Finite Group Databasebeta

Subgroups — #14521865

Order 1984.

248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 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	#11758567	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758564	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#137	C₂	1	1	yes	maximal, characteristic, complemented
4	#11758527	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#92482	C₂	1	1	yes	maximal, characteristic, complemented
6	#11758495	C₂	1	2	yes	maximal, complemented
7	Q₁₆×D₃₁	C₂	1	2	yes	maximal, complemented
8	#11758493	C₂	1	2	yes	maximal, complemented
9	#11758492	C₂	1	2	yes	maximal, complemented
10	#11758486	C₂	1	1	yes	maximal, characteristic, complemented
11	#11758484	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	#92499	—	2	1	no	—
15	#92499	C₂²	1	2	yes	—
16	Q₈×D₃₁	—	2	1	no	—
17	Q₈×D₃₁	C₂²	1	2	yes	—
18	Q₈×D₃₁	C₂²	1	2	yes	—
19	#92497	—	2	2	no	—
20	#92497	C₂²	1	2	yes	—
21	#92495	—	2	2	no	—
22	#92495	—	2	1	no	—
23	#92495	C₂²	1	1	yes	characteristic
24	#92495	C₂²	1	1	yes	characteristic
25	C₂×Dic₆₂	—	2	1	no	—
26	C₂×Dic₆₂	C₂²	1	1	yes	characteristic
27	Q₁₆×C₃₁	C₂²	1	2	yes	—
28	SD₁₆×C₃₁	C₂²	1	2	yes	—
29	M₁₆×C₃₁	C₂²	1	1	yes	characteristic
30	#92482	C₂²	1	2	yes	—
31	#92482	C₂²	1	2	yes	—
32	#92482	C₂²	1	1	yes	characteristic
33	#92481	C₂²	1	1	yes	characteristic
34	#92480	C₂²	1	1	yes	characteristic
35	#92479	C₂²	1	1	yes	characteristic
36	C₂×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
37	Dic₁₂₄	C₂²	1	2	yes	—
38	#92470	C₂²	1	2	yes	—
39	#92469	C₂²	1	2	yes	—
40	C₈×D₃₁	C₂²	1	2	yes	—
41	Q₈×C₃₁	—	2	1	no	—
42	Q₈×C₃₁	C₂³	1	1	yes	characteristic
43	Q₈×C₃₁	C₂³	1	2	yes	—
44	D₄×C₃₁	—	2	1	no	—
45	D₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₃₁	—	2	1	no	—
47	C₂×C₄×C₃₁	—	2	1	no	—
48	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²⋊D₃₁	—	2	1	no	—
50	C₂²⋊D₃₁	—	2	2	no	—
51	C₂²⋊D₃₁	D₄	1	2	yes	—
52	C₂×Dic₃₁	—	2	1	no	—
53	C₂×Dic₃₁	—	2	2	no	—
54	D₁₂₄	—	2	2	no	—
55	D₁₂₄	D₄	1	1	yes	characteristic
56	D₁₂₄	C₂³	1	1	yes	characteristic
57	C₄×D₃₁	—	2	2	no	—
58	C₄×D₃₁	—	2	2	no	—
59	C₄×D₃₁	—	2	1	no	—
60	C₄×D₃₁	—	2	2	no	—
61	C₄×D₃₁	C₂³	1	2	yes	—
62	Dic₆₂	—	2	1	no	—
63	Dic₆₂	—	2	2	no	—
64	Dic₆₂	—	2	2	no	—
65	Dic₆₂	D₄	1	1	yes	characteristic
66	Dic₆₂	C₂³	1	2	yes	—
67	Dic₆₂	C₂³	1	1	yes	characteristic
68	C₈×C₃₁	C₂³	1	2	yes	—
69	C₃₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₃₁⋊C₈	C₂³	1	1	yes	characteristic
71	C₂²×C₃₁	—	2	1	no	—
72	C₂²×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂×D₃₁	—	2	1	no	—
74	C₂×D₃₁	C₂×D₄	1	2	yes	—
75	C₄×C₃₁	—	2	1	no	—
76	C₄×C₃₁	—	2	2	no	—
77	C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₄×C₃₁	C₂⁴	1	1	yes	characteristic
79	Dic₃₁	—	2	1	no	—
80	Dic₃₁	—	2	2	no	—
81	Dic₃₁	C₂×D₄	1	2	yes	—
82	#578	—	31	1	no	maximal, Sylow
83	C₂×C₃₁	—	2	1	no	—
84	C₂×C₃₁	—	4	1	no	—
85	C₂×C₃₁	C₂²×D₄	1	1	yes	characteristic
86	D₃₁	—	2	2	no	—
87	D₃₁	—	4	1	no	—
88	2₋¹⁺⁴	—	31	1	no	—
89	2₋¹⁺⁴	—	31	1	no	—
90	#137	—	31	2	no	—
91	#137	—	31	2	no	—
92	#137	—	31	1	no	—
93	#137	D₃₁	1	1	yes	characteristic, complemented
94	#135	—	31	1	no	—
95	#135	—	31	2	no	—
96	C₂×Q₁₆	—	31	1	no	—
97	C₂×Q₁₆	—	31	2	no	—
98	#131	—	31	1	no	—
99	C₃₁	#578	1	1	yes	characteristic, complemented, Sylow
100	#41	—	31	1	no	—
101	#41	—	31	2	no	—
102	#41	—	31	2	no	—
103	#41	—	31	1	no	—
104	#41	—	62	2	no	—
105	#41	—	62	1	no	—
106	#41	—	62	2	no	—
107	#41	—	62	1	no	—
108	#41	C₂×D₃₁	1	1	yes	characteristic
109	C₂×Q₈	—	31	2	no	—
110	C₂×Q₈	—	31	2	no	—
111	C₂×Q₈	—	31	1	no	—
112	C₂×Q₈	—	62	1	no	—
113	C₂×Q₈	—	62	1	no	—
114	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic
115	Q₁₆	—	31	1	no	—
116	Q₁₆	—	31	2	no	—
117	Q₁₆	—	31	2	no	—
118	Q₁₆	—	31	2	no	—
119	Q₁₆	C₂×D₃₁	1	2	yes	—
120	SD₁₆	—	31	1	no	—
121	SD₁₆	—	31	1	no	—
122	SD₁₆	—	31	2	no	—
123	SD₁₆	C₂×D₃₁	1	2	yes	—
124	D₈	—	31	1	no	—
125	M₁₆	—	31	2	no	—
126	M₁₆	C₂×D₃₁	1	1	yes	characteristic
127	C₂×C₈	—	31	1	no	—
128	C₂×C₈	—	31	2	no	—
129	Q₈	—	2	1	no	—
130	Q₈	—	31	1	no	—
131	Q₈	—	31	2	no	—
132	Q₈	—	31	1	no	—
133	Q₈	—	62	2	no	—
134	Q₈	—	62	2	no	—
135	Q₈	—	62	1	no	—
136	Q₈	C₂²×D₃₁	1	1	yes	characteristic
137	Q₈	C₂²×D₃₁	1	2	yes	—
138	D₄	—	2	1	no	—
139	D₄	—	31	2	no	—
140	D₄	—	31	1	no	—
141	D₄	—	31	1	no	—
142	D₄	—	62	2	no	—
143	D₄	—	62	2	no	—
144	D₄	—	62	1	no	—
145	D₄	C₂²×D₃₁	1	1	yes	characteristic
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	31	2	no	—
149	C₂×C₄	—	62	2	no	—
150	C₂×C₄	—	62	2	no	—
151	C₂×C₄	—	62	2	no	—
152	C₂×C₄	—	62	2	no	—
153	C₂×C₄	—	62	1	no	—
154	C₂×C₄	—	62	1	no	—
155	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
156	C₈	—	31	1	no	—
157	C₈	—	31	1	no	—
158	C₈	C₂²×D₃₁	1	2	yes	—
159	C₂²	—	2	1	no	—
160	C₂²	—	31	2	no	—
161	C₂²	—	62	1	no	—
162	C₂²	D₄×D₃₁	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	2	no	—
165	C₄	—	31	2	no	—
166	C₄	—	62	1	no	—
167	C₄	—	62	2	no	—
168	C₄	D₄×D₃₁	1	1	yes	characteristic
169	C₄	C₂³×D₃₁	1	1	yes	characteristic
170	C₂	—	2	1	no	—
171	C₂	—	4	1	no	—
172	C₂	—	62	2	no	—
173	C₂	—	124	1	no	—
174	C₂	C₂×D₄×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₃₁
commutator, cosocle	C₄×C₃₁	C₂⁴
Frattini	C₄	C₂³×D₃₁
Fitting	C₃₁×#137	C₂
socle	C₂×C₃₁	C₂²×D₄
hypercenter	#137	D₃₁
hypocenter	C₃₁	#578
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. #11758567	C₂	semidirect
2. #11758564	C₂	semidirect
3. C₃₁×#137	C₂	semidirect
4. #11758527	C₂	semidirect
5. C₂×#92482	C₂	semidirect
6. #11758495	C₂	semidirect
7. Q₁₆×D₃₁	C₂	semidirect
8. #11758493	C₂	semidirect
9. #11758492	C₂	semidirect
10. #11758486	C₂	semidirect
11. #11758484	C₂	semidirect
93. #137	D₃₁	semidirect
99. C₃₁	#578	semidirect

Maximal subgroups

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

subgroup	index
#11758567	2
#11758564	2
C₃₁×#137	2
#11758527	2
C₂×#92482	2
#11758495	2
Q₁₆×D₃₁	2
#11758493	2
#11758492	2
#11758486	2
#11758484	2
#578	31