Subgroups — #11783539

Order 1152.

447 conjugacy classes of subgroups.

The subgroup lattice has 247 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5122	C₂	1	1	yes	maximal, characteristic
2	#10591059	C₂	1	2	yes	maximal
3	#10589423	C₂	1	1	yes	maximal, characteristic
4	C₃×C₃⋊#338	C₂	1	1	yes	maximal, characteristic
5	#10589158	C₂	1	2	yes	maximal
6	#69895	—	3	1	no	maximal
7	#68717	—	3	1	no	maximal
8	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
9	#64135	C₂²	1	2	yes	—
10	C₃×#926	C₂²	1	1	yes	characteristic
11	C₂×C₃×#208	C₂²	1	2	yes	—
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₃×#815	C₄	1	2	yes	complemented
14	C₃×C₈×Dic₃	C₄	1	2	yes	complemented
15	#63555	C₄	1	4	yes	complemented
16	#63553	C₄	1	4	yes	complemented
17	C₃×#405	—	3	1	no	—
18	#5122	—	3	2	no	—
19	#5122	D₃	1	1	yes	characteristic
20	#5101	—	3	2	no	—
21	C₃×#338	—	3	1	no	—
22	C₃⋊#338	D₃	1	1	yes	characteristic
23	#4557	—	3	1	no	—
24	#4557	—	3	1	no	—
25	#4557	—	3	2	no	—
26	#4557	—	6	1	no	—
27	#4531	—	3	2	no	—
28	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
29	C₃²×M₁₆	—	2	2	no	—
30	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
31	#3757	—	2	2	no	—
32	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
33	C₄×C₃⋊Dic₃	C₂×C₄	1	4	yes	—
34	C₃×#217	—	2	1	no	—
35	C₃×C₄⋊Dic₃	—	2	1	no	—
36	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
37	C₃×#208	—	2	2	no	—
38	C₃×#208	—	2	2	no	—
39	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
40	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
44	#1747	—	9	1	no	Sylow
45	C₂×C₃×M₁₆	—	2	1	no	—
46	C₂×C₃×M₁₆	—	3	2	no	—
47	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
48	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₃×C₈	—	3	1	no	—
50	C₃×#117	—	3	1	no	—
51	#926	—	3	2	no	—
52	#926	—	3	2	no	—
53	#926	—	6	2	no	—
54	#926	C₂×D₃	1	1	yes	characteristic
55	C₂×#208	C₂×D₃	1	2	yes	—
56	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
57	C₃×#97	—	3	2	no	—
58	C₃×C₄×C₈	—	3	2	no	—
59	#815	—	3	4	no	—
60	#815	—	3	4	no	—
61	#815	Dic₃	1	2	yes	complemented
62	C₈×Dic₃	—	3	4	no	—
63	C₈×Dic₃	—	3	4	no	—
64	C₈×Dic₃	Dic₃	1	2	yes	complemented
65	C₂³×C₃²	#41	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
67	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	#41	1	1	yes	characteristic
69	C₂×C₃²×C₄	#41	1	2	yes	—
70	C₂×C₃⋊Dic₃	—	2	4	no	—
71	C₂×C₃×Dic₃	—	2	2	no	—
72	C₃²×C₈	—	2	2	no	—
73	C₃×C₃⋊C₈	—	2	4	no	—
74	C₃×C₃⋊C₈	C₄²	1	1	yes	characteristic
75	C₃×C₃⋊C₈	C₄²	1	2	yes	—
76	C₃×C₃⋊C₈	C₄²	1	1	yes	characteristic
77	#405	—	3	1	no	—
78	#405	—	9	2	no	—
79	#338	—	3	1	no	—
80	H(3,ℤ/4)	—	9	2	no	—
81	H(3,ℤ/4)	—	9	1	no	—
82	C₂²×C₃×C₄	—	2	1	no	—
83	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₃×M₁₆	—	2	2	no	—
86	C₃×M₁₆	—	2	2	no	—
87	C₃×M₁₆	—	4	2	no	—
88	C₃×M₁₆	—	6	2	no	—
89	C₃×M₁₆	—	6	2	no	—
90	C₂×C₃×C₈	—	2	2	no	—
91	C₂×C₃×C₈	—	3	2	no	—
92	C₂×C₃×C₈	—	3	2	no	—
93	C₂×C₃×C₈	—	3	1	no	—
94	C₂×C₃×C₈	—	3	1	no	—
95	C₂×C₃×C₈	—	3	4	no	—
96	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
97	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
98	C₃×C₄⋊C₄	—	6	1	no	—
99	C₃×C₂²⋊C₄	—	6	1	no	—
100	C₃×C₄²	—	3	2	no	—
101	#217	—	2	1	no	—
102	#217	—	6	2	no	—
103	#217	—	6	2	no	—
104	#217	—	12	2	no	—
105	C₄⋊Dic₃	—	2	1	no	—
106	C₄⋊Dic₃	—	6	2	no	—
107	C₄⋊Dic₃	—	6	2	no	—
108	C₄⋊Dic₃	—	12	2	no	—
109	C₄×Dic₃	—	3	4	no	—
110	C₄×Dic₃	—	3	4	no	—
111	C₄×Dic₃	—	6	4	no	—
112	C₄×Dic₃	C₂×Dic₃	1	2	yes	—
113	#208	—	2	2	no	—
114	#208	—	2	2	no	—
115	C₂×C₃⋊C₈	C₄×D₃	1	4	yes	—
116	C₂×C₃⋊C₈	C₄×D₃	1	2	yes	—
117	C₂×C₃⋊C₈	C₄×D₃	1	2	yes	—
118	C₂×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
119	C₂×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
120	C₂²×C₃²	—	4	1	no	—
121	C₂²×C₃²	#117	1	2	yes	—
122	C₂²×C₃²	#117	1	1	yes	characteristic
123	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
124	C₃²×C₄	#117	1	2	yes	—
125	C₃²×C₄	#117	1	1	yes	characteristic
126	C₃⋊Dic₃	—	4	4	no	—
127	C₃×Dic₃	—	4	2	no	—
128	C₂×M₁₆	—	3	2	no	—
129	C₂×M₁₆	D₃²	1	1	yes	characteristic
130	C₂²×C₈	—	3	1	no	—
131	#117	—	3	1	no	—
132	#117	—	9	2	no	—
133	#97	—	3	2	no	—
134	#97	—	9	4	no	—
135	C₄×C₈	—	3	2	no	—
136	C₄×C₈	—	9	4	no	—
137	C₂³×C₃	—	2	1	no	—
138	C₂³×C₃	#235	1	1	yes	characteristic
139	C₂³×C₃	#239	1	1	yes	characteristic
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	6	2	no	—
145	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
146	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
147	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
148	C₂×C₃×C₄	#235	1	1	yes	characteristic
149	C₂×C₃×C₄	#235	1	2	yes	—
150	C₂×C₃×C₄	#237	1	1	yes	characteristic
151	C₂×C₃×C₄	#237	1	2	yes	—
152	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
153	C₂×Dic₃	—	2	2	no	—
154	C₂×Dic₃	—	6	4	no	—
155	C₂×Dic₃	—	6	4	no	—
156	C₂×Dic₃	—	12	4	no	—
157	C₃×C₈	—	2	2	no	—
158	C₃×C₈	—	2	2	no	—
159	C₃×C₈	—	3	1	no	—
160	C₃×C₈	—	3	2	no	—
161	C₃×C₈	—	3	1	no	—
162	C₃×C₈	—	4	2	no	—
163	C₃×C₈	—	6	4	no	—
164	C₃⋊C₈	—	2	4	no	—
165	C₃⋊C₈	C₄×Dic₃	1	1	yes	characteristic
166	C₃⋊C₈	C₄×Dic₃	1	2	yes	—
167	C₃⋊C₈	C₄×Dic₃	1	1	yes	characteristic
168	C₂×C₃²	—	2	1	no	—
169	C₂×C₃²	—	2	2	no	—
170	C₂×C₃²	#376	1	1	yes	characteristic
171	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
172	M₁₆	—	2	2	no	—
173	M₁₆	—	6	2	no	—
174	M₁₆	—	6	2	no	—
175	C₂×C₈	—	3	2	no	—
176	C₂×C₈	—	3	2	no	—
177	C₂×C₈	—	3	1	no	—
178	C₂×C₈	—	3	1	no	—
179	C₂×C₈	—	3	4	no	—
180	C₂×C₈	D₃×Dic₃	1	2	yes	—
181	C₄⋊C₄	—	6	1	no	—
182	C₄⋊C₄	—	18	2	no	—
183	C₂²⋊C₄	—	6	1	no	—
184	C₂²⋊C₄	—	18	2	no	—
185	C₄²	—	3	2	no	—
186	C₄²	—	9	4	no	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	—	4	1	no	—
190	C₂²×C₃	—	4	1	no	—
191	C₂²×C₃	—	8	1	no	—
192	C₂²×C₃	#872	1	2	yes	—
193	C₂²×C₃	#892	1	2	yes	—
194	C₂²×C₃	#892	1	1	yes	characteristic
195	C₂²×C₃	#926	1	1	yes	characteristic
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	12	2	no	—
200	C₃×C₄	C₄²×D₃	1	1	yes	characteristic
201	C₃×C₄	#872	1	2	yes	—
202	C₃×C₄	#877	1	2	yes	—
203	C₃×C₄	#877	1	1	yes	characteristic
204	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
205	C₃×C₄	#926	1	1	yes	characteristic
206	Dic₃	—	4	2	no	—
207	Dic₃	—	12	4	no	—
208	Dic₃	—	12	4	no	—
209	Dic₃	—	24	4	no	—
210	C₃²	#1747	1	1	yes	characteristic, complemented, Sylow
211	C₂³	#3799	1	1	yes	characteristic
212	C₂×C₄	—	6	2	no	—
213	C₂×C₄	—	18	4	no	—
214	C₂×C₄	#3795	1	1	yes	characteristic
215	C₂×C₄	C₄×D₃²	1	2	yes	—
216	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
217	C₂×C₄	#3804	1	2	yes	—
218	C₈	—	2	2	no	—
219	C₈	—	3	1	no	—
220	C₈	—	3	2	no	—
221	C₈	—	3	1	no	—
222	C₈	—	6	4	no	—
223	C₂×C₃	—	2	1	no	—
224	C₂×C₃	—	2	2	no	—
225	C₂×C₃	—	2	2	no	—
226	C₂×C₃	—	2	1	no	—
227	C₂×C₃	—	2	1	no	—
228	C₂×C₃	—	4	2	no	—
229	C₂×C₃	—	4	1	no	—
230	C₂×C₃	#4648	1	1	yes	characteristic
231	C₂×C₃	#4932	1	1	yes	characteristic
232	C₂²	—	4	1	no	—
233	C₂²	#63836	1	2	yes	—
234	C₂²	#63841	1	1	yes	characteristic
235	C₄	—	12	2	no	—
236	C₄	—	36	4	no	—
237	C₄	C₄×D₃×Dic₃	1	1	yes	characteristic
238	C₄	#63877	1	2	yes	—
239	C₄	#63955	1	1	yes	characteristic
240	C₃	—	2	1	no	—
241	C₃	#68717	1	1	yes	characteristic, complemented
242	C₃	#69895	1	1	yes	characteristic, complemented
243	C₂	—	2	1	no	—
244	C₂	—	2	2	no	—
245	C₂	#10591239	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×D₃×Dic₃
commutator	C₃²×C₄	C₂×C₄²
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂×C₃²×M₁₆	C₂²
socle	C₂×C₃²	#376
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×M₁₆	D₃²
hypocenter	C₃²	#1747
LCS	G ▷ C₃²×C₄ ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
13. C₃×#815	C₄	semidirect
14. C₃×C₈×Dic₃	C₄	semidirect
15. #63555	C₄	semidirect
16. #63553	C₄	semidirect
61. #815	Dic₃	semidirect
64. C₈×Dic₃	Dic₃	semidirect
210. C₃²	#1747	semidirect
241. C₃	#68717	semidirect
242. C₃	#69895	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5122	2
#10591059	2
#10589423	2
C₃×C₃⋊#338	2
#10589158	2
#69895	3
#68717	3