Subgroups — #24983125

Order 11008.

414 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal
2	C₄²×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₄₃⋊C₈)	C₄	1	4	yes	—
4	C₂×C₄×C₄₃⋊C₈	—	2	2	no	—
5	#14582654	—	2	2	no	—
6	C₂×C₄×C₈×C₄₃	—	2	2	no	—
7	C₂×C₄×C₈×C₄₃	C₄	1	2	yes	—
8	C₄³×C₄₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄₃⋊C₈	—	2	4	no	—
10	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
11	C₄⋊(C₄₃⋊C₈)	C₈	1	8	yes	complemented
12	C₄×C₄₃⋊C₈	—	4	2	no	—
13	C₄×C₄₃⋊C₈	—	4	2	no	—
14	C₂²×C₈×C₄₃	—	2	2	no	—
15	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₄₃	D₄	1	2	yes	—
19	C₂×C₄²×C₄₃	Q₈	1	1	yes	characteristic
20	C₄×C₈×C₄₃	—	2	8	no	—
21	C₄×C₈×C₄₃	—	2	2	no	—
22	C₄×C₈×C₄₃	C₈	1	4	yes	complemented
23	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
24	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
25	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₄₃	—	2	8	no	—
29	C₂×C₈×C₄₃	—	2	4	no	—
30	C₄²×C₄₃	—	2	4	no	—
31	C₄²×C₄₃	—	2	2	no	—
32	C₄²×C₄₃	C₂×C₈	1	4	yes	—
33	C₄²×C₄₃	C₂×C₈	1	2	yes	—
34	C₄²×C₄₃	M₁₆	1	4	yes	—
35	C₄²×C₄₃	M₁₆	1	2	yes	—
36	C₄²×C₄₃	D₈	1	1	yes	characteristic
37	C₄²×C₄₃	SD₁₆	1	2	yes	—
38	C₄²×C₄₃	Q₁₆	1	1	yes	characteristic
39	C₂×C₄₃⋊C₈	—	2	8	no	—
40	C₂×C₄₃⋊C₈	—	4	4	no	—
41	C₂×C₄₃⋊C₈	—	4	4	no	—
42	C₂³×C₄₃	#95	1	1	yes	characteristic
43	C₂×C₄×C₄₃	—	2	2	no	—
44	C₂×C₄×C₄₃	—	2	2	no	—
45	C₂×C₄×C₄₃	—	2	2	no	—
46	C₂×C₄×C₄₃	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₄₃	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₄₃	#97	1	2	yes	—
49	C₂×C₄×C₄₃	#97	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂²⋊C₈	1	4	yes	—
51	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
52	C₂×C₄×C₄₃	#102	1	2	yes	—
53	C₂×C₄×C₄₃	#103	1	2	yes	—
54	C₂×C₄×C₄₃	#104	1	4	yes	—
55	C₂×C₄×C₄₃	C₄⋊C₈	1	4	yes	—
56	C₂×C₄×C₄₃	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
58	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
59	C₂×C₄×C₄₃	#108	1	2	yes	—
60	C₈×C₄₃	—	2	8	no	—
61	C₄₃⋊C₈	—	4	8	no	—
62	#7015	—	43	1	no	maximal, Sylow
63	C₂²×C₄₃	#336	1	2	yes	—
64	C₂²×C₄₃	#336	1	1	yes	characteristic
65	C₂²×C₄₃	#339	1	2	yes	—
66	C₂²×C₄₃	#340	1	1	yes	characteristic
67	C₂²×C₄₃	#341	1	1	yes	characteristic
68	C₄×C₄₃	—	2	2	no	—
69	C₄×C₄₃	—	2	4	no	—
70	C₄×C₄₃	C₈²	1	1	yes	characteristic
71	C₄×C₄₃	#322	1	1	yes	characteristic
72	C₄×C₄₃	#322	1	2	yes	—
73	C₄×C₄₃	D₄⋊C₈	1	4	yes	—
74	C₄×C₄₃	Q₈⋊C₈	1	4	yes	—
75	C₄×C₄₃	#334	1	2	yes	—
76	C₄×C₄₃	#335	1	2	yes	—
77	C₄×C₄⋊C₈	—	43	2	no	—
78	C₄²×C₈	D₄₃	1	1	yes	characteristic
79	C₂×C₄₃	#1273	1	1	yes	characteristic
80	C₂×C₄₃	#1276	1	1	yes	characteristic
81	C₂×C₄₃	#1277	1	1	yes	characteristic
82	C₂×C₄₃	#1278	1	2	yes	—
83	C₂×C₄₃	#1279	1	2	yes	—
84	C₂×C₄⋊C₈	—	43	4	no	—
85	C₂×C₄×C₈	—	2	2	no	—
86	C₂×C₄×C₈	—	86	2	no	—
87	C₂×C₄×C₈	Dic₄₃	1	2	yes	—
88	C₄³	C₂×D₄₃	1	1	yes	characteristic
89	#336	—	86	2	no	—
90	C₄₃	#7015	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	—	2	2	no	—
92	C₂²×C₈	—	86	4	no	—
93	C₂×C₄²	Dic₈₆	1	1	yes	characteristic
94	C₂×C₄²	C₄×D₄₃	1	2	yes	—
95	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
96	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
97	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
98	C₄⋊C₈	—	43	8	no	—
99	C₄⋊C₈	—	86	4	no	—
100	C₄×C₈	—	2	8	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	—	172	2	no	—
103	C₄×C₈	—	172	2	no	—
104	C₄×C₈	C₄₃⋊C₈	1	4	yes	complemented
105	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
106	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
107	C₂²×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
108	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
109	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
110	C₂×C₈	—	2	8	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	86	8	no	—
113	C₂×C₈	—	172	4	no	—
114	C₂×C₈	—	172	4	no	—
115	C₄²	—	2	4	no	—
116	C₄²	—	2	2	no	—
117	C₄²	C₈×D₄₃	1	4	yes	—
118	C₄²	#10621872	1	4	yes	—
119	C₄²	#10621873	1	2	yes	—
120	C₄²	D₃₄₄	1	1	yes	characteristic
121	C₄²	Dic₁₇₂	1	1	yes	characteristic
122	C₄²	C₂×C₄₃⋊C₈	1	2	yes	—
123	C₄²	#10621877	1	2	yes	—
124	C₂³	C₄₃⋊#95	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₈×Dic₄₃	1	2	yes	—
129	C₂×C₄	C₄×C₄₃⋊C₈	1	1	yes	characteristic
130	C₂×C₄	#13063235	1	2	yes	—
131	C₂×C₄	#13063236	1	1	yes	characteristic
132	C₂×C₄	(C₂×D₄₃)⋊C₈	1	4	yes	—
133	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
134	C₂×C₄	#13063247	1	2	yes	—
135	C₂×C₄	#13063250	1	2	yes	—
136	C₂×C₄	#13063253	1	4	yes	—
137	C₂×C₄	Dic₄₃⋊C₈	1	4	yes	—
138	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	2	yes	—
139	C₂×C₄	#13063257	1	1	yes	characteristic
140	C₂×C₄	#13063259	1	1	yes	characteristic
141	C₂×C₄	#13063261	1	2	yes	—
142	C₈	—	2	8	no	—
143	C₈	—	172	8	no	—
144	C₂²	#14582653	1	2	yes	—
145	C₂²	#14582654	1	1	yes	characteristic
146	C₂²	#14582659	1	2	yes	—
147	C₂²	#14582662	1	1	yes	characteristic
148	C₂²	#14582664	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	4	no	—
151	C₄	C₈×C₄₃⋊C₈	1	1	yes	characteristic
152	C₄	#14582620	1	1	yes	characteristic
153	C₄	#14582621	1	2	yes	—
154	C₄	#14582628	1	4	yes	—
155	C₄	#14582631	1	4	yes	—
156	C₄	#14582650	1	2	yes	—
157	C₄	#14582652	1	2	yes	—
158	C₂	C₄₃⋊#1273	1	1	yes	characteristic
159	C₂	#19132428	1	1	yes	characteristic
160	C₂	#19132430	1	1	yes	characteristic
161	C₂	#19132432	1	2	yes	—
162	C₂	#19132435	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	D₁₇₂
commutator	C₄×C₄₃	C₈²
Frattini	C₄³	C₂×D₄₃
Fitting	C₄²×C₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	#95
cosocle	C₄³×C₄₃	C₂²
hypercenter	C₄²×C₈	D₄₃
hypocenter	C₄₃	#7015
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
11. C₄⋊(C₄₃⋊C₈)	C₈	semidirect
22. C₄×C₈×C₄₃	C₈	semidirect
90. C₄₃	#7015	semidirect
104. C₄×C₈	C₄₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₄₃⋊C₈)	2
C₄²×C₈×C₄₃	2
#7015	43