Subgroups — #10694560

Order 768.

394 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#78442	C₂	1	1	yes	maximal, characteristic
2	C₃×#1846	C₂	1	1	yes	maximal, characteristic
3	#69724	C₂	1	1	yes	maximal, characteristic
4	#69546	C₂	1	4	yes	maximal
5	#9755	—	3	1	no	maximal, Sylow
6	#5470	C₂²	1	2	yes	—
7	C₂×C₃×#107	—	2	1	no	—
8	C₂×C₃×#106	—	2	1	no	—
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₃×#382	C₂²	1	2	yes	—
11	#5106	—	2	1	no	—
12	#5106	—	2	1	no	—
13	#4925	C₂²	1	2	yes	—
14	#4683	—	2	1	no	—
15	#4683	—	2	1	no	—
16	#4682	C₄	1	2	yes	complemented
17	#4680	C₄	1	2	yes	complemented
18	#4455	—	2	1	no	—
19	C₈×C₃⋊C₈	—	2	1	no	—
20	#4452	—	2	8	no	—
21	#3160	—	3	1	no	—
22	#1846	D₃	1	1	yes	characteristic
23	#1518	—	3	4	no	—
24	#1450	—	3	1	no	—
25	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄⋊C₄	—	2	2	no	—
28	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
29	#926	—	2	2	no	—
30	#924	—	2	4	no	—
31	#870	—	2	2	no	—
32	#869	C₂×C₄	1	4	yes	—
33	C₄×Dic₆	—	2	4	no	—
34	C₃×#107	—	4	2	no	—
35	C₃×#106	—	4	2	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	D₄	1	1	yes	characteristic
39	C₃×C₄×C₈	D₄	1	1	yes	characteristic
40	C₃×#95	—	2	4	no	—
41	#830	D₄	1	4	yes	—
42	#818	—	2	2	no	—
43	#818	—	2	2	no	—
44	#817	—	2	2	no	—
45	#817	—	2	2	no	—
46	C₄⋊(C₃⋊C₈)	—	2	2	no	—
47	C₄×C₃⋊C₈	—	2	2	no	—
48	#533	—	3	2	no	—
49	#500	—	3	2	no	—
50	#499	—	6	1	no	—
51	#499	—	6	1	no	—
52	#498	—	3	2	no	—
53	#431	—	3	2	no	—
54	#427	—	6	1	no	—
55	#427	—	6	1	no	—
56	C₂×#107	—	2	1	no	—
57	C₂×#106	—	2	1	no	—
58	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
59	#382	C₂×D₃	1	2	yes	—
60	#328	—	6	8	no	—
61	#322	—	6	1	no	—
62	C₈²	—	6	1	no	—
63	C₂²×C₃×C₄	—	4	2	no	—
64	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
65	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×Dic₆	—	2	4	no	—
67	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₈	SD₁₆	1	2	yes	—
72	C₂×C₃×C₈	SD₁₆	1	2	yes	—
73	C₂×C₃×C₈	Q₁₆	1	2	yes	—
74	C₂×C₃×C₈	Q₁₆	1	2	yes	—
75	C₃×C₄⋊C₄	—	4	2	no	—
76	C₃×C₄⋊C₄	—	4	2	no	—
77	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
78	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
79	C₃×C₄²	#41	1	2	yes	—
80	#217	—	4	2	no	—
81	C₄⋊Dic₃	—	2	4	no	—
82	C₄⋊Dic₃	—	2	4	no	—
83	Dic₃⋊C₄	—	4	4	no	—
84	C₄×Dic₃	—	4	2	no	—
85	C₂×C₃⋊C₈	—	2	4	no	—
86	C₂²×C₈	#67	1	1	yes	characteristic
87	C₂²×C₈	#67	1	1	yes	characteristic
88	C₄⋊Q₈	—	3	4	no	—
89	#125	—	6	2	no	—
90	C₂²⋊Q₈	—	6	4	no	—
91	C₄×Q₈	—	6	4	no	—
92	#117	—	6	2	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
95	#107	—	4	2	no	—
96	#107	—	6	2	no	—
97	#107	—	6	2	no	—
98	#106	—	4	2	no	—
99	#106	—	6	2	no	—
100	#106	—	6	2	no	—
101	C₄⋊C₈	—	6	2	no	—
102	C₂²⋊C₈	—	3	4	no	—
103	C₄×C₈	—	6	2	no	—
104	C₄×C₈	C₄×D₃	1	1	yes	characteristic
105	C₄×C₈	C₄×D₃	1	1	yes	characteristic
106	C₄×C₈	D₁₂	1	1	yes	characteristic
107	C₄×C₈	D₁₂	1	1	yes	characteristic
108	#95	—	2	4	no	—
109	C₂³×C₃	#127	1	1	yes	characteristic
110	C₂×C₃×C₄	—	4	2	no	—
111	C₂×C₃×C₄	—	8	4	no	—
112	C₂×C₃×C₄	—	8	2	no	—
113	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
115	C₂×C₃×C₄	#121	1	2	yes	—
116	C₂×C₃×C₄	#124	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
121	C₂×Dic₃	—	4	4	no	—
122	Dic₆	—	8	4	no	—
123	C₃×C₈	#103	1	2	yes	—
124	C₃×C₈	#103	1	2	yes	—
125	C₃×C₈	#103	1	2	yes	—
126	C₃×C₈	#103	1	2	yes	—
127	C₃⋊C₈	—	4	4	no	—
128	C₂×Q₈	—	6	4	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	D₃×D₄	1	2	yes	—
131	C₂²×C₄	C₂×#67	1	1	yes	characteristic
132	C₂×C₈	—	6	4	no	—
133	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
134	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
135	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
136	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
137	C₂×C₈	#214	1	2	yes	—
138	C₂×C₈	#214	1	2	yes	—
139	C₂×C₈	#216	1	2	yes	—
140	C₂×C₈	#216	1	2	yes	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	6	4	no	—
144	C₄⋊C₄	—	6	4	no	—
145	C₄⋊C₄	—	12	4	no	—
146	C₂²⋊C₄	—	12	2	no	—
147	C₄²	—	12	2	no	—
148	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
149	C₄²	C₂×D₁₂	1	1	yes	characteristic
150	C₄²	#239	1	2	yes	—
151	C₂²×C₃	—	2	2	no	—
152	C₂²×C₃	#390	1	1	yes	characteristic
153	C₂²×C₃	#492	1	1	yes	characteristic
154	C₂²×C₃	#494	1	1	yes	characteristic
155	C₃×C₄	—	2	2	no	—
156	C₃×C₄	—	2	2	no	—
157	C₃×C₄	—	16	2	no	—
158	C₃×C₄	C₂×#103	1	1	yes	characteristic
159	C₃×C₄	C₂×#103	1	1	yes	characteristic
160	C₃×C₄	#487	1	1	yes	characteristic
161	C₃×C₄	#487	1	1	yes	characteristic
162	Dic₃	—	8	4	no	—
163	C₂³	#940	1	1	yes	characteristic
164	Q₈	—	24	4	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	8	4	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	12	4	no	—
169	C₂×C₄	#893	1	2	yes	—
170	C₂×C₄	#895	1	2	yes	—
171	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
172	C₂×C₄	C₂×#214	1	1	yes	characteristic
173	C₂×C₄	C₂×#214	1	1	yes	characteristic
174	C₂×C₄	C₂×#216	1	1	yes	characteristic
175	C₂×C₄	C₂×#216	1	1	yes	characteristic
176	C₂×C₄	#947	1	1	yes	characteristic
177	C₈	—	12	4	no	—
178	C₈	#810	1	2	yes	—
179	C₈	#810	1	2	yes	—
180	C₈	#810	1	2	yes	—
181	C₈	#810	1	2	yes	—
182	C₂×C₃	—	2	1	no	—
183	C₂×C₃	—	2	1	no	—
184	C₂×C₃	#1956	1	1	yes	characteristic
185	C₂×C₃	#1976	1	2	yes	—
186	C₂²	—	2	2	no	—
187	C₂²	#4988	1	1	yes	characteristic
188	C₂²	#5076	1	1	yes	characteristic
189	C₂²	#5092	1	1	yes	characteristic
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	16	2	no	—
193	C₄	—	24	4	no	—
194	C₄	C₂×#810	1	1	yes	characteristic
195	C₄	C₂×#810	1	1	yes	characteristic
196	C₄	#5093	1	1	yes	characteristic
197	C₄	#5093	1	1	yes	characteristic
198	C₃	#9755	1	1	yes	characteristic, complemented, Sylow
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#69579	1	1	yes	characteristic
202	C₂	#70658	1	2	yes	—

	subgroup	G/H
center	C₂²	#4988
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1846	C₂
socle	C₂²×C₃	#390
prime radical	C₂³×C₃	#127
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1846	D₃
hypocenter	C₃	#9755
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
16. #4682	C₄	semidirect
17. #4680	C₄	semidirect
198. C₃	#9755	semidirect

Maximal subgroups

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

subgroup	index
#78442	2
C₃×#1846	2
#69724	2
#69546	2
#9755	3