₂²×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
207	C₄	C₂²×#10621873	1	1	yes	characteristic
208	C₂	—	344	8	no	—
209	C₂	C₂×#14582701	1	12	yes	direct product factor
210	C₂	C₂²×#13063247	1	1	yes	characteristic
211	C₂	C₂²×D₆₈₈	1	1	yes	characteristic
212	C₂	C₂²×#13063270	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₃₄₄
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂³×C₁₆×C₄₃	C₂
socle	C₂⁴×C₄₃	D₈
prime radical	C₂³×D₃₄₄	C₂
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂³×C₁₆	D₄₃
hypocenter	C₄₃	C₂²×#357
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂²×#13063259	C₂	semidirect
3. C₂×#14582701	C₂	direct (N × H)
4. C₂³×C₁₆×C₄₃	C₂	semidirect
8. C₂²×D₃₄₄	C₄	semidirect
12. C₂×#13063259	C₂²	semidirect
13. #14582701	C₂²	direct (N × H)
16. C₂²×C₁₆×C₄₃	C₂²	semidirect
26. C₂×D₃₄₄	C₂×C₄	semidirect
31. #13063259	C₂³	semidirect
38. C₂×C₁₆×C₄₃	C₂³	semidirect
64. D₃₄₄	C₂²×C₄	semidirect
89. C₂²×C₄₃	#357	semidirect
103. C₂³×C₁₆	D₄₃	semidirect
106. C₂×C₄₃	C₂×#357	semidirect
119. C₂²×C₁₆	C₂×D₄₃	semidirect
123. C₄₃	C₂²×#357	semidirect
143. C₂×C₁₆	C₂²×D₄₃	semidirect
199. C₂²	#14582701	direct (N × H)
209. C₂	C₂×#14582701	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₃₄₄	2
C₂²×#13063259	2
C₂×#14582701	2
C₂³×C₁₆×C₄₃	2
C₂²×#357	43

Subgroups — C22×#14582701 #25249470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14582701 #25249470