Subgroups — #10713338

Order 768.

280 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#1772	C₂	1	1	yes	maximal, characteristic
2	#70650	C₂	1	2	yes	maximal
3	#69792	C₂	1	1	yes	maximal, characteristic
4	C₂×#4463	C₂	1	1	yes	maximal, characteristic
5	#67733	C₂	1	2	yes	maximal
6	#13188	—	3	1	no	maximal, Sylow
7	C₂×C₃×#108	C₂²	1	1	yes	characteristic
8	C₃×#427	C₂²	1	1	yes	characteristic
9	C₃×#405	—	2	1	no	—
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×#819	—	2	1	no	—
12	C₂×#819	C₂²	1	2	yes	—
13	C₂×#812	C₂²	1	2	yes	—
14	#4924	—	2	1	no	—
15	C₃×#339	—	2	1	no	—
16	#4487	—	2	4	no	—
17	#4463	C₄	1	2	yes	complemented
18	#4463	C₄	1	2	yes	complemented
19	#2155	—	3	2	no	—
20	C₂×#364	—	3	1	no	—
21	#1848	—	3	1	no	—
22	#1772	D₃	1	1	yes	characteristic
23	#1388	—	3	2	no	—
24	C₂×C₃×M₁₆	—	2	1	no	—
25	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
27	C₃×#117	—	2	1	no	—
28	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
29	C₂×#208	—	2	2	no	—
30	C₃×#108	—	2	1	no	—
31	C₃×#108	Q₈	1	2	yes	—
32	C₃×#107	—	2	1	no	—
33	C₃×#106	D₄	1	2	yes	—
34	C₃×#97	—	4	1	no	—
35	C₃×C₄×C₈	—	4	1	no	—
36	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
39	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
40	#819	—	2	2	no	—
41	#819	—	2	2	no	—
42	#819	—	4	1	no	—
43	#819	—	4	1	no	—
44	#812	—	2	2	no	—
45	#812	C₂×C₄	1	4	yes	—
46	C₂×C₃⋊C₁₆	—	2	2	no	—
47	C₄⋊(C₃⋊C₈)	—	4	2	no	—
48	C₂×M₃₂	—	3	2	no	—
49	C₂×#108	—	3	2	no	—
50	C₂×#108	—	6	1	no	—
51	C₂×#108	C₂×D₃	1	1	yes	characteristic
52	#427	C₂×D₃	1	1	yes	characteristic
53	#423	—	6	1	no	—
54	#405	—	2	1	no	—
55	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
56	#365	—	6	4	no	—
57	#364	—	3	2	no	—
58	#364	—	3	2	no	—
59	#339	—	2	1	no	—
60	C₂²×C₃×C₄	—	2	1	no	—
61	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
62	C₃×M₁₆	—	2	2	no	—
63	C₃×M₁₆	—	4	1	no	—
64	C₂×C₃×C₈	—	4	1	no	—
65	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
68	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
69	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
70	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
71	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₃×C₈	#41	1	1	yes	characteristic
76	C₂×C₃×C₈	#41	1	1	yes	characteristic
77	C₃×C₄⋊C₄	—	2	2	no	—
78	C₃×C₂²⋊C₄	—	4	1	no	—
79	C₃×C₄²	—	2	1	no	—
80	C₃×C₄²	—	4	1	no	—
81	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
82	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
83	#208	—	4	2	no	—
84	#208	—	4	2	no	—
85	C₂×C₃⋊C₈	—	4	2	no	—
86	C₃⋊C₁₆	—	2	4	no	—
87	C₂×M₁₆	—	2	1	no	—
88	C₂×M₁₆	—	6	2	no	—
89	C₂²×C₈	#67	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
91	#117	—	2	1	no	—
92	C₂×C₄²	#67	1	1	yes	characteristic
93	M₃₂	—	3	4	no	—
94	M₃₂	—	6	2	no	—
95	C₂×C₁₆	—	6	2	no	—
96	#108	—	2	1	no	—
97	#108	—	6	2	no	—
98	#108	—	6	2	no	—
99	#108	—	12	1	no	—
100	#108	—	12	1	no	—
101	#108	Dic₆	1	2	yes	—
102	#107	—	2	1	no	—
103	#106	D₁₂	1	2	yes	—
104	C₄⋊C₈	—	12	2	no	—
105	#97	—	4	1	no	—
106	C₄×C₈	—	4	1	no	—
107	C₄×C₈	Dic₆	1	1	yes	characteristic
108	C₄×C₈	Dic₆	1	1	yes	characteristic
109	C₄×C₈	C₄×D₃	1	1	yes	characteristic
110	C₄×C₈	C₄×D₃	1	1	yes	characteristic
111	C₂³×C₃	#137	1	1	yes	characteristic
112	C₂×C₃×C₄	—	2	1	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	4	2	no	—
115	C₂×C₃×C₄	—	4	2	no	—
116	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₃×C₄	#121	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₃×C₈	—	4	2	no	—
123	C₃×C₈	#106	1	1	yes	characteristic
124	C₃×C₈	#106	1	1	yes	characteristic
125	C₃×C₈	#106	1	1	yes	characteristic
126	C₃×C₈	#106	1	1	yes	characteristic
127	C₃×C₈	C₄×D₄	1	1	yes	characteristic
128	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
129	C₃×C₈	#125	1	1	yes	characteristic
130	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
131	C₃⋊C₈	—	8	2	no	—
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	C₂×#67	1	1	yes	characteristic
134	M₁₆	—	2	2	no	—
135	M₁₆	—	4	1	no	—
136	M₁₆	—	12	2	no	—
137	M₁₆	—	12	2	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	12	2	no	—
140	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
141	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
142	C₂×C₈	#214	1	1	yes	characteristic
143	C₂×C₈	#214	1	1	yes	characteristic
144	C₂×C₈	#215	1	1	yes	characteristic
145	C₂×C₈	#215	1	1	yes	characteristic
146	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
147	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
148	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
149	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
150	C₂×C₈	#235	1	1	yes	characteristic
151	C₂×C₈	#235	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	4	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	4	1	no	—
156	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
157	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
158	C₁₆	—	6	4	no	—
159	C₂²×C₃	—	4	1	no	—
160	C₂²×C₃	#428	1	1	yes	characteristic
161	C₂²×C₃	#461	1	1	yes	characteristic
162	C₂²×C₃	#475	1	1	yes	characteristic
163	C₃×C₄	—	2	2	no	—
164	C₃×C₄	—	4	1	no	—
165	C₃×C₄	—	8	2	no	—
166	C₃×C₄	#389	1	1	yes	characteristic
167	C₃×C₄	C₂×#106	1	1	yes	characteristic
168	C₃×C₄	#460	1	1	yes	characteristic
169	C₃×C₄	#476	1	1	yes	characteristic
170	C₂³	#951	1	1	yes	characteristic
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
176	C₂×C₄	#924	1	1	yes	characteristic
177	C₂×C₄	#930	1	1	yes	characteristic
178	C₂×C₄	C₂×#214	1	1	yes	characteristic
179	C₂×C₄	C₂×#215	1	1	yes	characteristic
180	C₂×C₄	#949	1	1	yes	characteristic
181	C₈	—	4	2	no	—
182	C₈	—	24	2	no	—
183	C₈	#808	1	1	yes	characteristic
184	C₈	#808	1	1	yes	characteristic
185	C₈	#808	1	1	yes	characteristic
186	C₈	#808	1	1	yes	characteristic
187	C₈	C₄×Dic₆	1	1	yes	characteristic
188	C₈	#869	1	1	yes	characteristic
189	C₈	#870	1	1	yes	characteristic
190	C₈	C₄×D₁₂	1	1	yes	characteristic
191	C₂×C₃	—	2	1	no	—
192	C₂×C₃	—	2	1	no	—
193	C₂×C₃	—	2	1	no	—
194	C₂×C₃	#1948	1	1	yes	characteristic
195	C₂²	—	4	1	no	—
196	C₂²	#4995	1	1	yes	characteristic
197	C₂²	#5019	1	1	yes	characteristic
198	C₂²	#5020	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	4	1	no	—
201	C₄	—	8	2	no	—
202	C₄	#4928	1	1	yes	characteristic
203	C₄	C₂×#808	1	1	yes	characteristic
204	C₄	#5008	1	1	yes	characteristic
205	C₄	#5027	1	1	yes	characteristic
206	C₃	#13188	1	1	yes	characteristic, complemented, Sylow
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	#69467	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#5008
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#1772	C₂
socle	C₂×C₃	#1948
prime radical	C₂³×C₃	#137
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#1772	D₃
hypocenter	C₃	#13188
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
17. #4463	C₄	semidirect
18. #4463	C₄	semidirect
206. C₃	#13188	semidirect

Maximal subgroups

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

subgroup	index
C₃×#1772	2
#70650	2
#69792	2
C₂×#4463	2
#67733	2
#13188	3