Subgroups — C₂²×#14583012 #25960643

Order 11008.

3654 conjugacy classes of subgroups.

The subgroup lattice has 238 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×C₄×D₁₇₂	C₂	1	2	yes	maximal, complemented
3	C₂×#14583012	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13063257	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13063247	C₂	1	2	yes	maximal, complemented
6	C₂²×#97×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₇₂	C₂²	1	2	yes	—
8	C₂³×C₄×D₄₃	—	2	2	no	—
9	C₂²×D₃₄₄	C₄	1	16	yes	complemented
10	C₂²×D₃₄₄	C₂²	1	12	yes	—
11	C₂×C₄×D₁₇₂	—	2	12	no	—
12	C₂×C₄×D₁₇₂	C₂²	1	24	yes	complemented
13	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
14	C₂²×(C₂×D₄₃)⋊C₄	—	2	2	no	—
15	#14583012	C₂²	1	64	yes	direct product factor
16	C₂×#13063257	C₂²	1	12	yes	complemented
17	C₂×#13063247	C₂²	1	24	yes	complemented
18	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×#97×C₄₃	C₂²	1	12	yes	complemented
22	C₂⁴×D₄₃	—	2	2	no	—
23	C₂²×D₁₇₂	—	2	6	no	—
24	C₂²×D₁₇₂	—	2	8	no	—
25	C₂²×D₁₇₂	C₂×C₄	1	16	yes	complemented
26	C₂²×D₁₇₂	C₂³	1	12	yes	—
27	C₂³×Dic₄₃	—	2	2	no	—
28	C₂²×C₄×D₄₃	—	2	24	no	—
29	C₂²×C₄×D₄₃	—	4	16	no	—
30	C₂×D₃₄₄	C₂×C₄	1	96	yes	complemented
31	C₂×D₃₄₄	C₂³	1	16	yes	—
32	C₄×D₁₇₂	—	2	48	no	—
33	C₄×D₁₇₂	C₂³	1	32	yes	complemented
34	C₂×C₄⋊Dic₄₃	—	2	6	no	—
35	C₂×C₄⋊Dic₄₃	C₂³	1	12	yes	—
36	C₂×(C₂×D₄₃)⋊C₄	—	2	24	no	—
37	#13063257	C₂³	1	16	yes	complemented
38	#13063247	C₂³	1	32	yes	complemented
39	C₂³×C₄×C₄₃	D₄	1	2	yes	—
40	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₄₃	—	2	4	no	—
42	C₂²×C₈×C₄₃	C₂×C₄	1	8	yes	complemented
43	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
44	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
45	C₂×C₄²×C₄₃	D₄	1	6	yes	—
46	C₂×C₄²×C₄₃	C₂³	1	6	yes	—
47	#97×C₄₃	C₂³	1	16	yes	complemented
48	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂³×D₄₃	—	2	16	no	—
50	C₂³×D₄₃	—	2	12	no	—
51	C₂³×D₄₃	—	4	8	no	—
52	C₂³×D₄₃	—	4	8	no	—
53	C₂²×C₄×C₄₃	—	2	8	no	—
54	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₄₃	C₂×D₄	1	12	yes	—
56	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₄₃	#41	1	4	yes	—
58	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
59	C₂²×Dic₄₃	—	2	12	no	—
60	C₂²×Dic₄₃	—	4	8	no	—
61	C₂×D₁₇₂	—	2	48	no	—
62	C₂×D₁₇₂	—	2	12	no	—
63	C₂×D₁₇₂	—	2	24	no	—
64	C₂×D₁₇₂	C₂²×C₄	1	48	yes	complemented
65	C₂×D₁₇₂	C₂⁴	1	8	yes	—
66	C₂×C₄×D₄₃	—	2	32	no	—
67	C₂×C₄×D₄₃	—	4	96	no	—
68	C₂×C₈×C₄₃	—	2	12	no	—
69	C₂×C₈×C₄₃	C₂²×C₄	1	24	yes	complemented
70	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
71	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
72	C₄²×C₄₃	C₂×D₄	1	12	yes	—
73	C₄²×C₄₃	C₂⁴	1	4	yes	—
74	(C₂×D₄₃)⋊C₄	—	2	32	no	—
75	C₄⋊Dic₄₃	—	2	12	no	—
76	C₄⋊Dic₄₃	C₂⁴	1	8	yes	—
77	D₃₄₄	C₂²×C₄	1	64	yes	complemented
78	C₂³×C₄₃	C₄×D₄	1	4	yes	—
79	C₂³×C₄₃	AGL(1,ℤ/8)	1	8	yes	—
80	C₂³×C₄₃	C₂²×D₄	1	3	yes	—
81	C₂²×D₄₃	—	2	48	no	—
82	C₂²×D₄₃	—	2	8	no	—
83	C₂²×D₄₃	—	4	24	no	—
84	C₂²×D₄₃	—	4	24	no	—
85	C₂²×D₄₃	—	4	64	no	—
86	C₂×C₄×C₄₃	—	2	24	no	—
87	C₂×C₄×C₄₃	C₄×D₄	1	12	yes	—
88	C₂×C₄×C₄₃	C₂³×C₄	1	6	yes	—
89	C₂×C₄×C₄₃	C₂²×D₄	1	8	yes	—
90	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
91	C₂×C₄×C₄₃	C₂×#41	1	6	yes	—
92	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
93	C₂×Dic₄₃	—	2	8	no	—
94	C₂×Dic₄₃	—	4	24	no	—
95	D₁₇₂	—	2	48	no	—
96	D₁₇₂	—	2	8	no	—
97	D₁₇₂	C₂³×C₄	1	16	yes	complemented
98	C₄×D₄₃	—	4	64	no	—
99	C₈×C₄₃	—	2	4	no	—
100	C₈×C₄₃	C₂³×C₄	1	8	yes	complemented
101	C₂²×#442	—	43	1	no	maximal, Sylow
102	C₂²×C₄₃	#442	1	16	yes	complemented
103	C₂²×C₄₃	C₂×C₄×D₄	1	6	yes	—
104	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	12	yes	—
105	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
106	C₂×D₄₃	—	2	16	no	—
107	C₂×D₄₃	—	4	96	no	—
108	C₂×D₄₃	—	4	8	no	—
109	C₂×D₄₃	—	4	8	no	—
110	C₄×C₄₃	—	2	8	no	—
111	C₄×C₄₃	C₂×C₄×D₄	1	6	yes	—
112	C₄×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
113	C₄×C₄₃	C₂²×#41	1	1	yes	characteristic
114	Dic₄₃	—	4	8	no	—
115	C₂³×D₈	—	43	1	no	—
116	C₂²×C₄×D₄	—	43	2	no	—
117	C₂×#442	—	43	24	no	—
118	C₂²×#106	—	43	1	no	—
119	C₂²×#102	—	43	2	no	—
120	C₂²×#97	D₄₃	1	1	yes	characteristic, complemented
121	C₂×C₄₃	C₂×#442	1	12	yes	complemented
122	C₂×C₄₃	C₂²×C₄×D₄	1	1	yes	characteristic
123	C₂×C₄₃	C₂²×AGL(1,ℤ/8)	1	2	yes	—
124	D₄₃	—	4	16	no	—
125	C₂³×D₄	—	43	2	no	—
126	C₂⁴×C₄	—	86	2	no	—
127	C₂²×D₈	—	43	12	no	—
128	C₂²×D₈	—	43	16	no	—
129	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
130	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
131	C₂×C₄×D₄	—	43	24	no	—
132	C₂×C₄×D₄	—	86	12	no	—
133	C₂²×C₄⋊C₄	—	43	2	no	—
134	C₂²×C₂²⋊C₄	—	86	2	no	—
135	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
136	#442	—	43	64	no	—
137	C₂×#106	—	43	12	no	—
138	C₂×#102	—	43	24	no	—
139	C₂×#97	C₂×D₄₃	1	12	yes	complemented
140	C₄₃	C₂²×#442	1	1	yes	characteristic, complemented, Sylow
141	C₂⁵	—	86	2	no	—
142	C₂²×D₄	—	43	16	no	—
143	C₂²×D₄	—	43	12	no	—
144	C₂²×D₄	—	86	6	no	—
145	C₂²×D₄	—	86	8	no	—
146	C₂³×C₄	—	86	2	no	—
147	C₂³×C₄	—	86	24	no	—
148	C₂³×C₄	—	172	16	no	—
149	C₂³×C₄	D₁₇₂	1	2	yes	—
150	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
151	C₂×D₈	—	43	96	no	—
152	C₂×D₈	—	43	16	no	—
153	C₂²×C₈	—	2	4	no	—
154	C₂²×C₈	C₄×D₄₃	1	8	yes	complemented
155	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
156	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
157	C₄×D₄	—	43	32	no	—
158	C₄×D₄	—	86	48	no	—
159	C₂×C₄⋊C₄	—	43	12	no	—
160	C₂×C₄⋊C₄	—	86	6	no	—
161	C₂×C₂²⋊C₄	—	86	24	no	—
162	C₂×C₄²	D₁₇₂	1	6	yes	—
163	C₂×C₄²	C₂²×D₄₃	1	6	yes	—
164	#106	—	43	16	no	—
165	#102	—	43	32	no	—
166	#97	C₂²×D₄₃	1	16	yes	complemented
167	C₂⁴	—	86	16	no	—
168	C₂⁴	—	86	12	no	—
169	C₂⁴	—	172	8	no	—
170	C₂⁴	—	172	8	no	—
171	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
172	C₂×D₄	—	43	48	no	—
173	C₂×D₄	—	43	8	no	—
174	C₂×D₄	—	86	48	no	—
175	C₂×D₄	—	86	12	no	—
176	C₂×D₄	—	86	24	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	86	12	no	—
179	C₂²×C₄	—	86	32	no	—
180	C₂²×C₄	—	172	96	no	—
181	C₂²×C₄	—	172	8	no	—
182	C₂²×C₄	C₂×C₄×D₄₃	1	4	yes	—
183	C₂²×C₄	C₂×D₁₇₂	1	12	yes	—
184	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
185	C₂²×C₄	#10621898	1	4	yes	—
186	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
187	D₈	—	43	64	no	—
188	C₂×C₈	—	2	12	no	—
189	C₂×C₈	C₂×C₄×D₄₃	1	24	yes	complemented
190	C₂×C₈	C₂³×D₄₃	1	4	yes	—
191	C₂×C₈	C₂³×D₄₃	1	4	yes	—
192	C₄⋊C₄	—	43	8	no	—
193	C₄⋊C₄	—	86	12	no	—
194	C₂²⋊C₄	—	86	32	no	—
195	C₄²	C₂×D₁₇₂	1	12	yes	—
196	C₄²	C₂³×D₄₃	1	4	yes	—
197	C₂³	—	86	48	no	—
198	C₂³	—	86	8	no	—
199	C₂³	—	172	24	no	—
200	C₂³	—	172	24	no	—
201	C₂³	—	172	64	no	—
202	C₂³	C₄×D₁₇₂	1	4	yes	—
203	C₂³	#13063353	1	8	yes	—
204	C₂³	C₂²×D₁₇₂	1	3	yes	—
205	D₄	—	43	16	no	—
206	D₄	—	86	48	no	—
207	D₄	—	86	8	no	—
208	C₂×C₄	—	2	24	no	—
209	C₂×C₄	—	86	8	no	—
210	C₂×C₄	—	172	64	no	—
211	C₂×C₄	—	172	24	no	—
212	C₂×C₄	C₄×D₁₇₂	1	12	yes	—
213	C₂×C₄	C₂²×C₄×D₄₃	1	6	yes	—
214	C₂×C₄	C₂²×D₁₇₂	1	8	yes	—
215	C₂×C₄	C₂²×D₁₇₂	1	3	yes	—
216	C₂×C₄	C₂×#10621898	1	6	yes	—
217	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
218	C₈	—	2	4	no	—
219	C₈	C₂²×C₄×D₄₃	1	8	yes	complemented
220	C₂²	—	86	16	no	—
221	C₂²	—	172	8	no	—
222	C₂²	—	172	8	no	—
223	C₂²	—	172	96	no	—
224	C₂²	#14583012	1	16	yes	direct product factor
225	C₂²	C₂×C₄×D₁₇₂	1	6	yes	—
226	C₂²	C₂×#13063353	1	12	yes	—
227	C₂²	C₂³×D₁₇₂	1	1	yes	characteristic
228	C₄	—	2	8	no	—
229	C₄	—	172	8	no	—
230	C₄	C₂×C₄×D₁₇₂	1	6	yes	—
231	C₄	C₂³×C₄×D₄₃	1	1	yes	characteristic
232	C₄	C₂²×#10621898	1	1	yes	characteristic
233	C₂	—	172	16	no	—
234	C₂	C₂×#14583012	1	12	yes	direct product factor
235	C₂	C₂²×C₄×D₁₇₂	1	1	yes	characteristic
236	C₂	C₂²×#13063353	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₇₂
commutator	C₄×C₄₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂²×#97×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂×D₄
prime radical	C₂³×D₃₄₄	C₂
cosocle	C₂×C₄×C₄₃	C₂⁵
hypercenter	C₂²×#97	D₄₃
hypocenter	C₄₃	C₂²×#442
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₁₇₂	C₂	semidirect
3. C₂×#14583012	C₂	direct (N × H)
4. C₂²×#13063257	C₂	semidirect
5. C₂²×#13063247	C₂	semidirect
6. C₂²×#97×C₄₃	C₂	semidirect
9. C₂²×D₃₄₄	C₄	semidirect
12. C₂×C₄×D₁₇₂	C₂²	semidirect
15. #14583012	C₂²	direct (N × H)
16. C₂×#13063257	C₂²	semidirect
17. C₂×#13063247	C₂²	semidirect
21. C₂×#97×C₄₃	C₂²	semidirect
25. C₂²×D₁₇₂	C₂×C₄	semidirect
30. C₂×D₃₄₄	C₂×C₄	semidirect
33. C₄×D₁₇₂	C₂³	semidirect
37. #13063257	C₂³	semidirect
38. #13063247	C₂³	semidirect
42. C₂²×C₈×C₄₃	C₂×C₄	semidirect
47. #97×C₄₃	C₂³	semidirect
64. C₂×D₁₇₂	C₂²×C₄	semidirect
69. C₂×C₈×C₄₃	C₂²×C₄	semidirect
77. D₃₄₄	C₂²×C₄	semidirect
97. D₁₇₂	C₂³×C₄	semidirect
100. C₈×C₄₃	C₂³×C₄	semidirect
102. C₂²×C₄₃	#442	semidirect
120. C₂²×#97	D₄₃	semidirect
121. C₂×C₄₃	C₂×#442	semidirect
139. C₂×#97	C₂×D₄₃	semidirect
140. C₄₃	C₂²×#442	semidirect
154. C₂²×C₈	C₄×D₄₃	semidirect
166. #97	C₂²×D₄₃	semidirect
189. C₂×C₈	C₂×C₄×D₄₃	semidirect
219. C₈	C₂²×C₄×D₄₃	semidirect
224. C₂²	#14583012	direct (N × H)
234. C₂	C₂×#14583012	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×C₄×D₁₇₂	2
C₂×#14583012	2
C₂²×#13063257	2
C₂²×#13063247	2
C₂²×#97×C₄₃	2
C₂²×#442	43

Subgroups — C22×#14583012 #25960643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14583012 #25960643