Subgroups — #11786244

Order 1152.

372 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	C₉×#382	C₂	1	1	yes	maximal, characteristic
2	C₂×#63367	C₂	1	2	yes	maximal
3	C₂×#63366	C₂	1	2	yes	maximal
4	#10588616	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#69549	—	3	1	no	maximal
7	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
10	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₉×#95	—	2	2	no	—
13	C₉⋊#95	—	2	2	no	—
14	#63367	—	2	4	no	—
15	#63366	—	2	4	no	—
16	C₄×C₉⋊C₈	C₄	1	4	yes	complemented
17	C₃×#382	D₃	1	1	yes	characteristic
18	C₂×#808	—	3	2	no	—
19	C₂×#807	—	3	2	no	—
20	#4936	—	3	1	no	—
21	C₂×C₄×C₃⋊C₈	—	3	1	no	—
22	C₂²×C₄×C₉	—	2	2	no	—
23	C₂²×C₄×C₉	D₄	1	2	yes	—
24	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
25	C₂²×Dic₉	—	2	2	no	—
26	C₉×C₄⋊C₄	—	2	2	no	—
27	C₉×C₄⋊C₄	—	2	2	no	—
28	C₄²×C₉	C₂×C₄	1	2	yes	—
29	C₄²×C₉	Q₈	1	2	yes	—
30	C₄⋊Dic₉	—	2	2	no	—
31	C₄⋊Dic₉	—	2	2	no	—
32	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
33	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
34	C₂×C₉⋊C₈	D₄	1	2	yes	—
35	C₂×C₉⋊C₈	D₄	1	2	yes	—
36	C₂×C₉⋊C₈	Q₈	1	2	yes	—
37	C₂×C₉⋊C₈	Q₈	1	2	yes	—
38	#1846	—	9	1	no	Sylow
39	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
40	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
41	C₂×C₄⋊Dic₃	—	3	2	no	—
42	C₂²×C₃⋊C₈	—	3	1	no	—
43	C₂²×C₃⋊C₈	—	3	1	no	—
44	C₃×#95	—	2	2	no	—
45	C₃⋊#95	—	6	2	no	—
46	#808	—	6	4	no	—
47	#807	—	6	4	no	—
48	C₄×C₃⋊C₈	—	3	4	no	—
49	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₉	—	2	4	no	—
51	C₂×C₄×C₉	—	4	2	no	—
52	C₂×C₄×C₉	—	4	4	no	—
53	C₂×C₄×C₉	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×Dic₉	—	4	2	no	—
61	C₂×Dic₉	—	4	4	no	—
62	C₉⋊C₈	C₄⋊C₄	1	4	yes	—
63	C₉⋊C₈	C₄⋊C₄	1	4	yes	—
64	C₂×#107	—	9	2	no	—
65	C₂×#106	—	9	2	no	—
66	C₂×C₄×C₈	—	9	1	no	—
67	#382	—	9	1	no	—
68	#382	D₉	1	1	yes	characteristic
69	C₂²×C₃×C₄	—	2	2	no	—
70	C₂²×C₃×C₄	#67	1	2	yes	—
71	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
72	C₂²×Dic₃	—	6	2	no	—
73	C₃×C₄⋊C₄	—	2	2	no	—
74	C₃×C₄⋊C₄	—	2	2	no	—
75	C₃×C₄²	Dic₆	1	2	yes	—
76	C₃×C₄²	C₄×D₃	1	2	yes	—
77	C₄⋊Dic₃	—	6	2	no	—
78	C₄⋊Dic₃	—	6	2	no	—
79	C₂×C₃⋊C₈	—	3	2	no	—
80	C₂×C₃⋊C₈	—	3	2	no	—
81	C₂×C₃⋊C₈	—	3	2	no	—
82	C₂×C₃⋊C₈	—	3	2	no	—
83	C₂×C₃⋊C₈	—	3	2	no	—
84	C₂×C₃⋊C₈	—	3	2	no	—
85	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₉	#127	1	1	yes	characteristic
87	C₂²×C₉	C₄⋊Q₈	1	1	yes	characteristic
88	C₂²×C₉	#135	1	4	yes	—
89	C₄×C₉	—	2	4	no	—
90	C₄×C₉	—	8	2	no	—
91	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
94	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
95	Dic₉	—	8	2	no	—
96	C₂²×C₈	—	9	1	no	—
97	C₂²×C₈	—	9	1	no	—
98	C₂×C₄⋊C₄	—	9	2	no	—
99	C₂×C₄⋊C₄	C₂×D₉	1	2	yes	—
100	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
101	#107	—	18	4	no	—
102	#106	—	18	4	no	—
103	C₄×C₈	—	9	4	no	—
104	#95	—	2	2	no	—
105	#95	—	18	2	no	—
106	C₂³×C₃	C₂×#67	1	1	yes	characteristic
107	C₂×C₃×C₄	—	2	4	no	—
108	C₂×C₃×C₄	—	4	2	no	—
109	C₂×C₃×C₄	—	4	4	no	—
110	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
111	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
113	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
114	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
115	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
116	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
117	C₂×Dic₃	—	12	2	no	—
118	C₂×Dic₃	—	12	4	no	—
119	C₃⋊C₈	—	3	4	no	—
120	C₃⋊C₈	—	3	4	no	—
121	C₂×C₉	#384	1	1	yes	characteristic
122	C₂×C₉	#427	1	2	yes	—
123	C₂×C₉	#495	1	2	yes	—
124	C₂×C₉	#499	1	2	yes	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	18	2	no	—
127	C₂²×C₄	#611	1	2	yes	—
128	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
129	C₂×C₈	—	9	2	no	—
130	C₂×C₈	—	9	2	no	—
131	C₂×C₈	—	9	2	no	—
132	C₂×C₈	—	9	2	no	—
133	C₂×C₈	—	9	2	no	—
134	C₂×C₈	—	9	2	no	—
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	18	2	no	—
138	C₄⋊C₄	—	18	2	no	—
139	C₄²	Dic₁₈	1	2	yes	—
140	C₄²	C₄×D₉	1	2	yes	—
141	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
142	C₂²×C₃	#940	1	1	yes	characteristic
143	C₂²×C₃	#944	1	1	yes	characteristic
144	C₂²×C₃	#950	1	4	yes	—
145	C₃×C₄	—	2	4	no	—
146	C₃×C₄	—	8	2	no	—
147	C₃×C₄	#888	1	1	yes	characteristic
148	C₃×C₄	#888	1	1	yes	characteristic
149	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
150	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
151	Dic₃	—	24	2	no	—
152	C₉	#1846	1	1	yes	characteristic, complemented, Sylow
153	C₂³	C₂×#611	1	1	yes	characteristic
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	4	no	—
157	C₂×C₄	—	36	2	no	—
158	C₂×C₄	—	36	4	no	—
159	C₂×C₄	Dic₉⋊C₄	1	4	yes	—
160	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
161	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
162	C₂×C₄	D₄×D₉	1	1	yes	characteristic
163	C₂×C₄	D₄×D₉	1	1	yes	characteristic
164	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
165	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
166	C₈	—	9	4	no	—
167	C₈	—	9	4	no	—
168	C₂×C₃	#4975	1	1	yes	characteristic
169	C₂×C₃	#4998	1	2	yes	—
170	C₂×C₃	#5059	1	2	yes	—
171	C₂×C₃	#5065	1	2	yes	—
172	C₂²	C₂×Dic₉⋊C₄	1	1	yes	characteristic
173	C₂²	#63502	1	1	yes	characteristic
174	C₂²	#63506	1	1	yes	characteristic
175	C₂²	#63513	1	4	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	8	2	no	—
178	C₄	—	72	2	no	—
179	C₄	#63450	1	1	yes	characteristic
180	C₄	#63450	1	1	yes	characteristic
181	C₄	C₄⋊C₄×D₉	1	1	yes	characteristic
182	C₄	C₄⋊C₄×D₉	1	1	yes	characteristic
183	C₃	#69549	1	1	yes	characteristic
184	C₂	#10588655	1	1	yes	characteristic
185	C₂	#10588676	1	2	yes	—
186	C₂	#10588740	1	2	yes	—
187	C₂	#10588744	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#611
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#382	C₂
socle, prime radical	C₂³×C₃	C₂×#67
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#382	D₉
hypocenter	C₉	#1846
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

Complements

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

N	H ≅ G/N	type
16. C₄×C₉⋊C₈	C₄	semidirect
152. C₉	#1846	semidirect

Maximal subgroups

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

subgroup	index
C₉×#382	2
C₂×#63367	2
C₂×#63366	2
#10588616	2
C₂×C₄×C₉⋊C₈	2
#69549	3