Subgroups — C₉×#1798 #11766519

Order 1152.

468 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₉×C₂²⋊C₈	C₂	1	2	yes	maximal
4	C₉×#336	C₂	1	2	yes	maximal, complemented
5	C₃×#1798	C₃	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₉	C₂²	1	4	yes	—
8	C₉×#117	—	2	4	no	—
9	C₂×C₉×C₄⋊C₄	C₄	1	2	yes	—
10	C₂×C₉×C₂²⋊C₄	C₄	1	2	yes	—
11	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₉×C₄⋊C₈	—	2	4	no	—
14	C₉×C₂²⋊C₈	—	2	4	no	—
15	C₂×C₃×#117	C₂×C₃	1	1	yes	characteristic
16	C₂×C₃×C₄⋊C₈	C₂×C₃	1	2	yes	—
17	C₂×C₃×C₂²⋊C₈	C₂×C₃	1	2	yes	—
18	C₃×#336	C₂×C₃	1	2	yes	—
19	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₉	—	2	2	no	—
21	C₂²×C₄×C₉	—	2	2	no	—
22	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₉	D₄	1	2	yes	—
26	C₂²×C₄×C₉	Q₈	1	2	yes	—
27	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₉	—	2	4	no	—
29	C₂×C₈×C₉	—	2	4	no	—
30	C₂×C₈×C₉	—	2	4	no	—
31	C₉×C₄⋊C₄	—	2	4	no	—
32	C₉×C₂²⋊C₄	—	2	4	no	—
33	C₄²×C₉	—	2	2	no	—
34	C₄²×C₉	D₄	1	4	yes	—
35	#1798	C₉	1	1	yes	characteristic, direct product factor, Sylow
36	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
37	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
38	C₃×#117	—	2	4	no	—
39	C₂×C₃×C₄⋊C₄	C₃×C₄	1	2	yes	—
40	C₂×C₃×C₂²⋊C₄	C₃×C₄	1	2	yes	—
41	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
43	C₃×C₄⋊C₈	—	2	4	no	—
44	C₃×C₂²⋊C₈	—	2	4	no	—
45	C₂³×C₉	—	2	1	no	—
46	C₂³×C₉	—	2	1	no	—
47	C₂³×C₉	—	2	4	no	—
48	C₂³×C₉	C₄⋊C₄	1	2	yes	—
49	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₉	—	2	1	no	—
51	C₂×C₄×C₉	—	2	1	no	—
52	C₂×C₄×C₉	—	2	4	no	—
53	C₂×C₄×C₉	—	2	4	no	—
54	C₂×C₄×C₉	—	2	4	no	—
55	C₂×C₄×C₉	—	2	2	no	—
56	C₂×C₄×C₉	—	2	4	no	—
57	C₂×C₄×C₉	C₂²⋊C₄	1	4	yes	—
58	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₉	#41	1	2	yes	—
63	C₈×C₉	—	4	4	no	—
64	C₂×#117	C₂×C₉	1	1	yes	characteristic
65	C₂×C₄⋊C₈	C₂×C₉	1	2	yes	complemented
66	C₂×C₂²⋊C₈	C₂×C₉	1	2	yes	—
67	#336	C₂×C₉	1	2	yes	complemented
68	C₂⁴×C₃	C₂×C₃×C₄	1	1	yes	characteristic
69	C₂²×C₃×C₄	—	2	2	no	—
70	C₂²×C₃×C₄	—	2	2	no	—
71	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
72	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
73	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
75	C₂²×C₃×C₄	C₃×Q₈	1	2	yes	—
76	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
77	C₂×C₃×C₈	—	2	4	no	—
78	C₂×C₃×C₈	—	2	4	no	—
79	C₂×C₃×C₈	—	2	4	no	—
80	C₃×C₄⋊C₄	—	2	4	no	—
81	C₃×C₂²⋊C₄	—	2	4	no	—
82	C₃×C₄²	—	2	2	no	—
83	C₃×C₄²	C₃×D₄	1	4	yes	—
84	C₂²×C₉	—	2	2	no	—
85	C₂²×C₉	—	2	4	no	—
86	C₂²×C₉	—	4	2	no	—
87	C₂²×C₉	—	4	2	no	—
88	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₉	#117	1	1	yes	characteristic
91	C₂²×C₉	#131	1	4	yes	—
92	C₄×C₉	—	2	2	no	—
93	C₄×C₉	—	2	4	no	—
94	C₄×C₉	—	4	2	no	—
95	C₄×C₉	#121	1	2	yes	—
96	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
97	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
98	C₂²×C₈	C₂²×C₉	1	4	yes	—
99	#117	—	2	4	no	—
100	C₂×C₄⋊C₄	C₄×C₉	1	2	yes	—
101	C₂×C₂²⋊C₄	C₄×C₉	1	2	yes	—
102	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
103	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
104	C₄⋊C₈	—	2	4	no	—
105	C₂²⋊C₈	—	2	4	no	—
106	C₂³×C₃	—	2	1	no	—
107	C₂³×C₃	—	2	1	no	—
108	C₂³×C₃	—	2	4	no	—
109	C₂³×C₃	C₃×C₄⋊C₄	1	2	yes	—
110	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
111	C₂×C₃×C₄	—	2	1	no	—
112	C₂×C₃×C₄	—	2	1	no	—
113	C₂×C₃×C₄	—	2	4	no	—
114	C₂×C₃×C₄	—	2	4	no	—
115	C₂×C₃×C₄	—	2	4	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	4	no	—
118	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	4	yes	—
119	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
120	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
122	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
123	C₂×C₃×C₄	C₃×#41	1	2	yes	—
124	C₃×C₈	—	4	4	no	—
125	C₂×C₉	—	4	2	no	—
126	C₂×C₉	#380	1	1	yes	characteristic
127	C₂×C₉	#408	1	2	yes	—
128	C₂×C₉	#424	1	2	yes	—
129	C₂×C₉	#433	1	2	yes	—
130	C₂⁴	C₂×C₄×C₉	1	1	yes	characteristic
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
134	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
135	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
136	C₂²×C₄	D₄×C₉	1	2	yes	—
137	C₂²×C₄	Q₈×C₉	1	2	yes	—
138	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₂²⋊C₄	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	D₄×C₉	1	4	yes	—
146	C₂²×C₃	—	2	2	no	—
147	C₂²×C₃	—	2	4	no	—
148	C₂²×C₃	—	4	2	no	—
149	C₂²×C₃	—	4	2	no	—
150	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₃	C₃×#117	1	1	yes	characteristic
153	C₂²×C₃	C₃×#131	1	4	yes	—
154	C₃×C₄	—	2	2	no	—
155	C₃×C₄	—	2	4	no	—
156	C₃×C₄	—	4	2	no	—
157	C₃×C₄	C₃×#121	1	2	yes	—
158	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
159	C₉	#1798	1	1	yes	characteristic, direct product factor, Sylow
160	C₂³	—	2	1	no	—
161	C₂³	—	2	1	no	—
162	C₂³	—	2	4	no	—
163	C₂³	C₉×C₄⋊C₄	1	2	yes	—
164	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	C₉×C₂²⋊C₄	1	4	yes	—
173	C₂×C₄	C₉×C₄⋊C₄	1	2	yes	—
174	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
175	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
176	C₂×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
177	C₂×C₄	C₉×#41	1	2	yes	—
178	C₈	—	4	4	no	—
179	C₂×C₃	—	4	2	no	—
180	C₂×C₃	C₃×#380	1	1	yes	characteristic
181	C₂×C₃	C₃×#408	1	2	yes	—
182	C₂×C₃	C₃×#424	1	2	yes	—
183	C₂×C₃	C₃×#433	1	2	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	2	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
189	C₂²	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
190	C₂²	C₉×#117	1	1	yes	characteristic
191	C₂²	C₉×#131	1	4	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	2	4	no	—
194	C₄	—	4	2	no	—
195	C₄	C₉×#121	1	2	yes	—
196	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
197	C₃	C₃×#1798	1	1	yes	characteristic
198	C₂	—	4	2	no	—
199	C₂	C₉×#380	1	1	yes	characteristic
200	C₂	C₉×#408	1	2	yes	—
201	C₂	C₉×#424	1	2	yes	—
202	C₂	C₉×#433	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₉	C₂³
commutator	C₂³	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂²×C₃×C₄
prime radical	C₂⁴×C₃	C₂×C₃×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×C₄⋊C₈	C₂	semidirect
4. C₉×#336	C₂	semidirect
35. #1798	C₉	direct (N × H)
65. C₂×C₄⋊C₈	C₂×C₉	semidirect
67. #336	C₂×C₉	semidirect
159. C₉	#1798	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×#117	2
C₂×C₉×C₄⋊C₈	2
C₂×C₉×C₂²⋊C₈	2
C₉×#336	2
C₃×#1798	3

Subgroups — C9×#1798 #11766519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1798 #11766519