Subgroups — #25015202

Order 11008.

390 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	C₂×#14583304	C₂	1	1	yes	maximal, characteristic
2	#19134759	C₂	1	1	yes	maximal, characteristic
3	#19132687	C₂	1	4	yes	maximal
4	C₂×C₄₃×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063325	—	2	1	no	—
6	#14583304	—	2	2	no	—
7	#14583304	C₄	1	4	yes	complemented
8	C₂×#13063259	—	2	1	no	—
9	C₂×#13063259	C₂²	1	2	yes	—
10	C₂×#13063259	C₂²	1	2	yes	—
11	C₂×#13063257	—	2	1	no	—
12	#14582765	—	2	1	no	—
13	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#346	—	2	2	no	—
16	C₂²×Dic₈₆	—	4	1	no	—
17	#13063325	—	2	4	no	—
18	#13063325	—	4	2	no	—
19	C₂×C₄⋊Dic₄₃	—	2	2	no	—
20	C₂×C₄⋊Dic₄₃	—	2	2	no	—
21	C₂×C₄⋊Dic₄₃	—	4	1	no	—
22	#13063259	—	2	4	no	—
23	#13063259	—	2	2	no	—
24	#13063259	—	2	4	no	—
25	#13063259	C₂×C₄	1	4	yes	—
26	#13063257	—	4	2	no	—
27	C₄₃⋊#95	—	4	1	no	—
28	C₄₃⋊#95	—	4	1	no	—
29	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₄₃	—	2	2	no	—
33	C₂×C₁₆×C₄₃	—	2	2	no	—
34	C₂×C₁₆×C₄₃	Q₈	1	4	yes	complemented
35	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₄₃	D₄	1	2	yes	—
37	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₄₃	—	4	2	no	—
41	C₂²×Dic₄₃	—	4	2	no	—
42	C₂×Dic₈₆	—	4	4	no	—
43	C₂×Dic₈₆	—	8	4	no	—
44	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
46	C₂×C₈×C₄₃	Q₁₆	1	2	yes	—
47	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₄₃	#41	1	2	yes	—
51	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
52	C₄²×C₄₃	SD₁₆	1	2	yes	—
53	C₄⋊Dic₄₃	—	2	4	no	—
54	C₄⋊Dic₄₃	—	4	4	no	—
55	C₄⋊Dic₄₃	—	4	2	no	—
56	C₄⋊Dic₄₃	—	4	2	no	—
57	C₄⋊Dic₄₃	—	4	2	no	—
58	C₁₆×C₄₃	—	2	4	no	—
59	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₄₃	—	2	2	no	—
61	C₂×C₄×C₄₃	—	2	2	no	—
62	C₂×C₄×C₄₃	#102	1	2	yes	—
63	C₂×C₄×C₄₃	#102	1	2	yes	—
64	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
66	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×Dic₄₃	—	4	4	no	—
71	C₂×Dic₄₃	—	8	4	no	—
72	C₂×Dic₄₃	—	8	2	no	—
73	C₂×Dic₄₃	—	8	2	no	—
74	C₂×Dic₄₃	—	8	2	no	—
75	Dic₈₆	—	8	8	no	—
76	C₈×C₄₃	#103	1	4	yes	—
77	C₈×C₄₃	C₄×Q₈	1	2	yes	—
78	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
79	#13283	—	43	1	no	maximal, Sylow
80	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
81	C₂²×C₄₃	#486	1	1	yes	characteristic
82	C₂²×C₄₃	#500	1	1	yes	characteristic
83	C₂²×C₄₃	#509	1	2	yes	—
84	C₂²×C₄₃	#510	1	2	yes	—
85	C₄×C₄₃	—	2	2	no	—
86	C₄×C₄₃	—	2	2	no	—
87	C₄×C₄₃	#391	1	1	yes	characteristic
88	C₄×C₄₃	C₂×#103	1	1	yes	characteristic
89	C₄×C₄₃	#487	1	1	yes	characteristic
90	C₄×C₄₃	#498	1	1	yes	characteristic
91	Dic₄₃	—	8	4	no	—
92	Dic₄₃	—	16	2	no	—
93	C₂×#500	—	43	1	no	—
94	C₂×#346	D₄₃	1	1	yes	characteristic
95	#1846	—	43	1	no	—
96	#1380	—	43	4	no	—
97	C₂×C₄₃	#1990	1	1	yes	characteristic
98	C₂×C₄₃	#2140	1	1	yes	characteristic
99	C₂×C₄₃	#2141	1	1	yes	characteristic
100	C₂×C₄₃	#2243	1	2	yes	—
101	C₂×C₄₃	#2255	1	2	yes	—
102	C₂×C₄⋊Q₈	—	86	1	no	—
103	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
104	#500	—	43	4	no	—
105	#500	—	86	2	no	—
106	C₂×#107	—	43	2	no	—
107	C₂×#107	—	43	2	no	—
108	C₂×#107	—	86	1	no	—
109	C₂×#106	—	86	1	no	—
110	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
111	#382	—	86	1	no	—
112	#346	—	2	2	no	—
113	C₄₃	#13283	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	172	1	no	—
115	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
117	C₄⋊Q₈	—	86	4	no	—
118	C₄⋊Q₈	—	172	2	no	—
119	C₂×C₄⋊C₄	—	86	2	no	—
120	C₂×C₄⋊C₄	—	86	2	no	—
121	C₂×C₄⋊C₄	—	172	1	no	—
122	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	Dic₈₆	1	4	yes	complemented
126	#107	—	43	4	no	—
127	#107	—	86	4	no	—
128	#107	—	86	2	no	—
129	#107	—	86	4	no	—
130	#106	—	172	2	no	—
131	C₄×C₈	C₄×D₄₃	1	2	yes	—
132	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
133	#95	—	172	1	no	—
134	#95	—	172	1	no	—
135	C₂×Q₈	—	344	4	no	—
136	C₂×Q₈	—	172	4	no	—
137	C₂²×C₄	—	172	2	no	—
138	C₂²×C₄	—	172	2	no	—
139	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
140	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
142	C₂×C₈	#10621873	1	2	yes	—
143	C₂×C₈	Dic₁₇₂	1	2	yes	—
144	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
145	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
146	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
147	C₂×C₈	#10621898	1	2	yes	—
148	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
149	C₄⋊C₄	—	86	4	no	—
150	C₄⋊C₄	—	172	2	no	—
151	C₄⋊C₄	—	172	2	no	—
152	C₄⋊C₄	—	172	4	no	—
153	C₄⋊C₄	—	172	2	no	—
154	C₄²	#10621873	1	2	yes	—
155	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
156	C₁₆	—	2	4	no	—
157	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
158	Q₈	—	344	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	344	4	no	—
162	C₂×C₄	—	344	2	no	—
163	C₂×C₄	—	344	2	no	—
164	C₂×C₄	—	344	2	no	—
165	C₂×C₄	—	172	4	no	—
166	C₂×C₄	#13063247	1	2	yes	—
167	C₂×C₄	#13063247	1	2	yes	—
168	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
169	C₂×C₄	#13063313	1	1	yes	characteristic
170	C₂×C₄	#13063325	1	1	yes	characteristic
171	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
172	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
173	C₂×C₄	C₂×Dic₁₇₂	1	1	yes	characteristic
174	C₈	#13063250	1	4	yes	—
175	C₈	C₄×Dic₈₆	1	2	yes	—
176	C₈	#13063304	1	2	yes	—
177	C₂²	C₂×#13063247	1	1	yes	characteristic
178	C₂²	#14583246	1	1	yes	characteristic
179	C₂²	#14583304	1	1	yes	characteristic
180	C₂²	#14583337	1	2	yes	—
181	C₂²	#14583342	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	344	4	no	—
185	C₄	—	688	2	no	—
186	C₄	#14582805	1	1	yes	characteristic
187	C₄	C₂×#13063250	1	1	yes	characteristic
188	C₄	#14583250	1	1	yes	characteristic
189	C₄	#14583297	1	1	yes	characteristic
190	C₂	#19135556	1	1	yes	characteristic
191	C₂	#19136312	1	1	yes	characteristic
192	C₂	#19136316	1	1	yes	characteristic
193	C₂	#19136832	1	2	yes	—
194	C₂	#19136881	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₄₄
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#346	C₂
socle, prime radical	C₂³×C₄₃	C₂×D₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#346	D₄₃
hypocenter	C₄₃	#13283
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
7. #14583304	C₄	semidirect
34. C₂×C₁₆×C₄₃	Q₈	semidirect
113. C₄₃	#13283	semidirect
125. C₂×C₁₆	Dic₈₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583304	2
#19134759	2
#19132687	2
C₂×C₄₃×#346	2
#13283	43