Subgroups — C₂²×#14582704 #25249474

Order 11008.

1214 conjugacy classes of subgroups.

The subgroup lattice has 196 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×#13063259	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582704	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₈₆	—	2	1	no	—
6	C₂²×Dic₁₇₂	—	2	3	no	—
7	C₂²×Dic₁₇₂	—	2	4	no	—
8	C₂²×Dic₁₇₂	C₄	1	8	yes	complemented
9	C₂²×Dic₁₇₂	C₂²	1	6	yes	—
10	C₂²×C₄⋊Dic₄₃	—	2	1	no	—
11	C₂×#13063259	—	2	3	no	—
12	C₂×#13063259	C₂²	1	6	yes	—
13	#14582704	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	—
17	C₂²×Dic₈₆	—	2	8	no	—
18	C₂²×Dic₈₆	—	2	6	no	—
19	C₂²×Dic₈₆	—	4	3	no	—
20	C₂²×Dic₈₆	—	4	4	no	—
21	C₂³×Dic₄₃	—	4	1	no	—
22	C₂³×Dic₄₃	—	4	1	no	—
23	C₂×Dic₁₇₂	—	2	24	no	—
24	C₂×Dic₁₇₂	—	2	6	no	—
25	C₂×Dic₁₇₂	—	2	12	no	—
26	C₂×Dic₁₇₂	C₂×C₄	1	24	yes	complemented
27	C₂×Dic₁₇₂	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	—
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	—
39	C₂⁴×C₄₃	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
41	C₂²×C₄×C₄₃	D₈	1	3	yes	—
42	C₂²×C₄×C₄₃	SD₁₆	1	4	yes	—
43	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
44	C₂²×Dic₄₃	—	4	8	no	—
45	C₂²×Dic₄₃	—	4	6	no	—
46	C₂²×Dic₄₃	—	4	6	no	—
47	C₂²×Dic₄₃	—	8	4	no	—
48	C₂×Dic₈₆	—	2	24	no	—
49	C₂×Dic₈₆	—	2	4	no	—
50	C₂×Dic₈₆	—	4	24	no	—
51	C₂×Dic₈₆	—	4	6	no	—
52	C₂×Dic₈₆	—	4	12	no	—
53	C₂×C₈×C₄₃	C₂²⋊C₄	1	12	yes	—
54	C₂×C₈×C₄₃	C₂²×C₄	1	6	yes	—
55	C₂×C₈×C₄₃	C₂×D₄	1	3	yes	—
56	C₂×C₈×C₄₃	C₂×D₄	1	6	yes	—
57	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
58	C₄⋊Dic₄₃	—	2	4	no	—
59	C₄⋊Dic₄₃	—	4	6	no	—
60	Dic₁₇₂	—	2	24	no	—
61	Dic₁₇₂	—	2	4	no	—
62	Dic₁₇₂	C₂²×C₄	1	8	yes	complemented
63	C₁₆×C₄₃	—	2	4	no	—
64	C₂³×C₄₃	#102	1	4	yes	—
65	C₂³×C₄₃	SD₃₂	1	4	yes	—
66	C₂³×C₄₃	Q₃₂	1	4	yes	—
67	C₂³×C₄₃	C₂×D₈	1	3	yes	—
68	C₂×C₄×C₄₃	#102	1	12	yes	—
69	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	6	yes	—
70	C₂×C₄×C₄₃	C₂×D₈	1	3	yes	—
71	C₂×C₄×C₄₃	C₂×SD₁₆	1	6	yes	—
72	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
73	C₂×Dic₄₃	—	4	4	no	—
74	C₂×Dic₄₃	—	4	24	no	—
75	C₂×Dic₄₃	—	4	4	no	—
76	C₂×Dic₄₃	—	8	12	no	—
77	Dic₈₆	—	2	8	no	—
78	Dic₈₆	—	4	4	no	—
79	Dic₈₆	—	4	24	no	—
80	C₈×C₄₃	C₂×C₂²⋊C₄	1	6	yes	—
81	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
82	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
83	C₂²×#358	—	43	1	no	maximal, Sylow
84	C₂²×C₄₃	#358	1	16	yes	complemented
85	C₂²×C₄₃	C₂×#102	1	6	yes	—
86	C₂²×C₄₃	C₂×SD₃₂	1	6	yes	—
87	C₂²×C₄₃	C₂×Q₃₂	1	6	yes	—
88	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
89	C₄×C₄₃	C₂×#102	1	6	yes	—
90	C₄×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
91	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
92	Dic₄₃	—	4	8	no	—
93	Dic₄₃	—	8	4	no	—
94	C₂³×Q₁₆	—	43	1	no	—
95	C₂³×C₁₆	D₄₃	1	1	yes	characteristic
96	C₂²×#107	—	43	1	no	—
97	C₂×#358	—	43	12	no	—
98	C₂×C₄₃	C₂×#358	1	12	yes	complemented
99	C₂×C₄₃	C₂²×#102	1	1	yes	characteristic
100	C₂×C₄₃	C₂²×SD₃₂	1	1	yes	characteristic
101	C₂×C₄₃	C₂²×Q₃₂	1	1	yes	characteristic
102	C₂³×Q₈	—	86	1	no	—
103	C₂²×Q₁₆	—	43	6	no	—
104	C₂²×Q₁₆	—	43	8	no	—
105	C₂²×Q₁₆	—	86	3	no	—
106	C₂²×Q₁₆	—	86	4	no	—
107	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
108	C₂²×C₄⋊C₄	—	86	1	no	—
109	C₂²×C₁₆	—	2	4	no	—
110	C₂²×C₁₆	C₂×D₄₃	1	6	yes	—
111	C₂×#107	—	43	6	no	—
112	C₂×#107	—	86	3	no	—
113	#358	—	43	16	no	—
114	C₄₃	C₂²×#358	1	1	yes	characteristic, complemented, Sylow
115	C₂²×Q₈	—	86	8	no	—
116	C₂²×Q₈	—	86	6	no	—
117	C₂²×Q₈	—	172	4	no	—
118	C₂²×Q₈	—	172	3	no	—
119	C₂³×C₄	—	172	1	no	—
120	C₂³×C₄	—	172	1	no	—
121	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
122	C₂×Q₁₆	—	43	24	no	—
123	C₂×Q₁₆	—	43	4	no	—
124	C₂×Q₁₆	—	86	24	no	—
125	C₂×Q₁₆	—	86	6	no	—
126	C₂×Q₁₆	—	86	12	no	—
127	C₂²×C₈	C₄×D₄₃	1	4	yes	—
128	C₂²×C₈	D₁₇₂	1	3	yes	—
129	C₂²×C₈	C₂²⋊D₄₃	1	4	yes	—
130	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
131	C₂×C₄⋊C₄	—	86	6	no	—
132	C₂×C₄⋊C₄	—	172	3	no	—
133	C₂×C₁₆	—	2	12	no	—
134	C₂×C₁₆	C₂²×D₄₃	1	4	yes	—
135	#107	—	43	4	no	—
136	#107	—	86	6	no	—
137	C₂⁴	D₃₄₄	1	1	yes	characteristic
138	C₂×Q₈	—	86	24	no	—
139	C₂×Q₈	—	86	4	no	—
140	C₂×Q₈	—	172	24	no	—
141	C₂×Q₈	—	172	6	no	—
142	C₂×Q₈	—	172	12	no	—
143	C₂²×C₄	—	344	4	no	—
144	C₂²×C₄	—	172	6	no	—
145	C₂²×C₄	—	172	8	no	—
146	C₂²×C₄	—	172	6	no	—
147	C₂²×C₄	#10621873	1	4	yes	—
148	C₂²×C₄	D₃₄₄	1	3	yes	—
149	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	4	yes	—
150	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
151	Q₁₆	—	43	8	no	—
152	Q₁₆	—	86	24	no	—
153	Q₁₆	—	86	4	no	—
154	C₂×C₈	(C₂×D₄₃)⋊C₄	1	12	yes	—
155	C₂×C₈	C₂×C₄×D₄₃	1	6	yes	—
156	C₂×C₈	C₂×D₁₇₂	1	3	yes	—
157	C₂×C₈	C₂×C₂²⋊D₄₃	1	6	yes	—
158	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
159	C₄⋊C₄	—	86	4	no	—
160	C₄⋊C₄	—	172	6	no	—
161	C₁₆	—	2	4	no	—
162	C₂³	#13063247	1	4	yes	—
163	C₂³	#13063270	1	4	yes	—
164	C₂³	Dic₃₄₄	1	4	yes	—
165	C₂³	C₂×D₃₄₄	1	3	yes	—
166	Q₈	—	86	8	no	—
167	Q₈	—	172	24	no	—
168	Q₈	—	172	4	no	—
169	C₂×C₄	—	344	12	no	—
170	C₂×C₄	—	172	4	no	—
171	C₂×C₄	—	172	24	no	—
172	C₂×C₄	—	172	4	no	—
173	C₂×C₄	#13063247	1	12	yes	—
174	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	6	yes	—
175	C₂×C₄	C₂×D₃₄₄	1	3	yes	—
176	C₂×C₄	C₂×#10621873	1	6	yes	—
177	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
178	C₈	C₂×(C₂×D₄₃)⋊C₄	1	6	yes	—
179	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
180	C₈	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
181	C₂²	#14582704	1	16	yes	direct product factor
182	C₂²	C₂×#13063247	1	6	yes	—
183	C₂²	C₂×#13063270	1	6	yes	—
184	C₂²	C₂×Dic₃₄₄	1	6	yes	—
185	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
186	C₄	—	344	4	no	—
187	C₄	—	172	8	no	—
188	C₄	C₂×#13063247	1	6	yes	—
189	C₄	C₂²×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
190	C₄	C₂²×#10621873	1	1	yes	characteristic
191	C₂	C₂×#14582704	1	12	yes	direct product factor
192	C₂	C₂²×#13063247	1	1	yes	characteristic
193	C₂	C₂²×#13063270	1	1	yes	characteristic
194	C₂	C₂²×Dic₃₄₄	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, prime radical	C₂⁴×C₄₃	D₈
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂³×C₁₆	D₄₃
hypocenter	C₄₃	C₂²×#358
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
3. C₂×#14582704	C₂	direct (N × H)
8. C₂²×Dic₁₇₂	C₄	semidirect
13. #14582704	C₂²	direct (N × H)
26. C₂×Dic₁₇₂	C₂×C₄	semidirect
62. Dic₁₇₂	C₂²×C₄	semidirect
84. C₂²×C₄₃	#358	semidirect
98. C₂×C₄₃	C₂×#358	semidirect
114. C₄₃	C₂²×#358	semidirect
181. C₂²	#14582704	direct (N × H)
191. C₂	C₂×#14582704	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×#13063259	2
C₂×#14582704	2
C₂³×C₁₆×C₄₃	2
C₂²×#358	43

Subgroups — C22×#14582704 #25249474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14582704 #25249474