Subgroups — C₂²×#14583272 #25965995

Order 11008.

3950 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#10621873	C₂	1	4	yes	maximal, complemented
2	C₂²×#13063325	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13063322	C₂	1	1	yes	maximal, characteristic
4	C₂×#14583272	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×Dic₈₆	C₂²	1	2	yes	—
7	C₂³×D₁₇₂	—	2	2	no	—
8	C₂³×D₁₇₂	C₂²	1	2	yes	—
9	C₂²×#10621873	—	2	32	no	—
10	C₂²×#10621873	C₂²	1	48	yes	complemented
11	C₂×#13063325	—	2	6	no	—
12	C₂×#13063325	C₂²	1	12	yes	complemented
13	C₂×#13063322	—	2	6	no	—
14	C₂×#13063322	C₂²	1	12	yes	—
15	C₂²×C₄⋊Dic₄₃	—	2	2	no	—
16	#14583272	C₂²	1	64	yes	direct product factor
17	C₂³×C₈×C₄₃	C₂²	1	2	yes	—
18	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₄₃	C₂²	1	12	yes	complemented
20	C₂⁴×D₄₃	—	2	2	no	—
21	C₂²×Dic₈₆	—	2	16	no	—
22	C₂²×Dic₈₆	—	2	6	no	—
23	C₂²×Dic₈₆	C₂³	1	12	yes	—
24	C₂²×D₁₇₂	—	2	16	no	—
25	C₂²×D₁₇₂	—	2	6	no	—
26	C₂²×D₁₇₂	—	2	24	no	—
27	C₂²×D₁₇₂	—	4	16	no	—
28	C₂²×D₁₇₂	C₂³	1	12	yes	—
29	C₂³×Dic₄₃	—	2	2	no	—
30	C₂×#10621873	—	2	192	no	—
31	C₂×#10621873	C₂³	1	64	yes	complemented
32	#13063325	—	2	24	no	—
33	#13063325	C₂³	1	16	yes	complemented
34	#13063322	—	2	24	no	—
35	#13063322	C₂³	1	16	yes	—
36	C₂×C₄⋊Dic₄₃	—	2	24	no	—
37	C₂³×C₄×C₄₃	D₄	1	2	yes	—
38	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₄₃	D₄	1	16	yes	complemented
40	C₂²×C₈×C₄₃	C₂³	1	12	yes	—
41	C₂×C₄²×C₄₃	D₄	1	6	yes	—
42	C₂×C₄²×C₄₃	C₂³	1	6	yes	—
43	C₄×C₈×C₄₃	C₂³	1	16	yes	complemented
44	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
45	C₂³×D₄₃	—	2	12	no	—
46	C₂³×D₄₃	—	4	8	no	—
47	C₂³×D₄₃	—	4	16	no	—
48	C₂²×C₄×C₄₃	SD₁₆	1	16	yes	complemented
49	C₂²×C₄×C₄₃	C₂×D₄	1	8	yes	—
50	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
51	C₂²×C₄×C₄₃	C₂×D₄	1	12	yes	—
52	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
53	C₂²×Dic₄₃	—	2	12	no	—
54	C₂²×Dic₄₃	—	4	8	no	—
55	C₂×D₁₇₂	—	2	12	no	—
56	C₂×D₁₇₂	—	2	32	no	—
57	C₂×D₁₇₂	—	2	48	no	—
58	C₂×D₁₇₂	—	4	24	no	—
59	C₂×D₁₇₂	—	4	96	no	—
60	C₂×D₁₇₂	C₂⁴	1	8	yes	—
61	C₂×Dic₈₆	—	2	48	no	—
62	C₂×Dic₈₆	—	2	12	no	—
63	C₂×Dic₈₆	—	4	24	no	—
64	C₂×Dic₈₆	C₂⁴	1	8	yes	—
65	C₂×C₈×C₄₃	C₂×D₄	1	48	yes	complemented
66	C₂×C₈×C₄₃	C₂⁴	1	8	yes	—
67	C₄²×C₄₃	C₂×D₄	1	12	yes	—
68	C₄²×C₄₃	C₂⁴	1	4	yes	—
69	C₄⋊Dic₄₃	—	2	32	no	—
70	#10621873	—	2	128	no	—
71	C₂³×C₄₃	#127	1	4	yes	—
72	C₂³×C₄₃	C₂×SD₁₆	1	8	yes	—
73	C₂³×C₄₃	C₂²×D₄	1	3	yes	—
74	C₂²×D₄₃	—	2	8	no	—
75	C₂²×D₄₃	—	4	24	no	—
76	C₂²×D₄₃	—	4	48	no	—
77	C₂²×D₄₃	—	8	32	no	—
78	C₂×C₄×C₄₃	#127	1	12	yes	—
79	C₂×C₄×C₄₃	C₂×SD₁₆	1	48	yes	complemented
80	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
81	C₂×C₄×C₄₃	C₂²×D₄	1	12	yes	—
82	C₂×C₄×C₄₃	C₂²×D₄	1	8	yes	—
83	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
84	C₂×Dic₄₃	—	2	8	no	—
85	C₂×Dic₄₃	—	4	24	no	—
86	D₁₇₂	—	2	16	no	—
87	D₁₇₂	—	4	24	no	—
88	D₁₇₂	—	4	64	no	—
89	Dic₈₆	—	2	16	no	—
90	Dic₈₆	—	4	24	no	—
91	C₈×C₄₃	C₂²×D₄	1	16	yes	complemented
92	C₂²×#492	—	43	1	no	maximal, Sylow
93	C₂²×C₄₃	#492	1	16	yes	complemented
94	C₂²×C₄₃	C₂×#127	1	6	yes	—
95	C₂²×C₄₃	C₂²×SD₁₆	1	12	yes	—
96	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
97	C₂×D₄₃	—	4	8	no	—
98	C₂×D₄₃	—	4	16	no	—
99	C₂×D₄₃	—	8	48	no	—
100	C₄×C₄₃	C₂×#127	1	6	yes	—
101	C₄×C₄₃	C₂²×SD₁₆	1	16	yes	complemented
102	C₄×C₄₃	C₂³×D₄	1	2	yes	—
103	Dic₄₃	—	4	8	no	—
104	C₂³×SD₁₆	—	43	4	no	—
105	C₂²×C₄⋊Q₈	—	43	1	no	—
106	C₂²×#127	—	43	1	no	—
107	C₂×#492	—	43	24	no	—
108	C₂²×C₄×C₈	D₄₃	1	1	yes	characteristic, complemented
109	C₂×C₄₃	C₂×#492	1	12	yes	complemented
110	C₂×C₄₃	C₂²×#127	1	1	yes	characteristic
111	C₂×C₄₃	C₂³×SD₁₆	1	2	yes	—
112	D₄₃	—	8	8	no	—
113	C₂³×Q₈	—	43	2	no	—
114	C₂³×D₄	—	43	2	no	—
115	C₂³×D₄	—	86	2	no	—
116	C₂²×SD₁₆	—	43	48	no	—
117	C₂²×SD₁₆	—	86	32	no	—
118	C₂³×C₈	C₂×D₄₃	1	2	yes	—
119	C₂×C₄⋊Q₈	—	43	12	no	—
120	C₂×C₄⋊Q₈	—	86	6	no	—
121	C₂×#127	—	43	12	no	—
122	C₂×#127	—	86	6	no	—
123	C₂²×C₄⋊C₄	—	86	2	no	—
124	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
125	#492	—	43	64	no	—
126	C₂×C₄×C₈	C₂×D₄₃	1	12	yes	complemented
127	C₄₃	C₂²×#492	1	1	yes	characteristic, complemented, Sylow
128	C₂⁵	—	86	2	no	—
129	C₂²×Q₈	—	43	12	no	—
130	C₂²×Q₈	—	86	6	no	—
131	C₂²×Q₈	—	86	16	no	—
132	C₂²×D₄	—	43	12	no	—
133	C₂²×D₄	—	86	16	no	—
134	C₂²×D₄	—	86	24	no	—
135	C₂²×D₄	—	86	6	no	—
136	C₂²×D₄	—	172	16	no	—
137	C₂³×C₄	—	86	2	no	—
138	C₂³×C₄	D₁₇₂	1	2	yes	—
139	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
140	C₂×SD₁₆	—	43	64	no	—
141	C₂×SD₁₆	—	86	192	no	—
142	C₂²×C₈	D₁₇₂	1	16	yes	complemented
143	C₂²×C₈	C₂²×D₄₃	1	12	yes	—
144	C₄⋊Q₈	—	43	16	no	—
145	C₄⋊Q₈	—	86	24	no	—
146	#127	—	43	16	no	—
147	#127	—	86	24	no	—
148	C₂×C₄⋊C₄	—	86	24	no	—
149	C₂×C₄²	D₁₇₂	1	6	yes	—
150	C₂×C₄²	C₂²×D₄₃	1	6	yes	—
151	C₄×C₈	C₂²×D₄₃	1	16	yes	complemented
152	C₂⁴	—	86	12	no	—
153	C₂⁴	—	172	8	no	—
154	C₂⁴	—	172	16	no	—
155	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
156	C₂×Q₈	—	43	8	no	—
157	C₂×Q₈	—	86	12	no	—
158	C₂×Q₈	—	86	48	no	—
159	C₂×Q₈	—	172	24	no	—
160	C₂×D₄	—	43	8	no	—
161	C₂×D₄	—	86	48	no	—
162	C₂×D₄	—	86	32	no	—
163	C₂×D₄	—	86	12	no	—
164	C₂×D₄	—	172	24	no	—
165	C₂×D₄	—	172	96	no	—
166	C₂²×C₄	—	86	12	no	—
167	C₂²×C₄	—	172	8	no	—
168	C₂²×C₄	#10621873	1	16	yes	complemented
169	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
170	C₂²×C₄	C₂×D₁₇₂	1	8	yes	—
171	C₂²×C₄	C₂×D₁₇₂	1	12	yes	—
172	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
173	SD₁₆	—	86	128	no	—
174	C₂×C₈	C₂×D₁₇₂	1	48	yes	complemented
175	C₂×C₈	C₂³×D₄₃	1	8	yes	—
176	C₄⋊C₄	—	86	32	no	—
177	C₄²	C₂×D₁₇₂	1	12	yes	—
178	C₄²	C₂³×D₄₃	1	4	yes	—
179	C₂³	—	86	8	no	—
180	C₂³	—	344	32	no	—
181	C₂³	—	172	48	no	—
182	C₂³	—	172	24	no	—
183	C₂³	#13063322	1	4	yes	—
184	C₂³	C₂×#10621873	1	8	yes	—
185	C₂³	C₂²×D₁₇₂	1	3	yes	—
186	Q₈	—	86	16	no	—
187	Q₈	—	172	24	no	—
188	D₄	—	86	16	no	—
189	D₄	—	172	64	no	—
190	D₄	—	172	24	no	—
191	C₂×C₄	—	86	8	no	—
192	C₂×C₄	—	172	24	no	—
193	C₂×C₄	#13063322	1	12	yes	—
194	C₂×C₄	C₂×#10621873	1	48	yes	complemented
195	C₂×C₄	C₂²×D₁₇₂	1	8	yes	—
196	C₂×C₄	C₂²×D₁₇₂	1	3	yes	—
197	C₂×C₄	C₂²×D₁₇₂	1	12	yes	—
198	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
199	C₈	C₂²×D₁₇₂	1	16	yes	complemented
200	C₂²	—	344	48	no	—
201	C₂²	—	172	16	no	—
202	C₂²	—	172	8	no	—
203	C₂²	#14583272	1	16	yes	direct product factor
204	C₂²	C₂×#13063322	1	6	yes	—
205	C₂²	C₂²×#10621873	1	12	yes	—
206	C₂²	C₂³×D₁₇₂	1	1	yes	characteristic
207	C₄	—	172	8	no	—
208	C₄	C₂×#13063322	1	6	yes	—
209	C₄	C₂²×#10621873	1	16	yes	complemented
210	C₄	C₂³×D₁₇₂	1	2	yes	—
211	C₂	—	344	8	no	—
212	C₂	C₂×#14583272	1	12	yes	direct product factor
213	C₂	C₂²×#13063322	1	1	yes	characteristic
214	C₂	C₂³×#10621873	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₇₂
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂²×C₄×C₈×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂×D₄
prime radical	C₂²×#13063322	C₂
hypercenter	C₂²×C₄×C₈	D₄₃
hypocenter	C₄₃	C₂²×#492
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×#10621873	C₂	semidirect
2. C₂²×#13063325	C₂	semidirect
4. C₂×#14583272	C₂	direct (N × H)
5. C₂²×C₄×C₈×C₄₃	C₂	semidirect
10. C₂²×#10621873	C₂²	semidirect
12. C₂×#13063325	C₂²	semidirect
16. #14583272	C₂²	direct (N × H)
19. C₂×C₄×C₈×C₄₃	C₂²	semidirect
31. C₂×#10621873	C₂³	semidirect
33. #13063325	C₂³	semidirect
39. C₂²×C₈×C₄₃	D₄	semidirect
43. C₄×C₈×C₄₃	C₂³	semidirect
48. C₂²×C₄×C₄₃	SD₁₆	semidirect
65. C₂×C₈×C₄₃	C₂×D₄	semidirect
79. C₂×C₄×C₄₃	C₂×SD₁₆	semidirect
91. C₈×C₄₃	C₂²×D₄	semidirect
93. C₂²×C₄₃	#492	semidirect
101. C₄×C₄₃	C₂²×SD₁₆	semidirect
108. C₂²×C₄×C₈	D₄₃	semidirect
109. C₂×C₄₃	C₂×#492	semidirect
126. C₂×C₄×C₈	C₂×D₄₃	semidirect
127. C₄₃	C₂²×#492	semidirect
142. C₂²×C₈	D₁₇₂	semidirect
151. C₄×C₈	C₂²×D₄₃	semidirect
168. C₂²×C₄	#10621873	semidirect
174. C₂×C₈	C₂×D₁₇₂	semidirect
194. C₂×C₄	C₂×#10621873	semidirect
199. C₈	C₂²×D₁₇₂	semidirect
203. C₂²	#14583272	direct (N × H)
209. C₄	C₂²×#10621873	semidirect
212. C₂	C₂×#14583272	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×#10621873	2
C₂²×#13063325	2
C₂²×#13063322	2
C₂×#14583272	2
C₂²×C₄×C₈×C₄₃	2
C₂²×#492	43

Subgroups — C22×#14583272 #25965995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14583272 #25965995