Subgroups — #25249976

Order 11008.

646 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₄₄	C₂	1	1	yes	maximal, characteristic
2	#19143464	C₂	1	2	yes	maximal
3	#19136467	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₃₄₄	C₂	1	4	yes	maximal, complemented
5	#19136379	C₂	1	1	yes	maximal, characteristic
6	#19136316	C₂	1	2	yes	maximal
7	C₄₃×#2108	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₇₂	C₂²	1	2	yes	—
9	#14583376	—	2	2	no	—
10	C₂×Dic₃₄₄	C₄	1	8	yes	complemented
11	C₂×Dic₃₄₄	C₂²	1	2	yes	—
12	C₂×Dic₃₄₄	C₂²	1	2	yes	—
13	#14583007	—	2	4	no	—
14	C₄×Dic₁₇₂	C₂²	1	8	yes	—
15	#14582937	C₂²	1	2	yes	—
16	#14582909	—	2	2	no	—
17	#14582727	C₂²	1	2	yes	—
18	#14582725	C₂²	1	2	yes	—
19	#14582704	C₂²	1	8	yes	—
20	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#405	C₂²	1	1	yes	characteristic
23	C₄₃×#346	C₂²	1	2	yes	—
24	C₄×C₁₆×C₄₃	C₂²	1	2	yes	—
25	C₂²×Dic₈₆	—	2	2	no	—
26	C₂×Dic₁₇₂	—	2	2	no	—
27	C₂×Dic₁₇₂	—	2	4	no	—
28	C₂×Dic₁₇₂	C₂×C₄	1	8	yes	—
29	C₂×Dic₁₇₂	C₂³	1	4	yes	—
30	C₄×Dic₈₆	—	2	8	no	—
31	#13063290	—	4	4	no	—
32	#13063284	—	2	2	no	—
33	#13063281	—	4	4	no	—
34	Dic₃₄₄	C₂×C₄	1	8	yes	—
35	Dic₃₄₄	C₂×C₄	1	8	yes	—
36	#13063259	C₂³	1	4	yes	—
37	#13063257	—	2	2	no	—
38	#13063250	—	2	8	no	—
39	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
41	#117×C₄₃	D₄	1	1	yes	characteristic
42	M₃₂×C₄₃	—	2	2	no	—
43	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	—
44	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
45	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
47	#97×C₄₃	D₄	1	2	yes	—
48	C₄×C₈×C₄₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₄₃	—	4	2	no	—
51	C₂×Dic₈₆	—	2	8	no	—
52	C₂×Dic₈₆	—	2	4	no	—
53	C₂×Dic₈₆	—	4	2	no	—
54	C₂×Dic₈₆	—	4	4	no	—
55	M₁₆×C₄₃	—	2	2	no	—
56	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₄₃	#41	1	2	yes	—
60	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₄₃	D₈	1	2	yes	—
62	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
63	C₄²×C₄₃	C₂×D₄	1	2	yes	—
64	C₂²⋊Dic₄₃	—	4	2	no	—
65	C₄⋊Dic₄₃	—	2	4	no	—
66	Dic₄₃⋊C₄	—	4	8	no	—
67	C₄×Dic₄₃	—	4	8	no	—
68	C₄×Dic₄₃	—	4	2	no	—
69	Dic₁₇₂	—	2	8	no	—
70	Dic₁₇₂	—	2	4	no	—
71	Dic₁₇₂	C₂²×C₄	1	8	yes	—
72	C₁₆×C₄₃	—	2	2	no	—
73	C₁₆×C₄₃	C₂²×C₄	1	2	yes	—
74	C₁₆×C₄₃	C₂²×C₄	1	2	yes	—
75	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₄₃	C₂×D₈	1	4	yes	—
78	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₄₃	#135	1	2	yes	—
80	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₄₃	—	4	4	no	—
82	C₂×Dic₄₃	—	4	8	no	—
83	C₂×Dic₄₃	—	4	4	no	—
84	Dic₈₆	—	2	8	no	—
85	Dic₈₆	—	4	4	no	—
86	Dic₈₆	—	4	8	no	—
87	C₈×C₄₃	—	2	2	no	—
88	C₈×C₄₃	C₄×D₄	1	2	yes	—
89	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
90	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
91	#33458	—	43	1	no	maximal, Sylow
92	C₂²×C₄₃	—	2	2	no	—
93	C₂²×C₄₃	C₄×D₈	1	2	yes	—
94	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
95	C₄×C₄₃	—	2	4	no	—
96	C₄×C₄₃	C₄×D₈	1	2	yes	—
97	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
98	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
99	Dic₄₃	—	4	8	no	—
100	Dic₄₃	—	8	4	no	—
101	C₂²×Q₃₂	—	43	1	no	—
102	#2950	—	43	2	no	—
103	#2176	—	43	4	no	—
104	C₄×Q₃₂	—	43	4	no	—
105	#2158	—	43	1	no	—
106	#2141	—	43	2	no	—
107	#2108	D₄₃	1	1	yes	characteristic
108	C₂×C₄₃	—	2	2	no	—
109	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
110	C₂×C₄₃	#3417	1	2	yes	—
111	C₂²×Q₁₆	—	43	2	no	—
112	#519	—	86	2	no	—
113	C₂×Q₃₂	—	43	2	no	—
114	C₂×Q₃₂	—	43	2	no	—
115	C₂×Q₃₂	—	43	8	no	—
116	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
117	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
118	#441	—	86	4	no	—
119	C₄×Q₁₆	—	43	8	no	—
120	#427	—	43	2	no	—
121	#419	—	86	2	no	—
122	#405	C₂×D₄₃	1	1	yes	characteristic
123	#367	—	43	2	no	—
124	#366	—	43	2	no	—
125	#358	—	43	8	no	—
126	#346	C₂×D₄₃	1	2	yes	—
127	C₄×C₁₆	C₂×D₄₃	1	2	yes	—
128	C₄₃	#33458	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	86	2	no	—
130	C₂×Q₁₆	—	43	8	no	—
131	C₂×Q₁₆	—	43	4	no	—
132	C₂×Q₁₆	—	86	2	no	—
133	C₂×Q₁₆	—	86	4	no	—
134	C₂×M₁₆	D₁₇₂	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
136	C₄×Q₈	—	86	8	no	—
137	C₄×Q₈	—	172	4	no	—
138	#117	—	86	2	no	—
139	#117	—	172	4	no	—
140	#117	D₁₇₂	1	1	yes	characteristic
141	Q₃₂	—	43	8	no	—
142	Q₃₂	—	43	8	no	—
143	M₃₂	—	2	2	no	—
144	C₂×C₁₆	C₄×D₄₃	1	4	yes	—
145	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
146	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
148	#107	—	43	4	no	—
149	#106	—	86	2	no	—
150	#103	—	86	8	no	—
151	#97	D₁₇₂	1	2	yes	—
152	C₄×C₈	C₂²×D₄₃	1	2	yes	—
153	C₂×Q₈	—	86	8	no	—
154	C₂×Q₈	—	86	4	no	—
155	C₂×Q₈	—	172	2	no	—
156	C₂×Q₈	—	172	4	no	—
157	C₂²×C₄	—	172	2	no	—
158	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
159	Q₁₆	—	43	8	no	—
160	Q₁₆	—	86	8	no	—
161	Q₁₆	—	86	4	no	—
162	M₁₆	—	2	2	no	—
163	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
164	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
165	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
166	C₂×C₈	#10621898	1	2	yes	—
167	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
168	C₄⋊C₄	—	86	4	no	—
169	C₄⋊C₄	—	172	8	no	—
170	C₄⋊C₄	D₃₄₄	1	2	yes	—
171	C₂²⋊C₄	—	172	2	no	—
172	C₂²⋊C₄	D₃₄₄	1	2	yes	—
173	C₄²	—	172	2	no	—
174	C₄²	—	172	8	no	—
175	C₄²	C₂×D₁₇₂	1	2	yes	—
176	C₁₆	—	2	2	no	—
177	C₁₆	C₂×C₄×D₄₃	1	2	yes	—
178	C₁₆	C₂×C₄×D₄₃	1	2	yes	—
179	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
180	Q₈	—	86	8	no	—
181	Q₈	—	172	4	no	—
182	Q₈	—	172	8	no	—
183	C₂×C₄	—	172	4	no	—
184	C₂×C₄	—	172	4	no	—
185	C₂×C₄	—	172	8	no	—
186	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
187	C₂×C₄	C₂×D₃₄₄	1	4	yes	—
188	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
189	C₂×C₄	#13063347	1	2	yes	—
190	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	C₄×D₁₇₂	1	2	yes	—
193	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
194	C₈	C₂×#10621898	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	C₄×D₃₄₄	1	2	yes	—
197	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	344	4	no	—
200	C₄	—	172	8	no	—
201	C₄	C₄×D₃₄₄	1	2	yes	—
202	C₄	C₂×C₄×D₁₇₂	1	1	yes	characteristic
203	C₄	C₂×#13063347	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×C₄×D₃₄₄	1	1	yes	characteristic
206	C₂	#19147748	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₃₄₄
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2108	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂³×C₄₃	C₂×D₈
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	#2108	D₄₃
hypocenter	C₄₃	#33458
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
3. #19136467	C₂	semidirect
4. C₄×Dic₃₄₄	C₂	semidirect
10. C₂×Dic₃₄₄	C₄	semidirect
128. C₄₃	#33458	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₃₄₄	2
#19143464	2
#19136467	2
C₄×Dic₃₄₄	2
#19136379	2
#19136316	2
C₄₃×#2108	2
#33458	43