Subgroups — #11859524

Order 1152.

726 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63464	C₂	1	2	yes	maximal, complemented
2	#10589579	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588791	C₂	1	2	yes	maximal, complemented
5	#10588604	C₂	1	1	yes	maximal, characteristic, complemented
6	#10588387	C₂	1	8	yes	maximal, complemented
7	#78491	—	3	1	no	maximal
8	C₂×#3697	C₂²	1	2	yes	—
9	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
12	#63492	C₄	1	4	yes	—
13	C₂×#3667	C₂²	1	2	yes	—
14	#63464	—	2	8	no	—
15	C₂×#3663	C₂²	1	4	yes	—
16	C₂×C₈×D₉	C₂²	1	4	yes	—
17	#63437	C₄	1	2	yes	—
18	#63436	C₄	1	1	yes	characteristic
19	C₄×D₃₆	C₂²	1	4	yes	—
20	#63431	C₄	1	1	yes	characteristic
21	C₉×#97	C₂²	1	4	yes	—
22	(C₂×D₉)⋊C₈	C₂²	1	8	yes	—
23	C₄⋊(C₉⋊C₈)	C₂²	1	4	yes	—
24	C₂×#901	—	3	2	no	—
25	#5477	—	3	1	no	—
26	C₂×C₃×#97	D₃	1	1	yes	characteristic
27	#5110	—	3	2	no	—
28	#4924	—	3	1	no	—
29	#4706	—	3	8	no	—
30	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂×#611	C₂×C₄	1	4	yes	—
33	#3697	—	2	4	no	—
34	#3697	—	2	4	no	—
35	C₂×D₃₆	C₂×C₄	1	2	yes	—
36	C₂×D₃₆	C₂×C₄	1	4	yes	—
37	C₂×C₄×D₉	C₂³	1	4	yes	—
38	C₂×Dic₁₈	C₂×C₄	1	2	yes	—
39	C₂×C₈×C₉	—	2	2	no	—
40	C₂×C₈×C₉	D₄	1	4	yes	complemented
41	C₂×C₈×C₉	C₂³	1	2	yes	—
42	C₂×C₈×C₉	C₂³	1	2	yes	—
43	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₉	C₂³	1	2	yes	—
46	(C₂×D₉)⋊C₄	C₂×C₄	1	8	yes	—
47	C₄⋊Dic₉	C₂×C₄	1	4	yes	—
48	#3667	—	2	4	no	—
49	C₂×C₉⋊C₈	C₂³	1	4	yes	—
50	#3663	—	2	8	no	—
51	C₈×D₉	—	2	8	no	—
52	#2932	—	9	1	no	Sylow
53	C₂×#235	—	3	2	no	—
54	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
55	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
57	#930	—	3	4	no	—
58	C₂×#208	—	3	2	no	—
59	#901	—	6	8	no	—
60	C₂×ΓL(1,𝔽₂₅)	—	3	4	no	—
61	C₂×D₃×C₈	—	3	4	no	—
62	#875	—	3	2	no	—
63	#874	—	3	1	no	—
64	C₄×D₁₂	—	3	4	no	—
65	#869	—	3	1	no	—
66	C₃×#97	C₂×D₃	1	4	yes	—
67	(C₂×D₃)⋊C₈	—	3	8	no	—
68	C₄⋊(C₃⋊C₈)	—	3	4	no	—
69	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₉	C₂²×C₄	1	4	yes	—
71	C₂×C₄×C₉	—	4	2	no	—
72	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₉	#41	1	2	yes	—
77	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
78	#611	—	2	8	no	—
79	C₂×Dic₉	C₂²×C₄	1	4	yes	—
80	D₃₆	—	2	4	no	—
81	D₃₆	—	4	4	no	—
82	C₄×D₉	—	2	4	no	—
83	C₄×D₉	—	2	4	no	—
84	Dic₁₈	—	2	4	no	—
85	C₈×C₉	—	2	2	no	—
86	C₈×C₉	C₂×D₄	1	4	yes	—
87	C₉⋊C₈	—	2	4	no	—
88	C₂×#131	—	9	2	no	—
89	#532	—	9	1	no	—
90	#435	—	9	8	no	—
91	#423	—	9	1	no	—
92	#408	—	9	2	no	—
93	C₂×#97	D₉	1	1	yes	characteristic, complemented
94	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
95	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
96	C₂×#67	—	3	4	no	—
97	#235	—	6	4	no	—
98	#235	—	6	4	no	—
99	C₂×D₁₂	—	3	2	no	—
100	C₂×D₁₂	—	3	4	no	—
101	C₂×C₄×D₃	—	3	4	no	—
102	C₂×Dic₆	—	3	2	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	D₁₂	1	4	yes	—
105	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
106	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
107	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
108	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
109	C₃×C₄²	C₂²×D₃	1	2	yes	—
110	(C₂×D₃)⋊C₄	—	3	8	no	—
111	C₄⋊Dic₃	—	3	4	no	—
112	#208	—	6	4	no	—
113	C₂×C₃⋊C₈	—	3	4	no	—
114	ΓL(1,𝔽₂₅)	—	6	8	no	—
115	D₃×C₈	—	6	8	no	—
116	C₂²×C₉	—	2	2	no	—
117	C₂²×C₉	C₄×D₄	1	2	yes	—
118	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
119	C₂×D₉	—	2	4	no	—
120	C₂×D₉	—	2	8	no	—
121	C₄×C₉	—	2	2	no	—
122	C₄×C₉	—	2	2	no	—
123	C₄×C₉	C₄×D₄	1	2	yes	—
124	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
125	C₄×C₉	C₂×#41	1	1	yes	characteristic
126	Dic₉	—	2	4	no	—
127	C₂×#41	—	9	2	no	—
128	#131	—	18	8	no	—
129	C₂×M₁₆	—	9	2	no	—
130	C₂×M₁₆	—	9	4	no	—
131	C₂²×C₈	—	9	4	no	—
132	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
133	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
134	C₄⋊Q₈	—	9	1	no	—
135	#127	—	9	1	no	—
136	#124	—	9	2	no	—
137	#121	—	9	4	no	—
138	C₄×D₄	—	9	4	no	—
139	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
140	C₄⋊C₈	—	9	4	no	—
141	C₂²⋊C₈	—	9	8	no	—
142	#97	C₂×D₉	1	4	yes	—
143	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
144	C₂²×D₃	—	3	4	no	—
145	C₂×C₃×C₄	—	4	2	no	—
146	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
147	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
148	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
150	C₂×C₃×C₄	#235	1	2	yes	—
151	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
152	#67	—	6	8	no	—
153	C₂×Dic₃	—	3	4	no	—
154	D₁₂	—	6	4	no	—
155	D₁₂	—	12	4	no	—
156	C₄×D₃	—	6	4	no	—
157	C₄×D₃	—	6	4	no	—
158	Dic₆	—	6	4	no	—
159	C₃×C₈	—	2	2	no	—
160	C₃×C₈	C₂×D₁₂	1	4	yes	—
161	C₃⋊C₈	—	6	4	no	—
162	C₂×C₉	—	2	2	no	—
163	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂×C₉	#568	1	2	yes	—
165	D₉	—	4	4	no	—
166	#41	—	18	4	no	—
167	#41	—	18	4	no	—
168	C₂×Q₈	—	9	2	no	—
169	C₂×D₄	—	9	4	no	—
170	C₂×D₄	—	9	2	no	—
171	C₂×D₄	—	9	4	no	—
172	C₂²×C₄	—	9	4	no	—
173	C₂²×C₄	C₄×D₉	1	2	yes	—
174	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
175	M₁₆	—	18	4	no	—
176	M₁₆	—	18	8	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	9	4	no	—
179	C₂×C₈	—	18	8	no	—
180	C₂×C₈	D₃₆	1	4	yes	complemented
181	C₂×C₈	C₂²×D₉	1	2	yes	—
182	C₂×C₈	C₂²×D₉	1	2	yes	—
183	C₄⋊C₄	—	9	4	no	—
184	C₂²⋊C₄	—	9	8	no	—
185	C₄²	C₄×D₉	1	1	yes	characteristic
186	C₄²	C₄×D₉	1	1	yes	characteristic
187	C₄²	C₂²×D₉	1	2	yes	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	C₄×D₁₂	1	2	yes	—
190	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
191	C₂×D₃	—	6	4	no	—
192	C₂×D₃	—	6	8	no	—
193	C₃×C₄	—	2	2	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	C₄×D₁₂	1	2	yes	—
196	C₃×C₄	C₂²×D₁₂	1	1	yes	characteristic
197	C₃×C₄	C₂×#235	1	1	yes	characteristic
198	Dic₃	—	6	4	no	—
199	C₉	#2932	1	1	yes	characteristic, complemented, Sylow
200	C₂³	—	9	4	no	—
201	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
202	Q₈	—	18	4	no	—
203	D₄	—	18	8	no	—
204	D₄	—	18	4	no	—
205	D₄	—	36	4	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	9	4	no	—
208	C₂×C₄	—	18	4	no	—
209	C₂×C₄	—	18	4	no	—
210	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
211	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
213	C₂×C₄	C₂×D₃₆	1	2	yes	—
214	C₂×C₄	#3697	1	2	yes	—
215	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	18	4	no	—
218	C₈	C₂×D₃₆	1	4	yes	—
219	C₂×C₃	—	2	2	no	—
220	C₂×C₃	C₂×C₄×D₁₂	1	1	yes	characteristic
221	C₂×C₃	#5745	1	2	yes	—
222	D₃	—	12	4	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	18	4	no	—
225	C₂²	—	18	8	no	—
226	C₂²	C₄×D₃₆	1	2	yes	—
227	C₂²	C₂²×C₄×D₉	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	18	4	no	—
231	C₄	C₄×D₃₆	1	2	yes	—
232	C₄	C₂²×D₃₆	1	1	yes	characteristic
233	C₄	C₂×#3697	1	1	yes	characteristic
234	C₃	#78491	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	—	36	4	no	—
237	C₂	C₂×C₄×D₃₆	1	1	yes	characteristic
238	C₂	#10589847	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₉×#97	C₂
socle	C₂²×C₃	C₂²×C₄×D₃
prime radical	#10589579	C₂
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂×#97	D₉
hypocenter	C₉	#2932
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
1. C₂×#63464	C₂	semidirect
3. C₂×C₉×#97	C₂	semidirect
4. #10588791	C₂	semidirect
5. #10588604	C₂	semidirect
6. #10588387	C₂	semidirect
40. C₂×C₈×C₉	D₄	semidirect
93. C₂×#97	D₉	semidirect
180. C₂×C₈	D₃₆	semidirect
199. C₉	#2932	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63464	2
#10589579	2
C₂×C₉×#97	2
#10588791	2
#10588604	2
#10588387	2
#78491	3