Subgroups — #11784405

Order 1152.

336 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×#108	C₂	1	1	yes	maximal, characteristic
2	#10588797	C₂	1	2	yes	maximal
3	#10588787	C₂	1	1	yes	maximal, characteristic
4	C₂×#63375	C₂	1	1	yes	maximal, characteristic
5	#10588235	C₂	1	2	yes	maximal
6	#70703	—	3	1	no	maximal
7	C₂×C₉×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
9	#63488	C₂²	1	2	yes	—
10	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
11	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₉×#108	C₄	1	4	yes	complemented
13	#63378	C₄	1	2	yes	complemented
14	#63377	C₄	1	2	yes	complemented
15	#63375	—	2	2	no	—
16	#63375	C₄	1	4	yes	complemented
17	C₈×Dic₉	—	2	2	no	—
18	C₂×C₃×#108	D₃	1	1	yes	characteristic
19	#5122	—	3	2	no	—
20	#5106	—	3	1	no	—
21	C₂×#815	—	3	1	no	—
22	#4555	—	3	2	no	—
23	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
24	C₂²×Dic₉	—	2	1	no	—
25	C₉×M₁₆	—	2	2	no	—
26	C₉×M₁₆	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₉	—	2	2	no	—
28	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
32	#3676	—	2	2	no	—
33	C₄⋊Dic₉	C₂×C₄	1	4	yes	—
34	C₄×Dic₉	—	2	2	no	—
35	C₄×Dic₉	—	2	1	no	—
36	C₄×Dic₉	C₂×C₄	1	2	yes	—
37	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
38	C₂×C₉⋊C₈	D₄	1	2	yes	—
39	C₂×C₉⋊C₈	Q₈	1	2	yes	—
40	#1773	—	9	1	no	Sylow
41	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
42	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
43	#926	—	3	2	no	—
44	C₂×C₄×Dic₃	—	3	1	no	—
45	C₂²×C₃⋊C₈	—	3	1	no	—
46	C₃×#108	Dic₃	1	4	yes	—
47	#818	—	3	2	no	—
48	#817	—	3	2	no	—
49	#815	—	3	4	no	—
50	#815	—	6	2	no	—
51	C₈×Dic₃	—	6	2	no	—
52	C₂³×C₉	#41	1	1	yes	characteristic
53	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₉	C₂²×C₄	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	#41	1	1	yes	characteristic
59	C₂×Dic₉	—	2	2	no	—
60	C₂×Dic₉	—	2	4	no	—
61	C₂×Dic₉	—	4	2	no	—
62	C₈×C₉	—	2	4	no	—
63	C₈×C₉	C₄²	1	2	yes	—
64	C₈×C₉	C₄²	1	2	yes	—
65	C₉⋊C₈	C₄⋊C₄	1	4	yes	—
66	C₂×#108	D₉	1	1	yes	characteristic
67	#427	—	9	1	no	—
68	#405	—	9	2	no	—
69	C₂×#97	—	9	1	no	—
70	#339	—	9	2	no	—
71	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
72	C₂²×Dic₃	—	6	1	no	—
73	C₃×M₁₆	—	2	2	no	—
74	C₃×M₁₆	C₂×Dic₃	1	4	yes	—
75	C₂×C₃×C₈	—	2	2	no	—
76	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
77	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
78	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
80	#217	—	6	2	no	—
81	C₄⋊Dic₃	—	3	4	no	—
82	C₄×Dic₃	—	3	2	no	—
83	C₄×Dic₃	—	6	2	no	—
84	C₄×Dic₃	—	6	1	no	—
85	C₂×C₃⋊C₈	—	3	2	no	—
86	C₂×C₃⋊C₈	—	3	2	no	—
87	C₂×C₃⋊C₈	—	3	2	no	—
88	C₂²×C₉	—	2	2	no	—
89	C₂²×C₉	#117	1	1	yes	characteristic
90	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
91	C₂²×C₉	C₄×Q₈	1	1	yes	characteristic
92	C₄×C₉	C₂×C₄²	1	1	yes	characteristic
93	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₄×C₉	C₄×D₄	1	1	yes	characteristic
95	C₄×C₉	C₄×Q₈	1	1	yes	characteristic
96	Dic₉	—	4	2	no	—
97	Dic₉	—	4	4	no	—
98	C₂×M₁₆	C₂×D₉	1	2	yes	—
99	C₂²×C₈	—	9	1	no	—
100	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
101	#117	—	9	2	no	—
102	C₂×C₄²	—	9	1	no	—
103	#108	Dic₉	1	4	yes	complemented
104	#107	—	9	2	no	—
105	#106	—	9	2	no	—
106	#97	—	9	4	no	—
107	#97	—	18	2	no	—
108	C₄×C₈	—	18	2	no	—
109	C₂³×C₃	#239	1	1	yes	characteristic
110	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
111	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
112	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
113	C₂×C₃×C₄	#237	1	1	yes	characteristic
114	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
116	C₂×Dic₃	—	6	2	no	—
117	C₂×Dic₃	—	6	4	no	—
118	C₂×Dic₃	—	12	2	no	—
119	C₃×C₈	—	2	4	no	—
120	C₃×C₈	C₄×Dic₃	1	2	yes	—
121	C₃×C₈	C₄×Dic₃	1	2	yes	—
122	C₃⋊C₈	—	3	4	no	—
123	C₂×C₉	—	2	1	no	—
124	C₂×C₉	—	2	1	no	—
125	C₂×C₉	C₄×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₉	#444	1	2	yes	—
127	C₂²×C₄	—	18	1	no	—
128	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
129	M₁₆	—	2	2	no	—
130	M₁₆	C₂×Dic₉	1	4	yes	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	9	2	no	—
133	C₂×C₈	—	9	2	no	—
134	C₂×C₈	—	9	2	no	—
135	C₂×C₈	C₄×D₉	1	2	yes	—
136	C₂×C₈	C₄×D₉	1	1	yes	characteristic
137	C₂×C₈	C₄×D₉	1	1	yes	characteristic
138	C₂×C₈	C₂×Dic₉	1	2	yes	—
139	C₄⋊C₄	—	9	4	no	—
140	C₂²⋊C₄	—	18	2	no	—
141	C₄²	—	9	2	no	—
142	C₄²	—	18	2	no	—
143	C₄²	—	18	1	no	—
144	C₂²×C₃	—	2	2	no	—
145	C₂²×C₃	#892	1	1	yes	characteristic
146	C₂²×C₃	#893	1	1	yes	characteristic
147	C₂²×C₃	Q₈×Dic₃	1	1	yes	characteristic
148	C₃×C₄	#887	1	1	yes	characteristic
149	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
150	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
151	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
152	Dic₃	—	12	2	no	—
153	Dic₃	—	12	4	no	—
154	C₉	#1773	1	1	yes	characteristic, complemented, Sylow
155	C₂³	#3701	1	1	yes	characteristic
156	C₂×C₄	—	18	2	no	—
157	C₂×C₄	—	18	4	no	—
158	C₂×C₄	—	36	2	no	—
159	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
160	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
161	C₂×C₄	D₄×D₉	1	1	yes	characteristic
162	C₂×C₄	#3699	1	1	yes	characteristic
163	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
164	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
165	C₈	—	2	4	no	—
166	C₈	—	9	4	no	—
167	C₈	C₄×Dic₉	1	2	yes	—
168	C₈	C₄×Dic₉	1	2	yes	—
169	C₂×C₃	—	2	1	no	—
170	C₂×C₃	—	2	1	no	—
171	C₂×C₃	#4894	1	2	yes	—
172	C₂×C₃	Dic₃×C₄⋊C₄	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	#63454	1	1	yes	characteristic
175	C₂²	#63455	1	1	yes	characteristic
176	C₂²	Q₈×Dic₉	1	1	yes	characteristic
177	C₄	—	36	2	no	—
178	C₄	—	36	4	no	—
179	C₄	#63449	1	1	yes	characteristic
180	C₄	C₄⋊C₄×D₉	1	1	yes	characteristic
181	C₄	C₂×C₄×Dic₉	1	1	yes	characteristic
182	C₄	D₄×Dic₉	1	1	yes	characteristic
183	C₃	#70703	1	1	yes	characteristic
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#10588574	1	2	yes	—
187	C₂	C₄⋊C₄×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₉
commutator	C₄×C₉	C₂×C₄²
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#108	C₂
socle	C₂²×C₃	#893
prime radical	C₂³×C₃	#239
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#108	D₉
hypocenter	C₉	#1773
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
12. C₉×#108	C₄	semidirect
13. #63378	C₄	semidirect
14. #63377	C₄	semidirect
16. #63375	C₄	semidirect
103. #108	Dic₉	semidirect
154. C₉	#1773	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#108	2
#10588797	2
#10588787	2
C₂×#63375	2
#10588235	2
#70703	3