Finite Group Databasebeta

Subgroups — #16830028

Order 3712.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#14271794	C₂	1	2	yes	maximal
2	#14271571	C₂	1	4	yes	maximal, complemented
3	#14271567	C₂	1	2	yes	maximal, complemented
4	#14271564	C₂	1	2	yes	maximal, complemented
5	#14271203	C₂	1	2	yes	maximal
6	#14271201	C₂	1	1	yes	maximal, characteristic
7	#14271118	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₉×#107	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
11	#11745288	C₂²	1	2	yes	—
12	#11745288	C₂²	1	1	yes	characteristic
13	#11745286	—	2	4	no	—
14	C₂×#90777	C₂²	1	1	yes	characteristic
15	#11745252	—	2	2	no	—
16	#11745252	C₂²	1	4	yes	—
17	#11745250	—	2	2	no	—
18	#11745250	C₂²	1	4	yes	—
19	C₂₉×#107	C₂²	1	2	yes	—
20	C₂₉×#107	C₂²	1	2	yes	—
21	#11745166	C₂²	1	1	yes	characteristic
22	#11745166	C₂²	1	1	yes	characteristic
23	#11745165	C₂²	1	2	yes	—
24	#11745163	C₂²	1	2	yes	—
25	C₈×Dic₂₉	C₂²	1	2	yes	—
26	#11745156	C₂²	1	4	yes	—
27	#11745155	C₂²	1	4	yes	—
28	C₂²×C₄×C₂₉	—	2	2	no	—
29	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₅₈	—	2	4	no	—
31	C₂×C₈×C₂₉	Q₈	1	4	yes	complemented
32	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₂₉	—	2	2	no	—
35	C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
36	C₂²⋊Dic₂₉	—	2	2	no	—
37	C₂²⋊Dic₂₉	D₄	1	2	yes	—
38	C₄⋊Dic₂₉	—	2	4	no	—
39	C₄⋊Dic₂₉	D₄	1	1	yes	characteristic
40	C₄⋊Dic₂₉	D₄	1	1	yes	characteristic
41	C₄⋊Dic₂₉	C₂³	1	2	yes	—
42	C₄⋊Dic₂₉	C₂³	1	2	yes	—
43	#90779	—	2	4	no	—
44	#90779	—	2	4	no	—
45	C₄×Dic₂₉	—	2	2	no	—
46	C₄×Dic₂₉	C₂³	1	2	yes	—
47	#90777	Q₈	1	4	yes	complemented
48	C₂×#6597	C₂³	1	2	yes	—
49	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₉	—	2	4	no	—
51	C₂×C₄×C₂₉	—	8	2	no	—
52	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₂₉	—	2	2	no	—
57	C₂×Dic₂₉	—	2	2	no	—
58	C₂×Dic₂₉	C₂×D₄	1	2	yes	—
59	C₂×Dic₂₉	C₂×D₄	1	2	yes	—
60	Dic₅₈	—	8	4	no	—
61	C₈×C₂₉	C₂×Q₈	1	2	yes	—
62	C₈×C₂₉	C₂×Q₈	1	2	yes	—
63	#6597	C₂×Q₈	1	4	yes	—
64	#3164	—	29	1	no	maximal, Sylow
65	C₂²×C₂₉	—	2	1	no	—
66	C₂²×C₂₉	—	2	1	no	—
67	C₂²×C₂₉	C₄⋊Q₈	1	2	yes	—
68	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
69	C₄×C₂₉	—	4	4	no	—
70	C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
71	C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
72	C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
73	Dic₂₉	—	2	2	no	—
74	Dic₂₉	—	2	2	no	—
75	Dic₂₉	—	4	2	no	—
76	Dic₂₉	—	4	2	no	—
77	#544	—	29	2	no	—
78	#501	—	29	4	no	—
79	#500	—	29	2	no	—
80	#499	—	29	2	no	—
81	#428	—	29	2	no	—
82	#427	—	29	1	no	—
83	C₂×#107	D₂₉	1	1	yes	characteristic
84	#405	—	29	1	no	—
85	C₂×C₂₉	—	2	2	no	—
86	C₂×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₂₉	#576	1	1	yes	characteristic
88	C₂×C₂₉	#578	1	1	yes	characteristic
89	C₂×M₁₆	—	29	1	no	—
90	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
91	C₄⋊Q₈	—	29	4	no	—
92	C₄⋊Q₈	—	58	2	no	—
93	#125	—	29	4	no	—
94	#125	—	58	2	no	—
95	C₂²⋊Q₈	—	58	4	no	—
96	#117	—	29	1	no	—
97	#117	—	29	2	no	—
98	C₂×C₄⋊C₄	C₂×D₂₉	1	2	yes	—
99	#107	—	29	1	no	—
100	#107	—	29	1	no	—
101	#107	—	29	4	no	—
102	#107	C₂×D₂₉	1	2	yes	—
103	#107	C₂×D₂₉	1	2	yes	—
104	#106	—	29	2	no	—
105	#106	—	29	4	no	—
106	#97	—	29	2	no	—
107	C₄×C₈	—	29	2	no	—
108	C₂₉	#3164	1	1	yes	characteristic, complemented, Sylow
109	C₂×Q₈	—	58	4	no	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
112	M₁₆	—	29	4	no	—
113	C₂×C₈	—	29	2	no	—
114	C₂×C₈	Dic₅₈	1	4	yes	complemented
115	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	29	1	no	—
119	C₄⋊C₄	—	29	1	no	—
120	C₄⋊C₄	—	29	2	no	—
121	C₄⋊C₄	—	29	2	no	—
122	C₄⋊C₄	—	58	4	no	—
123	C₄⋊C₄	—	58	4	no	—
124	C₄⋊C₄	—	58	4	no	—
125	C₄⋊C₄	C₂²×D₂₉	1	4	yes	—
126	C₂²⋊C₄	—	29	2	no	—
127	C₂²⋊C₄	—	58	2	no	—
128	C₄²	—	29	2	no	—
129	C₄²	—	58	2	no	—
130	C₂³	D₄×D₂₉	1	1	yes	characteristic
131	Q₈	—	232	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	8	2	no	—
134	C₂×C₄	—	29	2	no	—
135	C₂×C₄	—	29	2	no	—
136	C₂×C₄	—	58	2	no	—
137	C₂×C₄	—	58	2	no	—
138	C₂×C₄	C₂×Dic₅₈	1	2	yes	—
139	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
140	C₂×C₄	Q₈×D₂₉	1	2	yes	—
141	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
142	C₈	—	29	4	no	—
143	C₈	C₂×Dic₅₈	1	2	yes	—
144	C₈	C₂×Dic₅₈	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#11745250	1	2	yes	—
148	C₂²	C₂×D₄×D₂₉	1	1	yes	characteristic
149	C₄	—	4	4	no	—
150	C₄	—	58	2	no	—
151	C₄	—	58	2	no	—
152	C₄	—	116	2	no	—
153	C₄	—	116	2	no	—
154	C₄	#11745250	1	2	yes	—
155	C₄	C₂²×Dic₅₈	1	1	yes	characteristic
156	C₄	C₂×Q₈×D₂₉	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	C₂×#11745250	1	1	yes	characteristic
159	C₂	#14271964	1	1	yes	characteristic
160	C₂	#14271976	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₉
commutator, cosocle	C₂×C₄×C₂₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂×C₂₉×#107	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂³×C₂₉	C₂×D₄
hypercenter	C₂×#107	D₂₉
hypocenter	C₂₉	#3164
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
2. #14271571	C₂	semidirect
3. #14271567	C₂	semidirect
4. #14271564	C₂	semidirect
31. C₂×C₈×C₂₉	Q₈	semidirect
47. #90777	Q₈	semidirect
108. C₂₉	#3164	semidirect
114. C₂×C₈	Dic₅₈	semidirect

Maximal subgroups

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

subgroup	index
#14271794	2
#14271571	2
#14271567	2
#14271564	2
#14271203	2
#14271201	2
#14271118	2
C₂×C₂₉×#107	2
#3164	29