Subgroups — #11791061

Order 1152.

456 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×#63377	C₂	1	1	yes	maximal, characteristic
4	C₂×#63376	C₂	1	2	yes	maximal
5	#10588609	C₂	1	2	yes	maximal
6	#69824	—	3	1	no	maximal
7	C₂²×Dic₁₈	C₂²	1	2	yes	—
8	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
12	C₂×Dic₃₆	—	2	2	no	—
13	C₂×Dic₃₆	—	2	2	no	—
14	C₂×Dic₃₆	C₄	1	4	yes	complemented
15	C₉×#97	—	2	2	no	—
16	C₉⋊#95	—	2	4	no	—
17	#63377	—	2	2	no	—
18	#63376	—	2	4	no	—
19	C₂²×Dic₁₂	—	3	1	no	—
20	C₂×C₃×#97	D₃	1	1	yes	characteristic
21	C₂×#817	—	3	1	no	—
22	C₂×#816	—	3	2	no	—
23	#4929	—	3	2	no	—
24	C₂²×C₄×C₉	D₄	1	2	yes	—
25	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
26	C₂²×Dic₉	—	2	2	no	—
27	C₂²×Dic₉	—	2	2	no	—
28	C₂×Dic₁₈	—	2	2	no	—
29	C₂×Dic₁₈	—	2	2	no	—
30	C₂×Dic₁₈	—	2	2	no	—
31	C₂×Dic₁₈	—	2	2	no	—
32	C₂×Dic₁₈	—	2	2	no	—
33	C₂×Dic₁₈	C₂×C₄	1	4	yes	—
34	C₂×C₈×C₉	—	2	1	no	—
35	C₂×C₈×C₉	—	2	1	no	—
36	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
37	C₂×C₈×C₉	D₄	1	2	yes	—
38	C₂×C₈×C₉	D₄	1	2	yes	—
39	C₂×C₈×C₉	D₄	1	2	yes	—
40	C₄²×C₉	—	2	1	no	—
41	C₄²×C₉	—	2	1	no	—
42	C₄⋊Dic₉	—	2	2	no	—
43	C₄⋊Dic₉	—	2	2	no	—
44	Dic₃₆	—	2	8	no	—
45	#1971	—	9	1	no	Sylow
46	C₂²×Dic₆	—	3	2	no	—
47	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
49	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
50	C₂×C₄⋊Dic₃	—	3	2	no	—
51	C₂×Dic₁₂	—	3	4	no	—
52	C₂×Dic₁₂	—	6	2	no	—
53	C₂×Dic₁₂	—	6	2	no	—
54	C₃×#97	—	2	2	no	—
55	C₃⋊#95	—	6	4	no	—
56	#817	—	6	2	no	—
57	#816	—	6	4	no	—
58	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₉	—	2	2	no	—
60	C₂×C₄×C₉	—	2	4	no	—
61	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₉	#41	1	1	yes	characteristic
66	C₂×C₄×C₉	#41	1	1	yes	characteristic
67	C₂×Dic₉	—	2	4	no	—
68	C₂×Dic₉	—	4	2	no	—
69	C₂×Dic₉	—	4	2	no	—
70	C₂×Dic₉	—	4	2	no	—
71	C₂×Dic₉	—	4	4	no	—
72	Dic₁₈	—	2	4	no	—
73	Dic₁₈	—	2	4	no	—
74	Dic₁₈	—	4	2	no	—
75	Dic₁₈	—	4	2	no	—
76	C₈×C₉	—	2	2	no	—
77	C₈×C₉	C₂²⋊C₄	1	4	yes	—
78	C₂²×Q₁₆	—	9	1	no	—
79	C₂×#106	—	9	1	no	—
80	C₂×#103	—	9	2	no	—
81	C₂×#97	D₉	1	1	yes	characteristic
82	#391	—	9	2	no	—
83	C₂²×C₃×C₄	D₁₂	1	2	yes	—
84	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₂²×Dic₃	—	6	2	no	—
86	C₂²×Dic₃	—	6	2	no	—
87	C₂×Dic₆	—	3	4	no	—
88	C₂×Dic₆	—	6	2	no	—
89	C₂×Dic₆	—	6	2	no	—
90	C₂×Dic₆	—	6	2	no	—
91	C₂×Dic₆	—	6	2	no	—
92	C₂×Dic₆	—	6	2	no	—
93	C₂×C₃×C₈	—	2	1	no	—
94	C₂×C₃×C₈	—	2	1	no	—
95	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
96	C₂×C₃×C₈	D₁₂	1	2	yes	—
97	C₂×C₃×C₈	D₁₂	1	2	yes	—
98	C₂×C₃×C₈	#67	1	2	yes	—
99	C₃×C₄²	—	2	1	no	—
100	C₃×C₄²	—	2	1	no	—
101	C₄⋊Dic₃	—	6	2	no	—
102	C₄⋊Dic₃	—	6	2	no	—
103	Dic₁₂	—	6	8	no	—
104	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
105	C₂²×C₉	#121	1	1	yes	characteristic
106	C₂²×C₉	#127	1	1	yes	characteristic
107	C₂²×C₉	#137	1	4	yes	—
108	C₄×C₉	—	4	2	no	—
109	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₄×C₉	C₄×D₄	1	1	yes	characteristic
111	C₄×C₉	#121	1	1	yes	characteristic
112	C₄×C₉	#124	1	1	yes	characteristic
113	Dic₉	—	4	4	no	—
114	Dic₉	—	8	2	no	—
115	C₂²×Q₈	—	9	2	no	—
116	C₂×Q₁₆	—	9	4	no	—
117	C₂×Q₁₆	—	18	2	no	—
118	C₂×Q₁₆	—	18	2	no	—
119	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
120	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
121	C₂×C₄⋊C₄	—	9	2	no	—
122	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
123	#106	—	18	2	no	—
124	#103	—	18	4	no	—
125	#97	—	2	2	no	—
126	#95	—	18	4	no	—
127	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
128	C₂×C₃×C₄	—	2	4	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
133	C₂×C₃×C₄	#235	1	1	yes	characteristic
134	C₂×C₃×C₄	#235	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
136	C₂×Dic₃	—	6	4	no	—
137	C₂×Dic₃	—	12	2	no	—
138	C₂×Dic₃	—	12	2	no	—
139	C₂×Dic₃	—	12	2	no	—
140	C₂×Dic₃	—	12	4	no	—
141	Dic₆	—	6	4	no	—
142	Dic₆	—	6	4	no	—
143	Dic₆	—	12	2	no	—
144	Dic₆	—	12	2	no	—
145	C₃×C₈	—	2	2	no	—
146	C₃×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
147	C₂×C₉	#390	1	1	yes	characteristic
148	C₂×C₉	#441	1	2	yes	—
149	C₂×C₉	#470	1	2	yes	—
150	C₂×C₉	#497	1	2	yes	—
151	C₂×Q₈	—	9	4	no	—
152	C₂×Q₈	—	18	2	no	—
153	C₂×Q₈	—	18	2	no	—
154	C₂×Q₈	—	18	2	no	—
155	C₂×Q₈	—	18	2	no	—
156	C₂×Q₈	—	18	2	no	—
157	C₂²×C₄	—	18	2	no	—
158	C₂²×C₄	—	18	2	no	—
159	C₂²×C₄	D₃₆	1	2	yes	—
160	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
161	Q₁₆	—	18	8	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	C₄×D₉	1	2	yes	—
165	C₂×C₈	D₃₆	1	2	yes	—
166	C₂×C₈	D₃₆	1	2	yes	—
167	C₂×C₈	#611	1	2	yes	—
168	C₄⋊C₄	—	18	2	no	—
169	C₄⋊C₄	—	18	2	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	2	1	no	—
172	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
173	C₂²×C₃	#874	1	1	yes	characteristic
174	C₂²×C₃	#909	1	4	yes	—
175	C₂²×C₃	#930	1	1	yes	characteristic
176	C₃×C₄	—	4	2	no	—
177	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
178	C₃×C₄	#875	1	1	yes	characteristic
179	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	C₃×C₄	#930	1	1	yes	characteristic
181	Dic₃	—	12	4	no	—
182	Dic₃	—	24	2	no	—
183	C₉	#1971	1	1	yes	characteristic, complemented, Sylow
184	C₂³	C₂×D₃₆	1	1	yes	characteristic
185	Q₈	—	18	4	no	—
186	Q₈	—	18	4	no	—
187	Q₈	—	36	2	no	—
188	Q₈	—	36	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	18	4	no	—
192	C₂×C₄	—	36	2	no	—
193	C₂×C₄	—	36	2	no	—
194	C₂×C₄	—	36	2	no	—
195	C₂×C₄	—	36	4	no	—
196	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
197	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
198	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
199	C₂×C₄	#3697	1	1	yes	characteristic
200	C₂×C₄	#3697	1	1	yes	characteristic
201	C₂×C₄	C₂×#611	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	(C₂×D₉)⋊C₄	1	4	yes	—
204	C₂×C₃	#4715	1	2	yes	—
205	C₂×C₃	#4716	1	2	yes	—
206	C₂×C₃	#4939	1	1	yes	characteristic
207	C₂×C₃	#5137	1	2	yes	—
208	C₂²	C₄×D₃₆	1	1	yes	characteristic
209	C₂²	#63436	1	1	yes	characteristic
210	C₂²	#63471	1	4	yes	—
211	C₂²	#63492	1	1	yes	characteristic
212	C₄	—	4	2	no	—
213	C₄	—	36	4	no	—
214	C₄	—	72	2	no	—
215	C₄	C₄×D₃₆	1	1	yes	characteristic
216	C₄	#63437	1	1	yes	characteristic
217	C₄	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
218	C₄	#63492	1	1	yes	characteristic
219	C₃	#69824	1	1	yes	characteristic
220	C₂	#10588384	1	2	yes	—
221	C₂	#10588396	1	2	yes	—
222	C₂	#10588619	1	1	yes	characteristic
223	C₂	#10588807	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₆
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#97	C₂
socle, prime radical	C₂³×C₃	C₂×D₁₂
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#97	D₉
hypocenter	C₉	#1971
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
14. C₂×Dic₃₆	C₄	semidirect
183. C₉	#1971	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₃₆	2
C₂×C₉×#97	2
C₂×#63377	2
C₂×#63376	2
#10588609	2
#69824	3