Subgroups — #11786001

Order 1152.

258 conjugacy classes of subgroups.

The subgroup lattice has 198 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	#10588787	C₂	1	1	yes	maximal, characteristic
3	#10588782	C₂	1	1	yes	maximal, characteristic
4	#10588237	C₂	1	2	yes	maximal
5	#10588232	C₂	1	2	yes	maximal
6	#70698	—	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	#63488	C₂²	1	1	yes	characteristic
11	C₂×#3667	C₂²	1	1	yes	characteristic
12	C₉×#108	—	2	1	no	—
13	C₉×#108	—	2	1	no	—
14	#63378	—	2	1	no	—
15	#63377	—	2	1	no	—
16	#63375	C₄	1	2	yes	complemented
17	C₈×Dic₉	C₄	1	2	yes	complemented
18	C₂×C₃×#108	D₃	1	1	yes	characteristic
19	#5106	—	3	1	no	—
20	#5101	—	3	1	no	—
21	#4559	—	3	2	no	—
22	#4557	—	3	2	no	—
23	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
24	C₉×M₁₆	—	2	2	no	—
25	C₉×M₁₆	—	2	2	no	—
26	C₂×C₈×C₉	—	2	2	no	—
27	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₉	D₄	1	1	yes	characteristic
30	C₂×C₈×C₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₉	Q₈	1	1	yes	characteristic
32	C₂×C₈×C₉	Q₈	1	1	yes	characteristic
33	#3676	—	2	1	no	—
34	#3676	—	2	2	no	—
35	C₄⋊Dic₉	—	2	1	no	—
36	C₄⋊Dic₉	—	2	2	no	—
37	C₄×Dic₉	—	2	2	no	—
38	C₄×Dic₉	C₂×C₄	1	2	yes	—
39	#3667	—	2	1	no	—
40	#3667	—	2	1	no	—
41	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
42	#1833	—	9	1	no	Sylow
43	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
44	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
45	#926	—	3	1	no	—
46	#926	—	3	2	no	—
47	C₂×#208	—	3	1	no	—
48	C₃×#108	—	2	1	no	—
49	C₃×#108	—	2	1	no	—
50	#818	—	6	1	no	—
51	#817	—	6	1	no	—
52	#815	—	3	2	no	—
53	C₈×Dic₃	—	3	2	no	—
54	C₂³×C₉	#41	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×C₄×C₉	#41	1	1	yes	characteristic
60	C₂×C₄×C₉	#41	1	1	yes	characteristic
61	C₂×Dic₉	—	2	2	no	—
62	C₂×Dic₉	—	4	2	no	—
63	C₈×C₉	—	4	2	no	—
64	C₈×C₉	C₄⋊C₄	1	1	yes	characteristic
65	C₈×C₉	C₄⋊C₄	1	1	yes	characteristic
66	C₈×C₉	C₄⋊C₄	1	1	yes	characteristic
67	C₈×C₉	C₄⋊C₄	1	1	yes	characteristic
68	C₉⋊C₈	—	2	2	no	—
69	C₂×#108	D₉	1	1	yes	characteristic
70	#427	—	9	1	no	—
71	#405	—	9	1	no	—
72	#344	—	9	2	no	—
73	H(3,ℤ/4)	—	9	2	no	—
74	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
75	C₃×M₁₆	—	2	2	no	—
76	C₃×M₁₆	—	2	2	no	—
77	C₂×C₃×C₈	—	2	2	no	—
78	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
79	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
80	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
81	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
82	C₂×C₃×C₈	#67	1	1	yes	characteristic
83	C₂×C₃×C₈	#67	1	1	yes	characteristic
84	#217	—	6	1	no	—
85	#217	—	6	2	no	—
86	C₄⋊Dic₃	—	6	1	no	—
87	C₄⋊Dic₃	—	6	2	no	—
88	C₄×Dic₃	—	3	2	no	—
89	C₄×Dic₃	—	6	2	no	—
90	#208	—	6	1	no	—
91	#208	—	6	1	no	—
92	C₂×C₃⋊C₈	—	3	2	no	—
93	C₂²×C₉	—	4	1	no	—
94	C₂²×C₉	#117	1	1	yes	characteristic
95	C₂²×C₉	#124	1	1	yes	characteristic
96	C₂²×C₉	#125	1	1	yes	characteristic
97	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₉	#117	1	1	yes	characteristic
99	C₄×C₉	#124	1	1	yes	characteristic
100	C₄×C₉	#125	1	1	yes	characteristic
101	Dic₉	—	4	2	no	—
102	Dic₉	—	8	2	no	—
103	C₂×M₁₆	—	9	1	no	—
104	C₂×M₁₆	C₂×D₉	1	2	yes	—
105	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
106	#117	—	9	1	no	—
107	#117	—	9	2	no	—
108	#108	—	2	1	no	—
109	#108	—	2	1	no	—
110	#107	—	18	1	no	—
111	#106	—	18	1	no	—
112	#97	—	9	2	no	—
113	C₄×C₈	—	9	2	no	—
114	C₂³×C₃	#239	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
117	C₂×C₃×C₄	#237	1	1	yes	characteristic
118	C₂×C₃×C₄	#237	1	1	yes	characteristic
119	C₂×C₃×C₄	#239	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
121	C₂×Dic₃	—	6	2	no	—
122	C₂×Dic₃	—	12	2	no	—
123	C₃×C₈	—	4	2	no	—
124	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
125	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
126	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
127	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
128	C₃⋊C₈	—	6	2	no	—
129	C₂×C₉	—	2	1	no	—
130	C₂×C₉	—	2	1	no	—
131	C₂×C₉	—	2	1	no	—
132	C₂×C₉	#382	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
134	M₁₆	—	2	2	no	—
135	M₁₆	—	2	2	no	—
136	M₁₆	—	18	1	no	—
137	M₁₆	—	18	1	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	9	2	no	—
140	C₂×C₈	Dic₁₈	1	1	yes	characteristic
141	C₂×C₈	Dic₁₈	1	1	yes	characteristic
142	C₂×C₈	C₄×D₉	1	1	yes	characteristic
143	C₂×C₈	C₄×D₉	1	1	yes	characteristic
144	C₂×C₈	#611	1	1	yes	characteristic
145	C₂×C₈	#611	1	1	yes	characteristic
146	C₄⋊C₄	—	18	1	no	—
147	C₄⋊C₄	—	18	2	no	—
148	C₂²⋊C₄	—	18	1	no	—
149	C₂²⋊C₄	—	18	2	no	—
150	C₄²	—	9	2	no	—
151	C₄²	—	18	2	no	—
152	C₂²×C₃	—	4	1	no	—
153	C₂²×C₃	#890	1	1	yes	characteristic
154	C₂²×C₃	#892	1	1	yes	characteristic
155	C₂²×C₃	#947	1	1	yes	characteristic
156	C₃×C₄	#890	1	1	yes	characteristic
157	C₃×C₄	#892	1	1	yes	characteristic
158	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
159	C₃×C₄	#936	1	1	yes	characteristic
160	Dic₃	—	12	2	no	—
161	Dic₃	—	24	2	no	—
162	C₉	#1833	1	1	yes	characteristic, complemented, Sylow
163	C₂³	#3701	1	1	yes	characteristic
164	C₂×C₄	—	18	2	no	—
165	C₂×C₄	—	36	2	no	—
166	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
167	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
168	C₂×C₄	#3699	1	1	yes	characteristic
169	C₂×C₄	#3699	1	1	yes	characteristic
170	C₂×C₄	#3701	1	1	yes	characteristic
171	C₂×C₄	C₂×#611	1	1	yes	characteristic
172	C₈	—	4	2	no	—
173	C₈	—	18	2	no	—
174	C₈	Dic₉⋊C₄	1	1	yes	characteristic
175	C₈	Dic₉⋊C₄	1	1	yes	characteristic
176	C₈	Dic₉⋊C₄	1	1	yes	characteristic
177	C₈	Dic₉⋊C₄	1	1	yes	characteristic
178	C₂×C₃	—	2	1	no	—
179	C₂×C₃	—	2	1	no	—
180	C₂×C₃	—	2	1	no	—
181	C₂×C₃	#4977	1	1	yes	characteristic
182	C₂²	—	4	1	no	—
183	C₂²	#63452	1	1	yes	characteristic
184	C₂²	#63454	1	1	yes	characteristic
185	C₂²	#63509	1	1	yes	characteristic
186	C₄	—	36	2	no	—
187	C₄	—	72	2	no	—
188	C₄	#63452	1	1	yes	characteristic
189	C₄	#63454	1	1	yes	characteristic
190	C₄	C₂×Dic₉⋊C₄	1	1	yes	characteristic
191	C₄	#63498	1	1	yes	characteristic
192	C₃	#70698	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#10588657	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#63454
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#108	C₂
socle	C₂×C₃	#4977
prime radical	C₂³×C₃	#239
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#108	D₉
hypocenter	C₉	#1833
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
16. #63375	C₄	semidirect
17. C₈×Dic₉	C₄	semidirect
162. C₉	#1833	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
#10588787	2
#10588782	2
#10588237	2
#10588232	2
#70698	3