Finite Group Databasebeta

Subgroups — #16820276

Order 3712.

196 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂×#11745170	C₂	1	1	yes	maximal, characteristic
2	#14271205	C₂	1	1	yes	maximal, characteristic
3	#14271203	C₂	1	1	yes	maximal, characteristic
4	C₂×#11745155	C₂	1	1	yes	maximal, characteristic
5	#14271121	C₂	1	1	yes	maximal, characteristic
6	#14270931	C₂	1	1	yes	maximal, characteristic
7	C₂₉×#343	C₂	1	1	yes	maximal, characteristic
8	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
11	#11745288	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
13	C₂×#90777	C₂²	1	1	yes	characteristic
14	C₂²×#6597	C₂²	1	1	yes	characteristic
15	#11745170	C₄	1	4	yes	complemented
16	#11745166	—	2	1	no	—
17	#11745165	—	2	1	no	—
18	#11745163	—	2	1	no	—
19	C₈×Dic₂₉	—	2	1	no	—
20	#11745156	—	2	1	no	—
21	#11745155	—	2	1	no	—
22	#11745155	—	2	1	no	—
23	#11745155	—	2	1	no	—
24	C₂²×C₄×C₂₉	—	2	1	no	—
25	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
26	C₂²×Dic₂₉	—	2	1	no	—
27	M₁₆×C₂₉	—	2	1	no	—
28	M₁₆×C₂₉	—	2	1	no	—
29	M₁₆×C₂₉	C₂×C₄	1	2	yes	—
30	M₁₆×C₂₉	Q₈	1	2	yes	—
31	C₂×C₈×C₂₉	—	2	1	no	—
32	C₂×C₈×C₂₉	—	2	1	no	—
33	C₄⋊C₄×C₂₉	—	2	1	no	—
34	C₄⋊C₄×C₂₉	—	2	1	no	—
35	C₂²⋊Dic₂₉	—	2	1	no	—
36	C₄⋊Dic₂₉	—	2	1	no	—
37	C₄⋊Dic₂₉	—	2	1	no	—
38	C₄⋊Dic₂₉	D₄	1	1	yes	characteristic
39	C₄⋊Dic₂₉	D₄	1	1	yes	characteristic
40	C₄×Dic₂₉	—	2	1	no	—
41	#90777	—	2	1	no	—
42	#90777	C₂×C₄	1	2	yes	—
43	C₂×#6597	—	2	1	no	—
44	C₂×#6597	C₂×C₄	1	2	yes	—
45	C₂×#6597	D₄	1	1	yes	characteristic
46	C₂×#6597	D₄	1	1	yes	characteristic
47	C₂×#6597	Q₈	1	2	yes	—
48	C₂³×C₂₉	#41	1	1	yes	characteristic
49	C₂×C₄×C₂₉	—	4	1	no	—
50	C₂×C₄×C₂₉	—	8	1	no	—
51	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
57	C₂×Dic₂₉	—	2	1	no	—
58	C₂×Dic₂₉	—	2	1	no	—
59	C₂×Dic₂₉	—	4	1	no	—
60	C₂×Dic₂₉	—	8	1	no	—
61	C₈×C₂₉	—	2	2	no	—
62	C₈×C₂₉	—	2	2	no	—
63	#6597	—	2	2	no	—
64	#6597	C₄⋊C₄	1	2	yes	—
65	#6597	C₄⋊C₄	1	2	yes	—
66	#1952	—	29	1	no	maximal, Sylow
67	C₂²×C₂₉	—	2	1	no	—
68	C₂²×C₂₉	—	2	1	no	—
69	C₂²×C₂₉	C₄×Q₈	1	1	yes	characteristic
70	C₂²×C₂₉	#121	1	1	yes	characteristic
71	C₂²×C₂₉	#125	1	1	yes	characteristic
72	C₄×C₂₉	—	8	1	no	—
73	C₄×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
75	C₄×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₄×C₂₉	C₄⋊Q₈	1	1	yes	characteristic
77	Dic₂₉	—	4	1	no	—
78	Dic₂₉	—	4	1	no	—
79	Dic₂₉	—	8	1	no	—
80	C₂×#108	—	29	1	no	—
81	#428	—	29	1	no	—
82	#428	—	29	1	no	—
83	C₂×#107	—	29	1	no	—
84	#405	—	29	1	no	—
85	#343	—	29	1	no	—
86	#343	D₂₉	1	1	yes	characteristic
87	C₂×C₂₉	—	2	1	no	—
88	C₂×C₂₉	—	2	1	no	—
89	C₂×C₂₉	#389	1	1	yes	characteristic
90	C₂×C₂₉	#472	1	1	yes	characteristic
91	C₂×C₂₉	#473	1	1	yes	characteristic
92	C₂×M₁₆	—	29	1	no	—
93	C₂×M₁₆	C₂×D₂₉	1	1	yes	characteristic
94	C₂×M₁₆	C₂×D₂₉	1	1	yes	characteristic
95	C₂²×C₈	—	29	1	no	—
96	#117	—	29	1	no	—
97	C₂×C₄⋊C₄	—	29	1	no	—
98	C₂×C₄⋊C₄	C₂×D₂₉	1	1	yes	characteristic
99	#108	—	29	4	no	—
100	#107	—	58	1	no	—
101	#107	—	58	1	no	—
102	#107	—	58	1	no	—
103	#107	—	58	1	no	—
104	#106	—	58	1	no	—
105	#106	—	58	1	no	—
106	#97	—	58	1	no	—
107	C₄×C₈	—	58	1	no	—
108	C₂₉	#1952	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	58	1	no	—
111	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
112	M₁₆	—	2	1	no	—
113	M₁₆	—	2	1	no	—
114	M₁₆	—	29	2	no	—
115	M₁₆	—	58	1	no	—
116	M₁₆	Dic₅₈	1	2	yes	—
117	M₁₆	C₄×D₂₉	1	2	yes	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	29	2	no	—
121	C₂×C₈	—	29	1	no	—
122	C₂×C₈	—	29	1	no	—
123	C₂×C₈	—	29	2	no	—
124	C₂×C₈	—	58	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	29	1	no	—
128	C₄⋊C₄	—	29	1	no	—
129	C₄⋊C₄	—	58	1	no	—
130	C₄⋊C₄	—	58	1	no	—
131	C₂²⋊C₄	—	58	1	no	—
132	C₄²	—	58	1	no	—
133	C₂³	#90805	1	1	yes	characteristic
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	8	1	no	—
136	C₂×C₄	—	58	1	no	—
137	C₂×C₄	—	58	1	no	—
138	C₂×C₄	—	116	1	no	—
139	C₂×C₄	—	232	1	no	—
140	C₂×C₄	C₂×Dic₅₈	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
142	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
143	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
144	C₂×C₄	#90807	1	1	yes	characteristic
145	C₂×C₄	Q₈×D₂₉	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	2	2	no	—
148	C₈	—	29	2	no	—
149	C₈	—	29	2	no	—
150	C₈	—	58	2	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	C₄×Dic₅₈	1	1	yes	characteristic
154	C₂²	#11745246	1	1	yes	characteristic
155	C₂²	#11745251	1	1	yes	characteristic
156	C₄	—	8	1	no	—
157	C₄	—	116	1	no	—
158	C₄	—	116	1	no	—
159	C₄	—	232	1	no	—
160	C₄	#11745240	1	1	yes	characteristic
161	C₄	#11745243	1	1	yes	characteristic
162	C₄	#11745250	1	1	yes	characteristic
163	C₄	C₄⋊C₄×D₂₉	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	#14271060	1	1	yes	characteristic
167	C₂	#14271432	1	1	yes	characteristic
168	C₂	#14271437	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11745246
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₉
Fitting	C₂₉×#343	C₂
socle	C₂²×C₂₉	#121
prime radical	C₂³×C₂₉	#41
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	#343	D₂₉
hypocenter	C₂₉	#1952
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

Complements

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

N	H ≅ G/N	type
15. #11745170	C₄	semidirect
108. C₂₉	#1952	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11745170	2
#14271205	2
#14271203	2
C₂×#11745155	2
#14271121	2
#14270931	2
C₂₉×#343	2
#1952	29