Subgroups — #11783254

Order 1152.

636 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#63488	C₂	1	2	yes	maximal
3	#10588229	C₂	1	4	yes	maximal, complemented
4	#71969	—	3	1	no	maximal
5	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
6	C₂²×C₈×C₉	—	2	4	no	—
7	C₂²×C₈×C₉	C₄	1	4	yes	complemented
8	C₂²×C₈×C₉	C₂²	1	2	yes	—
9	C₂×#3676	—	2	2	no	—
10	#63488	—	2	4	no	—
11	#63488	C₄	1	8	yes	complemented
12	C₂×C₄⋊Dic₉	C₂²	1	4	yes	—
13	C₂×C₄×Dic₉	—	2	2	no	—
14	C₂³×C₃×C₈	D₃	1	1	yes	characteristic
15	C₂×#926	—	3	2	no	—
16	#4551	—	3	4	no	—
17	C₂⁴×C₉	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₉	D₄	1	4	yes	—
22	C₂²×C₄×C₉	Q₈	1	2	yes	—
23	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
24	C₂²×Dic₉	—	2	4	no	—
25	C₂×C₈×C₉	—	2	8	no	—
26	C₂×C₈×C₉	—	2	4	no	—
27	C₂×C₈×C₉	C₂×C₄	1	4	yes	—
28	#3676	—	2	8	no	—
29	C₄⋊Dic₉	—	2	4	no	—
30	C₄⋊Dic₉	C₂×C₄	1	8	yes	—
31	C₄×Dic₉	—	2	8	no	—
32	#1735	—	9	1	no	Sylow
33	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
34	C₂²×C₃×C₈	—	2	4	no	—
35	C₂²×C₃×C₈	Dic₃	1	4	yes	—
36	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
37	C₂×#217	—	6	2	no	—
38	#926	—	3	8	no	—
39	#926	—	6	4	no	—
40	C₂×C₄⋊Dic₃	—	3	4	no	—
41	C₂×C₄×Dic₃	—	6	2	no	—
42	C₂³×C₉	—	2	4	no	—
43	C₂³×C₉	C₂²⋊C₄	1	4	yes	—
44	C₂³×C₉	C₄⋊C₄	1	2	yes	—
45	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₉	C₄²	1	4	yes	—
47	C₂×C₄×C₉	C₂²⋊C₄	1	4	yes	—
48	C₂×C₄×C₉	C₂²⋊C₄	1	4	yes	—
49	C₂×C₄×C₉	C₄⋊C₄	1	8	yes	—
50	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
51	C₂×C₄×C₉	C₂²×C₄	1	2	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	1	yes	characteristic
55	C₂×Dic₉	—	2	8	no	—
56	C₂×Dic₉	—	4	8	no	—
57	C₈×C₉	—	2	4	no	—
58	C₂³×C₈	D₉	1	1	yes	characteristic
59	C₂×#117	—	9	2	no	—
60	#340	—	9	4	no	—
61	C₂⁴×C₃	D₁₂	1	1	yes	characteristic
62	C₂²×C₃×C₄	Dic₆	1	2	yes	—
63	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
64	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
66	C₂²×C₃×C₄	#67	1	4	yes	—
67	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
68	C₂²×Dic₃	—	6	4	no	—
69	C₂×C₃×C₈	—	2	8	no	—
70	C₂×C₃×C₈	—	2	4	no	—
71	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
72	#217	—	6	8	no	—
73	C₄⋊Dic₃	—	3	8	no	—
74	C₄⋊Dic₃	—	6	4	no	—
75	C₄×Dic₃	—	6	8	no	—
76	C₂²×C₉	—	2	8	no	—
77	C₂²×C₉	—	2	4	no	—
78	C₂²×C₉	#95	1	4	yes	—
79	C₂²×C₉	C₂×C₂²⋊C₄	1	2	yes	—
80	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₉	#135	1	4	yes	—
82	C₄×C₉	#95	1	4	yes	—
83	C₄×C₉	C₂×C₄²	1	1	yes	characteristic
84	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₄×C₉	C₂×C₄⋊C₄	1	2	yes	—
86	Dic₉	—	4	8	no	—
87	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
88	C₂²×C₈	—	2	4	no	—
89	C₂²×C₈	Dic₉	1	4	yes	complemented
90	C₂²×C₈	C₂×D₉	1	2	yes	—
91	#117	—	9	8	no	—
92	#117	—	18	4	no	—
93	C₂×C₄⋊C₄	—	9	4	no	—
94	C₂×C₂²⋊C₄	—	18	2	no	—
95	C₂×C₄²	—	18	2	no	—
96	C₂³×C₃	—	2	4	no	—
97	C₂³×C₃	C₄⋊Dic₃	1	2	yes	—
98	C₂³×C₃	(C₂×D₃)⋊C₄	1	4	yes	—
99	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
100	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
101	C₂×C₃×C₄	Dic₃⋊C₄	1	8	yes	—
102	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
103	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
104	C₂×C₃×C₄	#217	1	4	yes	—
105	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
106	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
107	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
108	C₂×C₃×C₄	C₂×#67	1	2	yes	—
109	C₂×Dic₃	—	6	8	no	—
110	C₂×Dic₃	—	12	8	no	—
111	C₃×C₈	—	2	4	no	—
112	C₂×C₉	—	2	4	no	—
113	C₂×C₉	C₂×#95	1	1	yes	characteristic
114	C₂×C₉	#416	1	4	yes	—
115	C₂×C₉	#427	1	2	yes	—
116	C₂⁴	D₃₆	1	1	yes	characteristic
117	C₂²×C₄	—	18	4	no	—
118	C₂²×C₄	Dic₁₈	1	2	yes	—
119	C₂²×C₄	C₄×D₉	1	4	yes	—
120	C₂²×C₄	D₃₆	1	1	yes	characteristic
121	C₂²×C₄	C₂×Dic₉	1	2	yes	—
122	C₂²×C₄	#611	1	4	yes	—
123	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
124	C₂×C₈	—	2	8	no	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	C₂×Dic₉	1	4	yes	—
127	C₄⋊C₄	—	9	8	no	—
128	C₄⋊C₄	—	18	4	no	—
129	C₂²⋊C₄	—	18	8	no	—
130	C₄²	—	18	8	no	—
131	C₂²×C₃	—	2	4	no	—
132	C₂²×C₃	—	2	8	no	—
133	C₂²×C₃	C₃⋊#95	1	4	yes	—
134	C₂²×C₃	#904	1	4	yes	—
135	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
136	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	2	yes	—
137	C₃×C₄	C₃⋊#95	1	4	yes	—
138	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
139	C₃×C₄	C₂×Dic₃⋊C₄	1	2	yes	—
140	C₃×C₄	C₂×#217	1	1	yes	characteristic
141	Dic₃	—	12	8	no	—
142	C₉	#1735	1	1	yes	characteristic, complemented, Sylow
143	C₂³	—	2	4	no	—
144	C₂³	C₄⋊Dic₉	1	2	yes	—
145	C₂³	(C₂×D₉)⋊C₄	1	4	yes	—
146	C₂³	C₂×D₃₆	1	1	yes	characteristic
147	C₂×C₄	—	18	8	no	—
148	C₂×C₄	—	36	8	no	—
149	C₂×C₄	C₄×Dic₉	1	4	yes	—
150	C₂×C₄	Dic₉⋊C₄	1	8	yes	—
151	C₂×C₄	C₄⋊Dic₉	1	2	yes	—
152	C₂×C₄	(C₂×D₉)⋊C₄	1	4	yes	—
153	C₂×C₄	#3676	1	4	yes	—
154	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
156	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
157	C₂×C₄	C₂×#611	1	2	yes	—
158	C₈	—	2	4	no	—
159	C₂×C₃	—	2	4	no	—
160	C₂×C₃	#5106	1	2	yes	—
161	C₂×C₃	#5113	1	4	yes	—
162	C₂×C₃	C₂×C₃⋊#95	1	1	yes	characteristic
163	C₂²	—	2	4	no	—
164	C₂²	—	2	8	no	—
165	C₂²	C₉⋊#95	1	4	yes	—
166	C₂²	#63467	1	4	yes	—
167	C₂²	C₂×C₄⋊Dic₉	1	1	yes	characteristic
168	C₂²	C₂×(C₂×D₉)⋊C₄	1	2	yes	—
169	C₄	—	36	8	no	—
170	C₄	C₉⋊#95	1	4	yes	—
171	C₄	C₂×C₄×Dic₉	1	1	yes	characteristic
172	C₄	C₂×Dic₉⋊C₄	1	2	yes	—
173	C₄	C₂×#3676	1	1	yes	characteristic
174	C₃	#71969	1	1	yes	characteristic
175	C₂	—	2	4	no	—
176	C₂	#10588787	1	2	yes	—
177	C₂	#10588794	1	4	yes	—
178	C₂	C₂×C₉⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₃₆
commutator	C₄×C₉	C₂×C₄²
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂³×C₈×C₉	C₂
socle	C₂³×C₃	C₂×D₁₂
prime radical	C₂⁴×C₃	D₁₂
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂³×C₈	D₉
hypocenter	C₉	#1735
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. #10588229	C₂	semidirect
7. C₂²×C₈×C₉	C₄	semidirect
11. #63488	C₄	semidirect
89. C₂²×C₈	Dic₉	semidirect
142. C₉	#1735	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₈×C₉	2
C₂×#63488	2
#10588229	2
#71969	3