Subgroups — #11774328

Order 1152.

372 conjugacy classes of subgroups.

The subgroup lattice has 237 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₈	C₂	1	1	yes	maximal, characteristic
2	C₉×#431	C₂	1	1	yes	maximal, characteristic
3	#10588605	C₂	1	1	yes	maximal, characteristic
4	#10588160	C₂	1	4	yes	maximal, complemented
5	#70766	—	3	1	no	maximal
6	C₂²×Dic₁₈	C₄	1	1	yes	characteristic
7	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₉	C₄	1	1	yes	characteristic
9	C₂×Dic₉⋊C₄	—	2	1	no	—
10	C₂×C₄×Dic₉	—	2	1	no	—
11	C₄×Dic₁₈	—	2	1	no	—
12	C₄×Dic₁₈	—	2	1	no	—
13	C₄×Dic₁₈	C₄	1	2	yes	—
14	C₄×Dic₁₈	C₂²	1	2	yes	—
15	C₉×C₄⋊C₈	C₂²	1	2	yes	—
16	C₉×C₂²⋊C₈	—	2	1	no	—
17	C₄×C₈×C₉	—	2	1	no	—
18	#63389	—	2	1	no	—
19	C₄⋊(C₉⋊C₈)	—	2	1	no	—
20	C₄×C₉⋊C₈	C₂²	1	2	yes	—
21	C₂×C₄×Dic₆	—	3	1	no	—
22	C₃×#431	D₃	1	1	yes	characteristic
23	#4925	—	3	1	no	—
24	#4484	—	3	4	no	—
25	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂²×Dic₉	—	2	1	no	—
29	C₂²×Dic₉	—	2	1	no	—
30	C₂×Dic₁₈	—	2	1	no	—
31	C₂×Dic₁₈	—	2	2	no	—
32	C₂×Dic₁₈	C₈	1	4	yes	complemented
33	C₂×Dic₁₈	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₉	—	2	2	no	—
35	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₉	D₄	1	1	yes	characteristic
37	C₄²×C₉	D₄	1	1	yes	characteristic
38	C₄²×C₉	C₂³	1	1	yes	characteristic
39	C₄⋊Dic₉	—	2	1	no	—
40	C₄⋊Dic₉	C₂×C₄	1	2	yes	—
41	Dic₉⋊C₄	—	2	2	no	—
42	Dic₉⋊C₄	—	2	2	no	—
43	C₄×Dic₉	—	2	2	no	—
44	C₄×Dic₉	—	2	2	no	—
45	C₂×C₉⋊C₈	—	2	2	no	—
46	#1479	—	9	1	no	Sylow
47	C₂²×Dic₆	—	3	1	no	—
48	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
49	C₂×C₄⋊Dic₃	—	3	1	no	—
50	C₂×Dic₃⋊C₄	—	6	1	no	—
51	C₂×C₄×Dic₃	—	6	1	no	—
52	C₄×Dic₆	—	3	2	no	—
53	C₄×Dic₆	—	3	2	no	—
54	C₄×Dic₆	—	6	1	no	—
55	C₄×Dic₆	—	6	1	no	—
56	C₃×C₄⋊C₈	C₂×D₃	1	2	yes	—
57	C₃×C₂²⋊C₈	—	2	1	no	—
58	C₃×C₄×C₈	—	2	1	no	—
59	#830	—	6	1	no	—
60	C₄⋊(C₃⋊C₈)	—	6	1	no	—
61	C₄×C₃⋊C₈	—	3	2	no	—
62	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₉	—	2	2	no	—
64	C₂×C₄×C₉	—	4	1	no	—
65	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
69	C₂×C₄×C₉	M₁₆	1	2	yes	—
70	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
73	C₂×Dic₉	—	2	4	no	—
74	C₂×Dic₉	—	2	2	no	—
75	C₂×Dic₉	—	2	2	no	—
76	C₂×Dic₉	—	4	2	no	—
77	Dic₁₈	—	2	4	no	—
78	Dic₁₈	—	2	2	no	—
79	Dic₁₈	C₂×C₈	1	4	yes	—
80	C₈×C₉	—	4	2	no	—
81	C₉⋊C₈	—	4	2	no	—
82	C₂×C₄×Q₈	—	9	1	no	—
83	#431	—	9	1	no	—
84	#431	D₉	1	1	yes	characteristic
85	Q₈⋊C₈	—	9	4	no	—
86	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
87	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
88	C₂²×C₃×C₄	#67	1	1	yes	characteristic
89	C₂²×Dic₃	—	6	1	no	—
90	C₂²×Dic₃	—	6	1	no	—
91	C₂×Dic₆	—	3	4	no	—
92	C₂×Dic₆	—	3	2	no	—
93	C₂×Dic₆	—	6	2	no	—
94	C₂×Dic₆	—	6	1	no	—
95	C₂×C₃×C₈	—	2	2	no	—
96	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
97	C₃×C₄²	D₁₂	1	1	yes	characteristic
98	C₃×C₄²	#67	1	1	yes	characteristic
99	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
100	C₄⋊Dic₃	—	3	2	no	—
101	C₄⋊Dic₃	—	6	1	no	—
102	Dic₃⋊C₄	—	6	2	no	—
103	Dic₃⋊C₄	—	6	2	no	—
104	C₄×Dic₃	—	6	2	no	—
105	C₄×Dic₃	—	6	2	no	—
106	C₂×C₃⋊C₈	—	6	2	no	—
107	C₂²×C₉	—	2	1	no	—
108	C₂²×C₉	—	2	1	no	—
109	C₂²×C₉	C₂²⋊C₈	1	2	yes	—
110	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₄×C₉	—	2	1	no	—
112	C₄×C₉	—	2	1	no	—
113	C₄×C₉	—	2	1	no	—
114	C₄×C₉	C₂²⋊C₈	1	2	yes	—
115	C₄×C₉	C₂²×C₈	1	1	yes	characteristic
116	C₄×C₉	C₂×M₁₆	1	1	yes	characteristic
117	C₄×C₉	#137	1	2	yes	—
118	Dic₉	—	2	4	no	—
119	Dic₉	—	4	2	no	—
120	C₂²×Q₈	—	9	1	no	—
121	C₄×Q₈	—	9	2	no	—
122	C₄×Q₈	—	9	2	no	—
123	C₄×Q₈	—	18	1	no	—
124	C₄×Q₈	—	18	1	no	—
125	C₂×C₄⋊C₄	—	9	1	no	—
126	C₂×C₄⋊C₄	—	18	1	no	—
127	C₂×C₄²	—	18	1	no	—
128	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
129	C₄⋊C₈	—	18	1	no	—
130	C₄⋊C₈	C₂×D₉	1	2	yes	—
131	C₂²⋊C₈	—	2	1	no	—
132	C₂²⋊C₈	—	18	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₄×C₈	—	9	2	no	—
135	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	4	1	no	—
138	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
139	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
140	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
141	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
142	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
146	C₂×Dic₃	—	6	4	no	—
147	C₂×Dic₃	—	6	2	no	—
148	C₂×Dic₃	—	6	2	no	—
149	C₂×Dic₃	—	12	2	no	—
150	Dic₆	—	3	4	no	—
151	Dic₆	—	6	4	no	—
152	Dic₆	—	6	2	no	—
153	C₃×C₈	—	4	2	no	—
154	C₃⋊C₈	—	12	2	no	—
155	C₂×C₉	—	2	2	no	—
156	C₂×C₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
157	C₂×C₉	#419	1	1	yes	characteristic
158	C₂×C₉	#421	1	1	yes	characteristic
159	C₂×Q₈	—	9	4	no	—
160	C₂×Q₈	—	9	2	no	—
161	C₂×Q₈	—	18	2	no	—
162	C₂×Q₈	—	18	1	no	—
163	C₂²×C₄	—	18	1	no	—
164	C₂²×C₄	—	18	1	no	—
165	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
166	C₂²×C₄	D₃₆	1	1	yes	characteristic
167	C₂²×C₄	#611	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	18	2	no	—
170	C₄⋊C₄	—	9	2	no	—
171	C₄⋊C₄	—	18	1	no	—
172	C₄⋊C₄	—	18	2	no	—
173	C₄⋊C₄	—	18	2	no	—
174	C₄²	—	18	2	no	—
175	C₄²	—	18	2	no	—
176	C₄²	C₄×D₉	1	1	yes	characteristic
177	C₄²	D₃₆	1	1	yes	characteristic
178	C₄²	#611	1	1	yes	characteristic
179	C₄²	C₂²×D₉	1	1	yes	characteristic
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	(C₂×D₃)⋊C₈	1	2	yes	—
183	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	(C₂×D₃)⋊C₈	1	2	yes	—
188	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
189	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
190	C₃×C₄	#951	1	2	yes	—
191	Dic₃	—	6	4	no	—
192	Dic₃	—	12	2	no	—
193	C₉	#1479	1	1	yes	characteristic, complemented, Sylow
194	C₂³	(C₂×D₉)⋊C₄	1	1	yes	characteristic
195	Q₈	—	9	4	no	—
196	Q₈	—	18	4	no	—
197	Q₈	—	18	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	18	2	no	—
201	C₂×C₄	—	18	4	no	—
202	C₂×C₄	—	18	2	no	—
203	C₂×C₄	—	36	2	no	—
204	C₂×C₄	C₈×D₉	1	2	yes	—
205	C₂×C₄	#3663	1	2	yes	—
206	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
207	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
208	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
209	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
210	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
211	C₂×C₄	C₂×#611	1	1	yes	characteristic
212	C₈	—	4	2	no	—
213	C₈	—	36	2	no	—
214	C₂×C₃	—	2	2	no	—
215	C₂×C₃	#5004	1	1	yes	characteristic
216	C₂×C₃	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
217	C₂×C₃	#5139	1	1	yes	characteristic
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	(C₂×D₉)⋊C₈	1	2	yes	—
221	C₂²	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	18	4	no	—
226	C₄	—	36	2	no	—
227	C₄	(C₂×D₉)⋊C₈	1	2	yes	—
228	C₄	C₂×C₈×D₉	1	1	yes	characteristic
229	C₄	C₂×#3663	1	1	yes	characteristic
230	C₄	#63511	1	2	yes	—
231	C₃	#70766	1	1	yes	characteristic
232	C₂	—	2	2	no	—
233	C₂	#10588681	1	1	yes	characteristic
234	C₂	C₂×(C₂×D₉)⋊C₈	1	1	yes	characteristic
235	C₂	#10588819	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#611
commutator	C₄×C₉	C₂²×C₈
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₉×#431	C₂
socle	C₂²×C₃	C₂×(C₂×D₃)⋊C₄
prime radical	C₂³×C₃	(C₂×D₃)⋊C₄
cosocle	C₄²×C₉	C₂³
hypercenter	#431	D₉
hypocenter	C₉	#1479
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
4. #10588160	C₂	semidirect
32. C₂×Dic₁₈	C₈	semidirect
193. C₉	#1479	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₈	2
C₉×#431	2
#10588605	2
#10588160	2
#70766	3