Finite Group Databasebeta

← #104×D₇₁ #17986010

Subgroups — #104×D₇₁ #17986010

Order 4544.

170 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×D₇₁	C₂	1	1	yes	maximal, characteristic
2	M₁₆×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#14556056	C₂	1	1	yes	maximal, characteristic, complemented
5	#14556055	C₂	1	1	yes	maximal, characteristic, complemented
6	#14556054	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#11765866	C₄	1	1	yes	characteristic, complemented
9	#11765865	—	2	1	no	—
10	#11765865	C₂²	1	1	yes	characteristic
11	#11765864	—	2	1	no	—
12	#11765864	C₄	1	1	yes	characteristic, complemented
13	Q₈×D₇₁	C₄	1	1	yes	characteristic, complemented
14	D₄×D₇₁	—	2	1	no	—
15	D₄×D₇₁	C₄	1	1	yes	characteristic, complemented
16	C₂×C₄×D₇₁	—	2	1	no	—
17	C₂×C₄×D₇₁	—	2	1	no	—
18	C₂×C₄×D₇₁	C₂²	1	1	yes	characteristic
19	#11765849	C₂²	1	1	yes	characteristic, complemented
20	#11765848	—	2	1	no	—
21	C₈×D₇₁	—	2	1	no	—
22	C₄×Dic₇₁	—	2	1	no	—
23	C₄×Dic₇₁	C₂²	1	1	yes	characteristic, complemented
24	#41×C₇₁	C₂²	1	1	yes	characteristic
25	M₁₆×C₇₁	C₂²	1	1	yes	characteristic, complemented
26	C₄²×C₇₁	C₂²	1	1	yes	characteristic, complemented
27	C₂²×D₇₁	—	2	1	no	—
28	C₂²×D₇₁	D₄	1	1	yes	characteristic
29	Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic, complemented
30	D₄×C₇₁	—	2	1	no	—
31	D₄×C₇₁	C₂×C₄	1	1	yes	characteristic, complemented
32	C₂×C₄×C₇₁	—	2	1	no	—
33	C₂×C₄×C₇₁	—	2	1	no	—
34	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
35	C₂²⋊D₇₁	—	2	1	no	—
36	C₂²⋊D₇₁	—	2	1	no	—
37	C₂²⋊D₇₁	—	2	1	no	—
38	C₂×Dic₇₁	—	2	1	no	—
39	C₂×Dic₇₁	—	2	1	no	—
40	C₂×Dic₇₁	D₄	1	1	yes	characteristic
41	D₂₈₄	—	2	1	no	—
42	D₂₈₄	C₂×C₄	1	1	yes	characteristic, complemented
43	C₄×D₇₁	—	2	4	no	—
44	C₄×D₇₁	—	2	1	no	—
45	C₄×D₇₁	—	2	1	no	—
46	C₄×D₇₁	—	2	1	no	—
47	C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
48	C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
49	C₄×D₇₁	D₄	1	1	yes	characteristic
50	C₄×D₇₁	D₄	1	1	yes	characteristic
51	Dic₁₄₂	—	2	1	no	—
52	Dic₁₄₂	C₂×C₄	1	1	yes	characteristic, complemented
53	C₈×C₇₁	—	2	1	no	—
54	C₇₁⋊C₈	—	2	1	no	—
55	C₂²×C₇₁	—	2	1	no	—
56	C₂²×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₂×D₇₁	—	2	2	no	—
58	C₂×D₇₁	—	2	1	no	—
59	C₂×D₇₁	—	4	2	no	—
60	C₂×D₇₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×D₇₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₇₁	—	2	2	no	—
63	C₄×C₇₁	—	2	1	no	—
64	C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
65	C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
66	Dic₇₁	—	2	2	no	—
67	Dic₇₁	—	2	1	no	—
68	Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
69	Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₇₁	—	2	1	no	—
71	C₂×C₇₁	—	4	1	no	—
72	C₂×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	D₇₁	—	2	1	no	—
74	D₇₁	—	4	1	no	—
75	D₇₁	#104	1	2	yes	direct product factor
76	C₇₁	C₂×#104	1	1	yes	characteristic, complemented, Sylow
77	C₂×#104	—	71	1	no	maximal, Sylow
78	C₂×#41	—	71	1	no	—
79	C₂×M₁₆	—	71	1	no	—
80	C₂×C₄²	—	71	1	no	—
81	#104	—	71	1	no	—
82	#104	—	71	1	no	—
83	#104	—	71	1	no	—
84	#104	D₇₁	1	1	yes	characteristic, direct product factor
85	#41	—	71	1	no	—
86	#41	—	71	1	no	—
87	#41	—	71	1	no	—
88	#41	—	142	1	no	—
89	#41	—	142	1	no	—
90	#41	C₂×D₇₁	1	1	yes	characteristic
91	C₂×Q₈	—	71	1	no	—
92	C₂×D₄	—	71	1	no	—
93	C₂×D₄	—	142	1	no	—
94	C₂²×C₄	—	71	1	no	—
95	C₂²×C₄	—	142	1	no	—
96	C₂²×C₄	—	142	1	no	—
97	M₁₆	—	71	1	no	—
98	M₁₆	—	142	1	no	—
99	M₁₆	C₂×D₇₁	1	1	yes	characteristic, complemented
100	C₂×C₈	—	142	1	no	—
101	C₄²	—	71	1	no	—
102	C₄²	—	142	1	no	—
103	C₄²	C₂×D₇₁	1	1	yes	characteristic, complemented
104	C₂³	—	71	1	no	—
105	C₂³	—	142	1	no	—
106	Q₈	—	71	1	no	—
107	Q₈	—	142	1	no	—
108	Q₈	C₄×D₇₁	1	1	yes	characteristic, complemented
109	D₄	—	2	1	no	—
110	D₄	—	71	1	no	—
111	D₄	—	142	1	no	—
112	D₄	—	142	1	no	—
113	D₄	—	142	1	no	—
114	D₄	—	142	1	no	—
115	D₄	C₄×D₇₁	1	1	yes	characteristic, complemented
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	71	1	no	—
119	C₂×C₄	—	71	1	no	—
120	C₂×C₄	—	71	1	no	—
121	C₂×C₄	—	71	1	no	—
122	C₂×C₄	—	71	1	no	—
123	C₂×C₄	—	142	1	no	—
124	C₂×C₄	—	142	1	no	—
125	C₂×C₄	—	142	4	no	—
126	C₂×C₄	—	142	1	no	—
127	C₂×C₄	—	142	1	no	—
128	C₂×C₄	—	142	1	no	—
129	C₂×C₄	C₂²×D₇₁	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	142	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	284	2	no	—
134	C₂²	—	71	1	no	—
135	C₂²	—	71	1	no	—
136	C₂²	—	142	2	no	—
137	C₂²	—	142	1	no	—
138	C₂²	D₄×D₇₁	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	1	no	—
141	C₄	—	71	1	no	—
142	C₄	—	71	1	no	—
143	C₄	—	142	2	no	—
144	C₄	—	142	1	no	—
145	C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
146	C₄	D₄×D₇₁	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	4	1	no	—
149	C₂	—	284	1	no	—
150	C₂	—	71	2	no	—
151	C₂	—	142	1	no	—
152	C₂	C₂²⋊C₄×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₇₁
commutator	C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇₁
Fitting	#104×C₇₁	C₂
socle	C₂×C₇₁	C₂×C₂²⋊C₄
prime radical	#41×D₇₁	C₂
cosocle	C₂×C₄×C₇₁	C₂³
hypercenter	#104	D₇₁
hypocenter	C₇₁	C₂×#104
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
2. M₁₆×D₇₁	C₂	semidirect
3. C₄²×D₇₁	C₂	semidirect
4. #14556056	C₂	semidirect
5. #14556055	C₂	semidirect
6. #14556054	C₂	semidirect
7. #104×C₇₁	C₂	semidirect
8. #11765866	C₄	semidirect
12. #11765864	C₄	semidirect
13. Q₈×D₇₁	C₄	semidirect
15. D₄×D₇₁	C₄	semidirect
19. #11765849	C₂²	semidirect
23. C₄×Dic₇₁	C₂²	semidirect
25. M₁₆×C₇₁	C₂²	semidirect
26. C₄²×C₇₁	C₂²	semidirect
29. Q₈×C₇₁	C₂×C₄	semidirect
31. D₄×C₇₁	C₂×C₄	semidirect
42. D₂₈₄	C₂×C₄	semidirect
52. Dic₁₄₂	C₂×C₄	semidirect
75. D₇₁	#104	direct (N × H)
76. C₇₁	C₂×#104	semidirect
84. #104	D₇₁	direct (N × H)
99. M₁₆	C₂×D₇₁	semidirect
103. C₄²	C₂×D₇₁	semidirect
108. Q₈	C₄×D₇₁	semidirect
115. D₄	C₄×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
#41×D₇₁	2
M₁₆×D₇₁	2
C₄²×D₇₁	2
#14556056	2
#14556055	2
#14556054	2
#104×C₇₁	2
C₂×#104	71