Subgroups — #11790989

Order 1152.

468 conjugacy classes of subgroups.

The subgroup lattice has 234 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×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₉	C₂	1	1	yes	maximal, characteristic
4	#10588653	C₂	1	2	yes	maximal
5	C₂×#63368	C₂	1	2	yes	maximal
6	#70627	—	3	1	no	maximal
7	C₂²×Dic₁₈	C₂²	1	2	yes	—
8	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
11	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×Dic₃₆	—	2	2	no	—
13	C₂×Dic₃₆	—	2	2	no	—
14	C₂×Dic₃₆	C₄	1	4	yes	complemented
15	C₉×#107	—	2	2	no	—
16	C₉⋊#95	—	2	4	no	—
17	C₈×Dic₉	—	2	2	no	—
18	#63368	—	2	4	no	—
19	C₂²×Dic₁₂	—	3	1	no	—
20	C₂×C₃×#107	D₃	1	1	yes	characteristic
21	C₂×C₈×Dic₃	—	3	1	no	—
22	#4973	—	3	2	no	—
23	C₂×#810	—	3	2	no	—
24	C₂²×C₄×C₉	—	2	2	no	—
25	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
26	C₂²×Dic₉	—	2	2	no	—
27	C₂²×Dic₉	D₄	1	2	yes	—
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	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₉	D₄	1	2	yes	—
36	C₂×C₈×C₉	D₄	1	2	yes	—
37	C₉×C₄⋊C₄	—	2	2	no	—
38	C₉×C₄⋊C₄	—	2	2	no	—
39	C₄×Dic₉	—	2	1	no	—
40	C₄×Dic₉	—	2	1	no	—
41	C₂×C₉⋊C₈	—	2	1	no	—
42	C₂×C₉⋊C₈	—	2	1	no	—
43	C₂×C₉⋊C₈	D₄	1	2	yes	—
44	Dic₃₆	—	2	8	no	—
45	#1969	—	9	1	no	Sylow
46	C₂²×Dic₆	—	3	2	no	—
47	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
48	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
49	C₂×C₄×Dic₃	—	3	1	no	—
50	C₂²×C₃⋊C₈	—	3	1	no	—
51	C₂×Dic₁₂	—	3	4	no	—
52	C₂×Dic₁₂	—	6	2	no	—
53	C₂×Dic₁₂	—	6	2	no	—
54	C₃×#107	—	2	2	no	—
55	C₃⋊#95	—	6	4	no	—
56	C₈×Dic₃	—	6	2	no	—
57	#810	—	6	4	no	—
58	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₉	—	4	2	no	—
60	C₂×C₄×C₉	—	4	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	2	no	—
68	C₂×Dic₉	—	2	2	no	—
69	C₂×Dic₉	—	2	4	no	—
70	C₂×Dic₉	—	4	2	no	—
71	C₂×Dic₉	—	4	2	no	—
72	C₂×Dic₉	Q₁₆	1	4	yes	complemented
73	Dic₁₈	—	2	4	no	—
74	Dic₁₈	—	2	4	no	—
75	Dic₁₈	—	4	2	no	—
76	Dic₁₈	—	4	2	no	—
77	C₈×C₉	C₂²⋊C₄	1	4	yes	—
78	C₉⋊C₈	—	2	2	no	—
79	C₂²×Q₁₆	—	9	1	no	—
80	C₂×#107	D₉	1	1	yes	characteristic
81	C₂×#103	—	9	2	no	—
82	C₂×C₄×C₈	—	9	1	no	—
83	#391	—	9	2	no	—
84	C₂²×C₃×C₄	—	2	2	no	—
85	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
86	C₂²×Dic₃	—	3	2	no	—
87	C₂²×Dic₃	—	6	2	no	—
88	C₂×Dic₆	—	3	4	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×Dic₆	—	6	2	no	—
94	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
95	C₂×C₃×C₈	D₁₂	1	2	yes	—
96	C₂×C₃×C₈	#67	1	2	yes	—
97	C₃×C₄⋊C₄	—	2	2	no	—
98	C₃×C₄⋊C₄	—	2	2	no	—
99	C₄×Dic₃	—	6	1	no	—
100	C₄×Dic₃	—	6	1	no	—
101	C₂×C₃⋊C₈	—	3	2	no	—
102	C₂×C₃⋊C₈	—	6	1	no	—
103	C₂×C₃⋊C₈	—	6	1	no	—
104	Dic₁₂	—	6	8	no	—
105	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
106	C₂²×C₉	#121	1	1	yes	characteristic
107	C₂²×C₉	#127	1	1	yes	characteristic
108	C₂²×C₉	C₂×Q₁₆	1	2	yes	—
109	C₂²×C₉	#135	1	2	yes	—
110	C₄×C₉	—	8	2	no	—
111	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₄×C₉	C₄×D₄	1	1	yes	characteristic
113	C₄×C₉	#121	1	1	yes	characteristic
114	C₄×C₉	#124	1	1	yes	characteristic
115	Dic₉	—	2	4	no	—
116	Dic₉	—	4	4	no	—
117	C₂²×Q₈	—	9	2	no	—
118	C₂×Q₁₆	—	9	4	no	—
119	C₂×Q₁₆	—	18	2	no	—
120	C₂×Q₁₆	—	18	2	no	—
121	C₂²×C₈	—	9	1	no	—
122	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
123	C₂×C₄⋊C₄	C₂×D₉	1	2	yes	—
124	C₂×C₄²	—	9	1	no	—
125	#107	—	2	2	no	—
126	#103	—	18	4	no	—
127	C₄×C₈	—	18	2	no	—
128	#95	—	18	4	no	—
129	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
130	C₂×C₃×C₄	—	4	4	no	—
131	C₂×C₃×C₄	—	4	2	no	—
132	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
134	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
135	C₂×C₃×C₄	#239	1	1	yes	characteristic
136	C₂×C₃×C₄	#239	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
138	C₂×Dic₃	—	3	4	no	—
139	C₂×Dic₃	—	6	2	no	—
140	C₂×Dic₃	—	6	2	no	—
141	C₂×Dic₃	—	6	4	no	—
142	C₂×Dic₃	—	12	2	no	—
143	C₂×Dic₃	—	12	2	no	—
144	Dic₆	—	6	4	no	—
145	Dic₆	—	6	4	no	—
146	Dic₆	—	12	2	no	—
147	Dic₆	—	12	2	no	—
148	C₃×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
149	C₃⋊C₈	—	6	2	no	—
150	C₂×C₉	#390	1	1	yes	characteristic
151	C₂×C₉	C₄×Q₁₆	1	2	yes	—
152	C₂×C₉	#467	1	2	yes	—
153	C₂×C₉	#494	1	1	yes	characteristic
154	C₂×C₉	#495	1	1	yes	characteristic
155	C₂×Q₈	—	9	4	no	—
156	C₂×Q₈	—	18	2	no	—
157	C₂×Q₈	—	18	2	no	—
158	C₂×Q₈	—	18	2	no	—
159	C₂×Q₈	—	18	2	no	—
160	C₂×Q₈	—	18	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	9	2	no	—
163	C₂²×C₄	—	18	2	no	—
164	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
165	Q₁₆	—	18	8	no	—
166	C₂×C₈	—	9	2	no	—
167	C₂×C₈	—	18	1	no	—
168	C₂×C₈	—	18	1	no	—
169	C₂×C₈	C₄×D₉	1	2	yes	—
170	C₂×C₈	D₃₆	1	2	yes	—
171	C₂×C₈	#611	1	2	yes	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄²	—	18	1	no	—
175	C₄²	—	18	1	no	—
176	C₂²×C₃	#893	1	1	yes	characteristic
177	C₂²×C₃	#895	1	1	yes	characteristic
178	C₂²×C₃	#912	1	2	yes	—
179	C₂²×C₃	D₃×Q₁₆	1	2	yes	—
180	C₂²×C₃	#940	1	1	yes	characteristic
181	C₃×C₄	—	8	2	no	—
182	C₃×C₄	#893	1	1	yes	characteristic
183	C₃×C₄	#895	1	1	yes	characteristic
184	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
185	C₃×C₄	#947	1	1	yes	characteristic
186	Dic₃	—	6	4	no	—
187	Dic₃	—	12	4	no	—
188	C₉	#1969	1	1	yes	characteristic, complemented, Sylow
189	C₂³	D₄×D₉	1	1	yes	characteristic
190	Q₈	—	18	4	no	—
191	Q₈	—	18	4	no	—
192	Q₈	—	36	2	no	—
193	Q₈	—	36	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	9	4	no	—
197	C₂×C₄	—	18	2	no	—
198	C₂×C₄	—	18	2	no	—
199	C₂×C₄	—	18	4	no	—
200	C₂×C₄	—	36	2	no	—
201	C₂×C₄	—	36	2	no	—
202	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
203	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
204	C₂×C₄	D₄×D₉	1	1	yes	characteristic
205	C₂×C₄	#3701	1	1	yes	characteristic
206	C₂×C₄	#3701	1	1	yes	characteristic
207	C₂×C₄	C₂×#611	1	1	yes	characteristic
208	C₈	—	18	2	no	—
209	C₈	(C₂×D₉)⋊C₄	1	4	yes	—
210	C₂×C₃	#4873	1	2	yes	—
211	C₂×C₃	#4887	1	2	yes	—
212	C₂×C₃	#4988	1	1	yes	characteristic
213	C₂×C₃	#5155	1	1	yes	characteristic
214	C₂×C₃	#5183	1	1	yes	characteristic
215	C₂²	#63455	1	1	yes	characteristic
216	C₂²	#63457	1	1	yes	characteristic
217	C₂²	#63474	1	2	yes	—
218	C₂²	Q₁₆×D₉	1	2	yes	—
219	C₂²	#63502	1	1	yes	characteristic
220	C₄	—	8	2	no	—
221	C₄	—	18	4	no	—
222	C₄	—	36	4	no	—
223	C₄	#63455	1	1	yes	characteristic
224	C₄	#63457	1	1	yes	characteristic
225	C₄	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
226	C₄	#63509	1	1	yes	characteristic
227	C₃	#70627	1	1	yes	characteristic
228	C₂	#10588553	1	2	yes	—
229	C₂	#10588567	1	2	yes	—
230	C₂	#10588668	1	1	yes	characteristic
231	C₂	#10588835	1	1	yes	characteristic
232	C₂	#10588863	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₉
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#107	C₂
socle, prime radical	C₂³×C₃	D₃×D₄
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#107	D₉
hypocenter	C₉	#1969
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
14. C₂×Dic₃₆	C₄	semidirect
72. C₂×Dic₉	Q₁₆	semidirect
188. C₉	#1969	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×#107	2
C₂×C₈×Dic₉	2
#10588653	2
C₂×#63368	2
#70627	3