Subgroups — #10713299

Order 768.

294 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#364	C₂	1	1	yes	maximal, characteristic
2	#71428	C₂	1	2	yes	maximal
3	#69779	C₂	1	1	yes	maximal, characteristic
4	#69766	C₂	1	1	yes	maximal, characteristic
5	#67817	C₂	1	2	yes	maximal
6	#13180	—	3	1	no	maximal, Sylow
7	C₂×C₃×M₃₂	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#819	C₂²	1	1	yes	characteristic
10	#5106	C₂²	1	1	yes	characteristic
11	C₂×#818	C₂²	1	2	yes	—
12	C₂×#817	—	2	1	no	—
13	#5101	—	2	1	no	—
14	#4936	—	2	1	no	—
15	C₃×#364	C₄	1	4	yes	complemented
16	#4523	—	2	1	no	—
17	#4505	—	2	2	no	—
18	#4504	—	2	2	no	—
19	#2159	—	3	2	no	—
20	C₂×#364	D₃	1	1	yes	characteristic
21	#1846	—	3	1	no	—
22	#1772	—	3	1	no	—
23	#1387	—	3	2	no	—
24	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
27	#926	—	2	1	no	—
28	C₂×C₄⋊Dic₃	—	2	2	no	—
29	C₂×#208	—	2	1	no	—
30	C₃×M₃₂	—	2	2	no	—
31	C₃×M₃₂	C₂×C₄	1	4	yes	—
32	C₂×C₃×C₁₆	—	2	2	no	—
33	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
34	C₃×C₄×C₈	Q₈	1	2	yes	—
35	C₃⋊#95	—	4	2	no	—
36	#819	—	2	1	no	—
37	#819	Q₈	1	2	yes	—
38	#818	—	2	2	no	—
39	#818	—	2	2	no	—
40	#818	D₄	1	2	yes	—
41	#817	—	2	1	no	—
42	#817	—	4	1	no	—
43	#817	—	4	1	no	—
44	#815	—	4	1	no	—
45	C₈×Dic₃	—	4	1	no	—
46	C₂×M₃₂	C₂×D₃	1	2	yes	—
47	C₂×#108	—	3	1	no	—
48	#427	—	3	1	no	—
49	C₂×#107	—	3	2	no	—
50	C₂×#106	—	6	1	no	—
51	#405	—	6	1	no	—
52	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
53	#382	—	6	1	no	—
54	#367	—	6	2	no	—
55	#366	—	6	2	no	—
56	#364	Dic₃	1	4	yes	complemented
57	#339	—	6	1	no	—
58	C₂²×C₃×C₄	—	2	1	no	—
59	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₃	—	4	2	no	—
61	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
62	C₂×C₃×C₈	D₈	1	1	yes	characteristic
63	C₂×C₃×C₈	D₈	1	1	yes	characteristic
64	C₂×C₃×C₈	Q₁₆	1	2	yes	—
65	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₃×C₈	#41	1	1	yes	characteristic
70	C₂×C₃×C₈	#41	1	1	yes	characteristic
71	C₃×C₄²	—	2	1	no	—
72	C₃×C₄²	C₄⋊C₄	1	2	yes	—
73	#217	—	4	1	no	—
74	C₄⋊Dic₃	—	2	2	no	—
75	C₄⋊Dic₃	—	4	2	no	—
76	C₄⋊Dic₃	—	4	2	no	—
77	C₄×Dic₃	—	4	1	no	—
78	#208	—	2	2	no	—
79	#208	—	4	1	no	—
80	C₂×C₃⋊C₈	—	4	1	no	—
81	C₃×C₁₆	—	2	4	no	—
82	C₂×M₁₆	—	6	1	no	—
83	C₂²×C₈	D₁₂	1	1	yes	characteristic
84	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
85	#117	—	6	1	no	—
86	C₂×C₄⋊C₄	—	6	2	no	—
87	C₂×C₄²	D₁₂	1	1	yes	characteristic
88	M₃₂	—	2	2	no	—
89	M₃₂	C₂×Dic₃	1	4	yes	—
90	C₂×C₁₆	—	2	2	no	—
91	#108	—	3	2	no	—
92	#108	—	6	1	no	—
93	#107	—	3	2	no	—
94	#107	—	6	2	no	—
95	#107	—	6	2	no	—
96	#106	—	6	1	no	—
97	#106	—	12	1	no	—
98	#106	—	12	1	no	—
99	#97	—	12	1	no	—
100	C₄×C₈	—	12	1	no	—
101	C₄×C₈	Dic₆	1	2	yes	—
102	C₄×C₈	C₂×Dic₃	1	2	yes	—
103	#95	—	12	2	no	—
104	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	4	2	no	—
107	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₃×C₄	#121	1	1	yes	characteristic
109	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
111	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₃×C₄	#137	1	1	yes	characteristic
113	C₂×Dic₃	—	4	2	no	—
114	C₂×Dic₃	—	8	4	no	—
115	C₂×Dic₃	—	8	2	no	—
116	C₃×C₈	#107	1	2	yes	—
117	C₃×C₈	#107	1	2	yes	—
118	C₃×C₈	C₄×D₄	1	1	yes	characteristic
119	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
120	C₃×C₈	#125	1	1	yes	characteristic
121	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
122	C₃⋊C₈	—	4	2	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	12	2	no	—
125	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
126	M₁₆	—	6	2	no	—
127	M₁₆	—	12	1	no	—
128	C₂×C₈	—	12	1	no	—
129	C₂×C₈	D₂₄	1	1	yes	characteristic
130	C₂×C₈	D₂₄	1	1	yes	characteristic
131	C₂×C₈	Dic₁₂	1	2	yes	—
132	C₂×C₈	C₄⋊Dic₃	1	2	yes	—
133	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
134	C₂×C₈	D₃×D₄	1	1	yes	characteristic
135	C₂×C₈	#237	1	1	yes	characteristic
136	C₂×C₈	D₃×Q₈	1	1	yes	characteristic
137	C₂×C₈	#239	1	1	yes	characteristic
138	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
139	C₄⋊C₄	—	6	2	no	—
140	C₄⋊C₄	—	12	2	no	—
141	C₄⋊C₄	—	12	2	no	—
142	C₂²⋊C₄	—	12	1	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	12	1	no	—
145	C₄²	C₄⋊Dic₃	1	2	yes	—
146	C₁₆	—	2	4	no	—
147	C₂²×C₃	—	2	2	no	—
148	C₂²×C₃	#428	1	1	yes	characteristic
149	C₂²×C₃	#459	1	1	yes	characteristic
150	C₂²×C₃	#477	1	1	yes	characteristic
151	C₃×C₄	—	4	2	no	—
152	C₃×C₄	#389	1	1	yes	characteristic
153	C₃×C₄	C₂×#107	1	1	yes	characteristic
154	C₃×C₄	#462	1	1	yes	characteristic
155	C₃×C₄	#474	1	1	yes	characteristic
156	Dic₃	—	8	2	no	—
157	Dic₃	—	16	2	no	—
158	C₂³	#908	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	12	2	no	—
162	C₂×C₄	—	24	4	no	—
163	C₂×C₄	—	24	2	no	—
164	C₂×C₄	#895	1	1	yes	characteristic
165	C₂×C₄	#897	1	1	yes	characteristic
166	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
167	C₂×C₄	C₂×Dic₁₂	1	1	yes	characteristic
168	C₂×C₄	#909	1	1	yes	characteristic
169	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
170	C₈	—	12	2	no	—
171	C₈	#818	1	2	yes	—
172	C₈	#818	1	2	yes	—
173	C₈	#888	1	1	yes	characteristic
174	C₈	#890	1	1	yes	characteristic
175	C₈	D₄×Dic₃	1	1	yes	characteristic
176	C₈	Q₈×Dic₃	1	1	yes	characteristic
177	C₂×C₃	—	2	1	no	—
178	C₂×C₃	—	2	1	no	—
179	C₂×C₃	#1947	1	1	yes	characteristic
180	C₂×C₃	#2212	1	2	yes	—
181	C₂²	—	2	2	no	—
182	C₂²	#4843	1	1	yes	characteristic
183	C₂²	#4850	1	1	yes	characteristic
184	C₂²	#5121	1	1	yes	characteristic
185	C₄	—	4	2	no	—
186	C₄	—	24	2	no	—
187	C₄	—	48	2	no	—
188	C₄	#4846	1	1	yes	characteristic
189	C₄	#4849	1	1	yes	characteristic
190	C₄	#4984	1	1	yes	characteristic
191	C₄	C₂×#818	1	1	yes	characteristic
192	C₃	#13180	1	1	yes	characteristic, complemented, Sylow
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#69304	1	2	yes	—
196	C₂	#70453	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#4843
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₂×C₃×#364	C₂
socle	C₂²×C₃	#459
prime radical	C₂³×C₃	AGL(1,ℤ/8)
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×#364	D₃
hypocenter	C₃	#13180
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
15. C₃×#364	C₄	semidirect
56. #364	Dic₃	semidirect
192. C₃	#13180	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#364	2
#71428	2
#69779	2
#69766	2
#67817	2
#13180	3