Subgroups — C₂²×#14583007 #25960637

Order 11008.

2790 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×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₈₆	C₂	1	2	yes	maximal
3	C₂×#14583007	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13063257	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13063250	C₂	1	2	yes	maximal
6	C₂²×#97×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₈₆	C₂²	1	2	yes	—
8	C₂²×Dic₁₇₂	C₄	1	16	yes	complemented
9	C₂²×Dic₁₇₂	C₂²	1	12	yes	—
10	C₂×C₄×Dic₈₆	—	2	12	no	—
11	C₂×C₄×Dic₈₆	C₂²	1	24	yes	—
12	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
13	C₂²×Dic₄₃⋊C₄	—	2	2	no	—
14	C₂²×C₄×Dic₄₃	—	2	2	no	—
15	#14583007	C₂²	1	64	yes	direct product factor
16	C₂×#13063257	C₂²	1	12	yes	—
17	C₂×#13063250	C₂²	1	24	yes	—
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	—
22	C₂²×Dic₈₆	—	2	6	no	—
23	C₂²×Dic₈₆	—	2	8	no	—
24	C₂²×Dic₈₆	C₂×C₄	1	16	yes	—
25	C₂²×Dic₈₆	C₂³	1	12	yes	—
26	C₂³×Dic₄₃	—	2	2	no	—
27	C₂³×Dic₄₃	—	2	2	no	—
28	C₂×Dic₁₇₂	C₂×C₄	1	96	yes	complemented
29	C₂×Dic₁₇₂	C₂³	1	16	yes	—
30	C₄×Dic₈₆	—	2	48	no	—
31	C₄×Dic₈₆	C₂³	1	32	yes	—
32	C₂×C₄⋊Dic₄₃	—	2	6	no	—
33	C₂×C₄⋊Dic₄₃	C₂³	1	12	yes	—
34	C₂×Dic₄₃⋊C₄	—	2	24	no	—
35	C₂×C₄×Dic₄₃	—	2	24	no	—
36	#13063257	C₂³	1	16	yes	—
37	#13063250	C₂³	1	32	yes	—
38	C₂³×C₄×C₄₃	D₄	1	2	yes	—
39	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₄₃	—	2	4	no	—
41	C₂²×C₈×C₄₃	C₂×C₄	1	8	yes	—
42	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
43	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
44	C₂×C₄²×C₄₃	D₄	1	6	yes	—
45	C₂×C₄²×C₄₃	C₂³	1	6	yes	—
46	#97×C₄₃	C₂³	1	16	yes	—
47	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	—	2	8	no	—
49	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₄₃	C₂×D₄	1	12	yes	—
51	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
52	C₂²×C₄×C₄₃	#41	1	4	yes	—
53	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
54	C₂²×Dic₄₃	—	2	16	no	—
55	C₂²×Dic₄₃	—	2	12	no	—
56	C₂²×Dic₄₃	—	2	12	no	—
57	C₂²×Dic₄₃	—	4	8	no	—
58	C₂×Dic₈₆	—	2	48	no	—
59	C₂×Dic₈₆	—	2	12	no	—
60	C₂×Dic₈₆	—	2	24	no	—
61	C₂×Dic₈₆	C₂²×C₄	1	48	yes	—
62	C₂×Dic₈₆	C₂⁴	1	8	yes	—
63	C₂×C₈×C₄₃	—	2	12	no	—
64	C₂×C₈×C₄₃	C₂²×C₄	1	24	yes	—
65	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
66	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
67	C₄²×C₄₃	C₂×D₄	1	12	yes	—
68	C₄²×C₄₃	C₂⁴	1	4	yes	—
69	C₄⋊Dic₄₃	—	2	12	no	—
70	C₄⋊Dic₄₃	C₂⁴	1	8	yes	—
71	Dic₄₃⋊C₄	—	2	32	no	—
72	C₄×Dic₄₃	—	2	32	no	—
73	Dic₁₇₂	C₂²×C₄	1	64	yes	complemented
74	C₂³×C₄₃	C₄×D₄	1	4	yes	—
75	C₂³×C₄₃	#137	1	8	yes	—
76	C₂³×C₄₃	C₂²×D₄	1	3	yes	—
77	C₂×C₄×C₄₃	—	2	24	no	—
78	C₂×C₄×C₄₃	C₄×D₄	1	12	yes	—
79	C₂×C₄×C₄₃	C₂³×C₄	1	6	yes	—
80	C₂×C₄×C₄₃	C₂²×D₄	1	8	yes	—
81	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
82	C₂×C₄×C₄₃	C₂×#41	1	6	yes	—
83	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
84	C₂×Dic₄₃	—	2	48	no	—
85	C₂×Dic₄₃	—	2	8	no	—
86	C₂×Dic₄₃	—	2	8	no	—
87	C₂×Dic₄₃	—	4	24	no	—
88	Dic₈₆	—	2	48	no	—
89	Dic₈₆	—	2	8	no	—
90	Dic₈₆	C₂³×C₄	1	16	yes	—
91	C₈×C₄₃	—	2	4	no	—
92	C₈×C₄₃	C₂³×C₄	1	8	yes	—
93	C₂²×#441	—	43	1	no	maximal, Sylow
94	C₂²×C₄₃	#441	1	16	yes	complemented
95	C₂²×C₄₃	C₂×C₄×D₄	1	6	yes	—
96	C₂²×C₄₃	C₂×#137	1	12	yes	—
97	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
98	C₄×C₄₃	—	2	8	no	—
99	C₄×C₄₃	C₂×C₄×D₄	1	6	yes	—
100	C₄×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
101	C₄×C₄₃	C₂²×#41	1	1	yes	characteristic
102	Dic₄₃	—	2	16	no	—
103	Dic₄₃	—	4	8	no	—
104	C₂³×Q₁₆	—	43	1	no	—
105	C₂²×C₄×Q₈	—	43	2	no	—
106	C₂×#441	—	43	24	no	—
107	C₂²×#106	—	43	1	no	—
108	C₂²×#103	—	43	2	no	—
109	C₂²×#97	D₄₃	1	1	yes	characteristic
110	C₂×C₄₃	C₂×#441	1	12	yes	complemented
111	C₂×C₄₃	C₂²×C₄×D₄	1	1	yes	characteristic
112	C₂×C₄₃	C₂²×#137	1	2	yes	—
113	C₂³×Q₈	—	43	2	no	—
114	C₂²×Q₁₆	—	43	12	no	—
115	C₂²×Q₁₆	—	43	16	no	—
116	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
117	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
118	C₂×C₄×Q₈	—	43	24	no	—
119	C₂×C₄×Q₈	—	86	12	no	—
120	C₂²×C₄⋊C₄	—	43	2	no	—
121	C₂²×C₄⋊C₄	—	86	2	no	—
122	C₂²×C₄²	—	86	2	no	—
123	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
124	#441	—	43	64	no	—
125	C₂×#106	—	43	12	no	—
126	C₂×#103	—	43	24	no	—
127	C₂×#97	C₂×D₄₃	1	12	yes	—
128	C₄₃	C₂²×#441	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	43	16	no	—
130	C₂²×Q₈	—	43	12	no	—
131	C₂²×Q₈	—	86	6	no	—
132	C₂²×Q₈	—	86	8	no	—
133	C₂³×C₄	—	86	2	no	—
134	C₂³×C₄	—	86	2	no	—
135	C₂³×C₄	D₁₇₂	1	2	yes	—
136	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
137	C₂×Q₁₆	—	43	96	no	—
138	C₂×Q₁₆	—	43	16	no	—
139	C₂²×C₈	—	2	4	no	—
140	C₂²×C₈	C₄×D₄₃	1	8	yes	—
141	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
142	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
143	C₄×Q₈	—	43	32	no	—
144	C₄×Q₈	—	86	48	no	—
145	C₂×C₄⋊C₄	—	43	12	no	—
146	C₂×C₄⋊C₄	—	86	6	no	—
147	C₂×C₄⋊C₄	—	86	24	no	—
148	C₂×C₄²	—	86	24	no	—
149	C₂×C₄²	D₁₇₂	1	6	yes	—
150	C₂×C₄²	C₂²×D₄₃	1	6	yes	—
151	#106	—	43	16	no	—
152	#103	—	43	32	no	—
153	#97	C₂²×D₄₃	1	16	yes	—
154	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
155	C₂×Q₈	—	43	48	no	—
156	C₂×Q₈	—	43	8	no	—
157	C₂×Q₈	—	86	48	no	—
158	C₂×Q₈	—	86	12	no	—
159	C₂×Q₈	—	86	24	no	—
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	86	16	no	—
162	C₂²×C₄	—	86	12	no	—
163	C₂²×C₄	—	86	12	no	—
164	C₂²×C₄	—	172	8	no	—
165	C₂²×C₄	C₂×C₄×D₄₃	1	4	yes	—
166	C₂²×C₄	C₂×D₁₇₂	1	12	yes	—
167	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
168	C₂²×C₄	#10621898	1	4	yes	—
169	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
170	Q₁₆	—	43	64	no	—
171	C₂×C₈	—	2	12	no	—
172	C₂×C₈	C₂×C₄×D₄₃	1	24	yes	—
173	C₂×C₈	C₂³×D₄₃	1	4	yes	—
174	C₂×C₈	C₂³×D₄₃	1	4	yes	—
175	C₄⋊C₄	—	43	8	no	—
176	C₄⋊C₄	—	86	12	no	—
177	C₄⋊C₄	—	86	32	no	—
178	C₄²	—	86	32	no	—
179	C₄²	C₂×D₁₇₂	1	12	yes	—
180	C₄²	C₂³×D₄₃	1	4	yes	—
181	C₂³	C₄×D₁₇₂	1	4	yes	—
182	C₂³	#13063358	1	8	yes	—
183	C₂³	C₂²×D₁₇₂	1	3	yes	—
184	Q₈	—	43	16	no	—
185	Q₈	—	86	48	no	—
186	Q₈	—	86	8	no	—
187	C₂×C₄	—	2	24	no	—
188	C₂×C₄	—	86	48	no	—
189	C₂×C₄	—	86	8	no	—
190	C₂×C₄	—	86	8	no	—
191	C₂×C₄	—	172	24	no	—
192	C₂×C₄	C₄×D₁₇₂	1	12	yes	—
193	C₂×C₄	C₂²×C₄×D₄₃	1	6	yes	—
194	C₂×C₄	C₂²×D₁₇₂	1	8	yes	—
195	C₂×C₄	C₂²×D₁₇₂	1	3	yes	—
196	C₂×C₄	C₂×#10621898	1	6	yes	—
197	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
198	C₈	—	2	4	no	—
199	C₈	C₂²×C₄×D₄₃	1	8	yes	—
200	C₂²	#14583007	1	16	yes	direct product factor
201	C₂²	C₂×C₄×D₁₇₂	1	6	yes	—
202	C₂²	C₂×#13063358	1	12	yes	—
203	C₂²	C₂³×D₁₇₂	1	1	yes	characteristic
204	C₄	—	2	8	no	—
205	C₄	—	86	16	no	—
206	C₄	—	172	8	no	—
207	C₄	C₂×C₄×D₁₇₂	1	6	yes	—
208	C₄	C₂³×C₄×D₄₃	1	1	yes	characteristic
209	C₄	C₂²×#10621898	1	1	yes	characteristic
210	C₂	C₂×#14583007	1	12	yes	direct product factor
211	C₂	C₂²×C₄×D₁₇₂	1	1	yes	characteristic
212	C₂	C₂²×#13063358	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, prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂×C₄×C₄₃	C₂⁵
hypercenter	C₂²×#97	D₄₃
hypocenter	C₄₃	C₂²×#441
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
3. C₂×#14583007	C₂	direct (N × H)
8. C₂²×Dic₁₇₂	C₄	semidirect
15. #14583007	C₂²	direct (N × H)
28. C₂×Dic₁₇₂	C₂×C₄	semidirect
73. Dic₁₇₂	C₂²×C₄	semidirect
94. C₂²×C₄₃	#441	semidirect
110. C₂×C₄₃	C₂×#441	semidirect
128. C₄₃	C₂²×#441	semidirect
200. C₂²	#14583007	direct (N × H)
210. C₂	C₂×#14583007	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×Dic₁₇₂	2
C₂²×C₄×Dic₈₆	2
C₂×#14583007	2
C₂²×#13063257	2
C₂²×#13063250	2
C₂²×#97×C₄₃	2
C₂²×#441	43

Subgroups — C22×#14583007 #25960637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14583007 #25960637