Subgroups — #20243085

Order 6016.

416 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10625329	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#15688878	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10625329	—	2	2	no	—
6	C₂×#10625329	—	2	2	no	—
7	C₂×#10625329	C₄	1	4	yes	complemented
8	C₂²×C₄₇⋊C₈	—	2	2	no	—
9	C₂²×C₄₇⋊C₈	—	2	2	no	—
10	C₂²×C₄₇⋊C₈	C₄	1	4	yes	complemented
11	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
13	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
14	#117×C₄₇	—	2	2	no	—
15	#117×C₄₇	—	2	2	no	—
16	C₂×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
17	C₂×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
18	C₂×C₄²×C₄₇	C₂²	1	2	yes	—
19	C₂⁴×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₄₇	D₄	1	2	yes	—
25	C₂²×C₄×C₄₇	D₄	1	2	yes	—
26	C₂²×C₄×C₄₇	D₄	1	2	yes	—
27	C₂²×C₄×C₄₇	Q₈	1	2	yes	—
28	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₄₇	—	2	4	no	—
30	C₂²⋊C₄×C₄₇	—	2	4	no	—
31	C₄²×C₄₇	—	2	2	no	—
32	C₄²×C₄₇	—	2	2	no	—
33	#10625329	—	2	8	no	—
34	C₂×C₄₇⋊C₈	—	2	8	no	—
35	C₂×C₄₇⋊C₈	—	2	2	no	—
36	C₂×C₄₇⋊C₈	—	2	2	no	—
37	C₂×C₄₇⋊C₈	—	2	2	no	—
38	C₂×C₄₇⋊C₈	—	2	2	no	—
39	C₂×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
40	C₂×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
41	C₂³×C₄₇	—	2	4	no	—
42	C₂³×C₄₇	C₄²	1	2	yes	—
43	C₂³×C₄₇	C₂²⋊C₄	1	2	yes	—
44	C₂³×C₄₇	C₄⋊C₄	1	2	yes	—
45	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	—	2	4	no	—
47	C₂×C₄×C₄₇	—	2	8	no	—
48	C₂×C₄×C₄₇	C₄²	1	2	yes	—
49	C₂×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
50	C₂×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
51	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
53	C₂×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
55	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
61	C₄₇⋊C₈	—	2	4	no	—
62	C₄₇⋊C₈	—	2	4	no	—
63	C₂²×C₄₇	—	2	4	no	—
64	C₂²×C₄₇	—	2	4	no	—
65	C₂²×C₄₇	—	2	4	no	—
66	C₂²×C₄₇	#95	1	4	yes	—
67	C₂²×C₄₇	C₂×C₄²	1	1	yes	characteristic
68	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄₇	#131	1	4	yes	—
71	C₄×C₄₇	—	4	4	no	—
72	C₄×C₄₇	#95	1	4	yes	—
73	C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	#1729	—	47	1	no	maximal, Sylow
78	C₂×C₄₇	—	2	4	no	—
79	C₂×C₄₇	C₂×#95	1	1	yes	characteristic
80	C₂×C₄₇	#405	1	2	yes	—
81	C₂×C₄₇	#408	1	2	yes	—
82	C₂×C₄₇	#424	1	2	yes	—
83	C₂²×M₁₆	—	47	1	no	—
84	C₂³×C₈	—	47	1	no	—
85	C₂×#117	D₄₇	1	1	yes	characteristic
86	#336	—	47	4	no	—
87	C₄₇	#1729	1	1	yes	characteristic, complemented, Sylow
88	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
89	C₂×M₁₆	—	47	4	no	—
90	C₂×M₁₆	—	94	2	no	—
91	C₂×M₁₆	—	94	2	no	—
92	C₂²×C₈	—	47	4	no	—
93	C₂²×C₈	—	47	2	no	—
94	C₂²×C₈	—	47	2	no	—
95	C₂²×C₈	—	94	2	no	—
96	C₂²×C₈	—	94	2	no	—
97	#117	—	2	2	no	—
98	#117	—	2	2	no	—
99	C₂×C₄⋊C₄	Dic₄₇	1	2	yes	—
100	C₂×C₂²⋊C₄	Dic₄₇	1	2	yes	—
101	C₂×C₄²	C₂×D₄₇	1	2	yes	—
102	C₂⁴	C₂×Dic₄₇	1	1	yes	characteristic
103	C₂²×C₄	Dic₉₄	1	2	yes	—
104	C₂²×C₄	C₄×D₄₇	1	2	yes	—
105	C₂²×C₄	C₄×D₄₇	1	2	yes	—
106	C₂²×C₄	D₁₈₈	1	2	yes	—
107	C₂²×C₄	C₂×Dic₄₇	1	4	yes	—
108	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
110	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
112	M₁₆	—	94	8	no	—
113	C₂×C₈	—	47	4	no	—
114	C₂×C₈	—	47	4	no	—
115	C₂×C₈	—	94	8	no	—
116	C₂×C₈	—	94	2	no	—
117	C₂×C₈	—	94	2	no	—
118	C₂×C₈	—	94	2	no	—
119	C₂×C₈	—	94	2	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₂²⋊C₄	—	2	4	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	2	no	—
124	C₂³	—	2	4	no	—
125	C₂³	C₄×Dic₄₇	1	2	yes	—
126	C₂³	C₄⋊Dic₄₇	1	2	yes	—
127	C₂³	C₂²⋊Dic₄₇	1	2	yes	—
128	C₂³	C₂²×Dic₄₇	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	C₄×Dic₄₇	1	2	yes	—
132	C₂×C₄	Dic₄₇⋊C₄	1	4	yes	—
133	C₂×C₄	Dic₄₇⋊C₄	1	4	yes	—
134	C₂×C₄	C₄⋊Dic₄₇	1	2	yes	—
135	C₂×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
136	C₂×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
137	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
138	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
141	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
142	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
144	C₈	—	94	4	no	—
145	C₈	—	94	4	no	—
146	C₂²	—	2	4	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	2	4	no	—
149	C₂²	C₄₇⋊#95	1	4	yes	—
150	C₂²	C₂×C₄×Dic₄₇	1	1	yes	characteristic
151	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
152	C₂²	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
153	C₂²	#13094031	1	4	yes	—
154	C₄	—	4	4	no	—
155	C₄	C₄₇⋊#95	1	4	yes	—
156	C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
157	C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
158	C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
159	C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
160	C₂	—	2	4	no	—
161	C₂	C₂×C₄₇⋊#95	1	1	yes	characteristic
162	C₂	#15689079	1	2	yes	—
163	C₂	#15689090	1	2	yes	—
164	C₂	#15689150	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄²
Fitting	C₂×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×C₄
hypercenter	C₂×#117	D₄₇
hypocenter	C₄₇	#1729
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #15688878	C₂	semidirect
7. C₂×#10625329	C₄	semidirect
10. C₂²×C₄₇⋊C₈	C₄	semidirect
87. C₄₇	#1729	semidirect