Subgroups — #16828424

Order 3712.

364 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271716	C₂	1	1	yes	maximal, characteristic
2	#14271291	C₂	1	4	yes	maximal, complemented
3	C₈×Dic₅₈	C₂	1	4	yes	maximal, complemented
4	#14271191	C₂	1	1	yes	maximal, characteristic
5	#14271190	C₂	1	1	yes	maximal, characteristic
6	#14271186	C₂	1	1	yes	maximal, characteristic
7	#14271118	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Dic₂₉	C₂	1	1	yes	maximal, characteristic
9	C₂₉×#405	C₂	1	1	yes	maximal, characteristic
10	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
12	C₂₉×#117	C₂²	1	1	yes	characteristic
13	#11745288	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
15	C₂×#90777	C₂²	1	1	yes	characteristic
16	C₂²×#6597	C₂²	1	1	yes	characteristic
17	#11745252	C₄	1	2	yes	—
18	#11745250	C₄	1	2	yes	—
19	#11745240	C₄	1	4	yes	—
20	C₄×Dic₅₈	C₂²	1	4	yes	—
21	C₂₉×#97	C₂²	1	2	yes	—
22	C₄×C₈×C₂₉	C₂²	1	2	yes	—
23	#11745163	C₂²	1	2	yes	—
24	#11745162	C₂²	1	4	yes	—
25	#11745162	C₂²	1	4	yes	—
26	C₈×Dic₂₉	C₂²	1	2	yes	—
27	C₈×Dic₂₉	C₂²	1	2	yes	—
28	C₈×Dic₂₉	C₂²	1	2	yes	—
29	C₄⋊#6597	C₂²	1	4	yes	—
30	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
31	C₂²×Dic₂₉	C₂×C₄	1	2	yes	—
32	C₂×Dic₅₈	C₂×C₄	1	4	yes	—
33	M₁₆×C₂₉	—	2	2	no	—
34	C₂×C₈×C₂₉	Q₈	1	4	yes	complemented
35	C₂×C₈×C₂₉	C₂³	1	2	yes	—
36	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
39	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
40	C₄²×C₂₉	C₂³	1	2	yes	—
41	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
43	C₄⋊Dic₂₉	C₂×C₄	1	4	yes	—
44	#90779	C₂×C₄	1	4	yes	—
45	#90779	C₂×C₄	1	4	yes	—
46	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₂₉	C₂³	1	2	yes	—
49	C₄×Dic₂₉	C₂³	1	2	yes	—
50	#90777	—	2	2	no	—
51	C₂×#6597	—	2	2	no	—
52	C₂×#6597	C₂³	1	2	yes	—
53	C₂×#6597	C₂³	1	2	yes	—
54	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₉	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₂₉	#41	1	2	yes	—
59	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₂₉	—	2	4	no	—
61	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
62	C₂×Dic₂₉	C₂²×C₄	1	4	yes	—
63	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
64	Dic₅₈	—	2	8	no	—
65	C₈×C₂₉	—	2	2	no	—
66	C₈×C₂₉	C₂×Q₈	1	2	yes	—
67	C₈×C₂₉	C₂×Q₈	1	2	yes	—
68	#6597	—	2	2	no	—
69	#6597	—	2	2	no	—
70	#2960	—	29	1	no	maximal, Sylow
71	C₂²×C₂₉	—	2	2	no	—
72	C₂²×C₂₉	C₄×Q₈	1	2	yes	—
73	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
74	C₄×C₂₉	—	2	4	no	—
75	C₄×C₂₉	C₄×Q₈	1	2	yes	—
76	C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
77	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
78	Dic₂₉	—	2	4	no	—
79	Dic₂₉	—	2	2	no	—
80	Dic₂₉	#131	1	4	yes	—
81	#533	—	29	1	no	—
82	#446	—	29	4	no	—
83	C₈×Q₈	—	29	4	no	—
84	#424	—	29	1	no	—
85	#424	—	29	1	no	—
86	#423	—	29	1	no	—
87	#405	—	29	1	no	—
88	#405	D₂₉	1	1	yes	characteristic
89	C₂×C₄×C₈	—	29	1	no	—
90	C₂×C₂₉	—	2	2	no	—
91	C₂×C₂₉	C₂×C₄×Q₈	1	1	yes	characteristic
92	C₂×C₂₉	C₂×#131	1	2	yes	—
93	C₂×M₁₆	—	29	1	no	—
94	C₂×M₁₆	C₂×D₂₉	1	1	yes	characteristic
95	C₂²×C₈	—	29	1	no	—
96	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
97	C₄⋊Q₈	—	29	2	no	—
98	#125	—	29	2	no	—
99	C₂²⋊Q₈	—	29	4	no	—
100	C₄×Q₈	—	29	4	no	—
101	#117	—	29	1	no	—
102	#117	C₂×D₂₉	1	1	yes	characteristic
103	C₂×C₄²	—	29	1	no	—
104	C₄⋊C₈	—	29	4	no	—
105	C₄⋊C₈	—	29	4	no	—
106	C₄⋊C₈	—	29	4	no	—
107	#97	—	29	2	no	—
108	#97	C₂×D₂₉	1	2	yes	—
109	C₄×C₈	—	29	2	no	—
110	C₄×C₈	—	29	2	no	—
111	C₄×C₈	—	29	2	no	—
112	C₄×C₈	C₂×D₂₉	1	2	yes	—
113	C₂₉	#2960	1	1	yes	characteristic, complemented, Sylow
114	C₂×Q₈	—	29	4	no	—
115	C₂²×C₄	—	29	2	no	—
116	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
117	M₁₆	—	2	2	no	—
118	M₁₆	—	58	2	no	—
119	C₂×C₈	—	29	2	no	—
120	C₂×C₈	—	29	2	no	—
121	C₂×C₈	—	58	2	no	—
122	C₂×C₈	Dic₅₈	1	4	yes	complemented
123	C₂×C₈	C₂²×D₂₉	1	2	yes	—
124	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
125	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
126	C₄⋊C₄	—	29	2	no	—
127	C₄⋊C₄	—	29	4	no	—
128	C₄⋊C₄	—	29	4	no	—
129	C₄⋊C₄	—	29	4	no	—
130	C₄⋊C₄	C₄×D₂₉	1	2	yes	—
131	C₂²⋊C₄	—	29	2	no	—
132	C₂²⋊C₄	C₄×D₂₉	1	2	yes	—
133	C₄²	—	29	1	no	—
134	C₄²	—	29	1	no	—
135	C₄²	—	29	2	no	—
136	C₄²	—	29	2	no	—
137	C₄²	C₂²×D₂₉	1	2	yes	—
138	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
139	Q₈	—	58	8	no	—
140	C₂×C₄	—	29	2	no	—
141	C₂×C₄	—	29	2	no	—
142	C₂×C₄	—	29	4	no	—
143	C₂×C₄	—	58	4	no	—
144	C₂×C₄	C₂×Dic₅₈	1	2	yes	—
145	C₂×C₄	C₂×C₄×D₂₉	1	4	yes	—
146	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
147	C₂×C₄	#90805	1	2	yes	—
148	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	58	2	no	—
151	C₈	—	58	2	no	—
152	C₈	C₂×Dic₅₈	1	2	yes	—
153	C₈	C₂×Dic₅₈	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	C₄×Dic₅₈	1	2	yes	—
156	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	29	4	no	—
159	C₄	—	58	2	no	—
160	C₄	—	58	4	no	—
161	C₄	C₄×Dic₅₈	1	2	yes	—
162	C₄	C₂²×Dic₅₈	1	1	yes	characteristic
163	C₄	C₂×#90805	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	C₂×C₄×Dic₅₈	1	1	yes	characteristic
166	C₂	#131×D₂₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂₉×#405	C₂
prime radical	C₂³×C₂₉	C₂²×C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#405	D₂₉
hypocenter	C₂₉	#2960
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #14271291	C₂	semidirect
3. C₈×Dic₅₈	C₂	semidirect
34. C₂×C₈×C₂₉	Q₈	semidirect
113. C₂₉	#2960	semidirect
122. 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
#14271716	2
#14271291	2
C₈×Dic₅₈	2
#14271191	2
#14271190	2
#14271186	2
#14271118	2
C₂×C₈×Dic₂₉	2
C₂₉×#405	2
#2960	29