Subgroups — #11786278

Order 1152.

384 conjugacy classes of subgroups.

The subgroup lattice has 189 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×#63378	C₂	1	2	yes	maximal
3	C₂×#63377	C₂	1	2	yes	maximal
4	#10588616	C₂	1	1	yes	maximal, characteristic
5	#10588615	C₂	1	1	yes	maximal, characteristic
6	#69816	—	3	1	no	maximal
7	C₂²×C₈×C₉	C₂²	1	2	yes	—
8	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₉	—	2	2	no	—
10	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
13	C₉×#97	C₄	1	4	yes	complemented
14	C₉⋊#95	—	2	2	no	—
15	#63378	—	2	4	no	—
16	#63377	—	2	4	no	—
17	C₂×C₃×#97	D₃	1	1	yes	characteristic
18	C₂×#818	—	3	2	no	—
19	C₂×#817	—	3	2	no	—
20	#4936	—	3	1	no	—
21	#4935	—	3	1	no	—
22	C₂²×C₄×C₉	D₄	1	2	yes	—
23	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
24	C₂²×Dic₉	—	2	1	no	—
25	C₂²×Dic₉	—	2	1	no	—
26	C₂²×Dic₉	—	2	2	no	—
27	C₂×C₈×C₉	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₉	D₄	1	4	yes	—
29	C₂×C₈×C₉	Q₈	1	4	yes	—
30	C₄²×C₉	C₂×C₄	1	2	yes	—
31	C₄²×C₉	Q₈	1	2	yes	—
32	C₄⋊Dic₉	—	2	2	no	—
33	C₄⋊Dic₉	—	2	2	no	—
34	C₄⋊Dic₉	—	2	2	no	—
35	C₄⋊Dic₉	—	2	2	no	—
36	C₄⋊Dic₉	—	4	4	no	—
37	#1847	—	9	1	no	Sylow
38	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
39	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
40	C₂×C₄⋊Dic₃	—	3	2	no	—
41	C₂×C₄⋊Dic₃	—	3	1	no	—
42	C₂×C₄⋊Dic₃	—	3	1	no	—
43	C₂×C₄⋊Dic₃	—	6	2	no	—
44	C₃×#97	Dic₃	1	4	yes	—
45	C₃⋊#95	—	6	2	no	—
46	#818	—	6	4	no	—
47	#817	—	6	4	no	—
48	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₉	—	2	2	no	—
50	C₂×C₄×C₉	—	2	2	no	—
51	C₂×C₄×C₉	C₄⋊C₄	1	4	yes	—
52	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
56	C₂×Dic₉	—	4	2	no	—
57	C₂×Dic₉	—	4	1	no	—
58	C₂×Dic₉	—	4	1	no	—
59	C₂×Dic₉	—	4	4	no	—
60	C₂×Dic₉	—	4	2	no	—
61	C₂×Dic₉	—	4	2	no	—
62	C₈×C₉	C₄⋊C₄	1	8	yes	complemented
63	C₂×#107	—	9	2	no	—
64	C₂×#106	—	9	2	no	—
65	C₂×#97	D₉	1	1	yes	characteristic
66	#384	—	9	1	no	—
67	#382	—	9	1	no	—
68	C₂²×C₃×C₄	D₁₂	1	2	yes	—
69	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
70	C₂²×Dic₃	—	6	1	no	—
71	C₂²×Dic₃	—	6	1	no	—
72	C₂²×Dic₃	—	6	2	no	—
73	C₂×C₃×C₈	Dic₆	1	4	yes	—
74	C₂×C₃×C₈	D₁₂	1	4	yes	—
75	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
76	C₃×C₄²	Dic₆	1	2	yes	—
77	C₃×C₄²	C₂×Dic₃	1	2	yes	—
78	C₄⋊Dic₃	—	6	2	no	—
79	C₄⋊Dic₃	—	6	2	no	—
80	C₄⋊Dic₃	—	6	2	no	—
81	C₄⋊Dic₃	—	6	2	no	—
82	C₄⋊Dic₃	—	12	4	no	—
83	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₉	#127	1	1	yes	characteristic
85	C₂²×C₉	C₄⋊Q₈	1	1	yes	characteristic
86	C₂²×C₉	AGL(1,ℤ/8)	1	2	yes	—
87	C₂²×C₉	#137	1	2	yes	—
88	C₄×C₉	—	2	4	no	—
89	C₄×C₉	C₂×C₄⋊C₄	1	2	yes	—
90	C₄×C₉	C₄⋊Q₈	1	2	yes	—
91	Dic₉	—	8	2	no	—
92	Dic₉	—	8	1	no	—
93	Dic₉	—	8	1	no	—
94	C₂²×C₈	C₂×D₉	1	2	yes	—
95	C₂×C₄⋊C₄	—	9	1	no	—
96	C₂×C₄⋊C₄	—	9	1	no	—
97	C₂×C₄⋊C₄	—	9	2	no	—
98	C₂×C₄⋊C₄	—	18	2	no	—
99	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
100	#107	—	18	4	no	—
101	#106	—	18	4	no	—
102	#97	Dic₉	1	4	yes	complemented
103	#95	—	18	2	no	—
104	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
108	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
109	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
110	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
111	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
112	C₂×Dic₃	—	12	2	no	—
113	C₂×Dic₃	—	12	1	no	—
114	C₂×Dic₃	—	12	1	no	—
115	C₂×Dic₃	—	12	4	no	—
116	C₂×Dic₃	—	12	2	no	—
117	C₂×Dic₃	—	12	2	no	—
118	C₃×C₈	C₄⋊Dic₃	1	8	yes	—
119	C₂×C₉	#384	1	1	yes	characteristic
120	C₂×C₉	#428	1	2	yes	—
121	C₂×C₉	#496	1	1	yes	characteristic
122	C₂×C₉	#497	1	1	yes	characteristic
123	C₂×C₉	#501	1	2	yes	—
124	C₂²×C₄	—	18	1	no	—
125	C₂²×C₄	—	18	1	no	—
126	C₂²×C₄	—	18	2	no	—
127	C₂²×C₄	D₃₆	1	2	yes	—
128	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
129	C₂×C₈	Dic₁₈	1	4	yes	—
130	C₂×C₈	D₃₆	1	4	yes	—
131	C₂×C₈	C₂×Dic₉	1	4	yes	—
132	C₄⋊C₄	—	18	2	no	—
133	C₄⋊C₄	—	18	2	no	—
134	C₄⋊C₄	—	18	2	no	—
135	C₄⋊C₄	—	18	2	no	—
136	C₄⋊C₄	—	36	4	no	—
137	C₄²	Dic₁₈	1	2	yes	—
138	C₄²	C₂×Dic₉	1	2	yes	—
139	C₂²×C₃	#869	1	1	yes	characteristic
140	C₂²×C₃	#874	1	1	yes	characteristic
141	C₂²×C₃	#908	1	2	yes	—
142	C₂²×C₃	#909	1	2	yes	—
143	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
144	C₃×C₄	—	2	4	no	—
145	C₃×C₄	#869	1	2	yes	—
146	C₃×C₄	C₂×C₄⋊Dic₃	1	2	yes	—
147	Dic₃	—	24	2	no	—
148	Dic₃	—	24	1	no	—
149	Dic₃	—	24	1	no	—
150	C₉	#1847	1	1	yes	characteristic, complemented, Sylow
151	C₂³	C₂×D₃₆	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	36	2	no	—
155	C₂×C₄	—	36	1	no	—
156	C₂×C₄	—	36	1	no	—
157	C₂×C₄	—	36	4	no	—
158	C₂×C₄	—	36	2	no	—
159	C₂×C₄	—	36	2	no	—
160	C₂×C₄	C₄⋊Dic₉	1	4	yes	—
161	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
162	C₂×C₄	C₂×Dic₁₈	1	2	yes	—
163	C₂×C₄	C₂×D₃₆	1	2	yes	—
164	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
165	C₈	C₄⋊Dic₉	1	8	yes	complemented
166	C₂×C₃	#4702	1	2	yes	—
167	C₂×C₃	#4712	1	1	yes	characteristic
168	C₂×C₃	#4715	1	1	yes	characteristic
169	C₂×C₃	#4935	1	1	yes	characteristic
170	C₂×C₃	#5121	1	2	yes	—
171	C₂²	#63431	1	1	yes	characteristic
172	C₂²	#63436	1	1	yes	characteristic
173	C₂²	#63470	1	2	yes	—
174	C₂²	#63471	1	2	yes	—
175	C₂²	C₂×C₄⋊Dic₉	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	72	2	no	—
178	C₄	—	72	1	no	—
179	C₄	—	72	1	no	—
180	C₄	#63431	1	2	yes	—
181	C₄	C₂×C₄⋊Dic₉	1	2	yes	—
182	C₃	#69816	1	1	yes	characteristic
183	C₂	#10588382	1	2	yes	—
184	C₂	#10588393	1	1	yes	characteristic
185	C₂	#10588396	1	1	yes	characteristic
186	C₂	#10588615	1	1	yes	characteristic
187	C₂	#10588805	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₆
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#97	C₂
socle, prime radical	C₂³×C₃	C₂×D₁₂
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#97	D₉
hypocenter	C₉	#1847
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
13. C₉×#97	C₄	semidirect
62. C₈×C₉	C₄⋊C₄	semidirect
102. #97	Dic₉	semidirect
150. C₉	#1847	semidirect
165. C₈	C₄⋊Dic₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#97	2
C₂×#63378	2
C₂×#63377	2
#10588616	2
#10588615	2
#69816	3