Subgroups — #16828593

Order 3712.

330 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271720	C₂	1	1	yes	maximal, characteristic
2	#14271293	C₂	1	2	yes	maximal, complemented
3	#14271290	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₅₈	C₂	1	2	yes	maximal, complemented
5	#14271287	C₂	1	2	yes	maximal, complemented
6	#14271221	C₂	1	4	yes	maximal
7	#14271216	C₂	1	1	yes	maximal, characteristic
8	C₂₉×#431	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
10	#11745288	C₂²	1	2	yes	—
11	#11745286	C₄	1	2	yes	—
12	#11745286	C₄	1	2	yes	—
13	#11745232	C₄	1	2	yes	—
14	#11745231	C₄	1	1	yes	characteristic
15	#11745231	C₄	1	1	yes	characteristic
16	C₄×Dic₅₈	C₂²	1	2	yes	—
17	C₄×Dic₅₈	C₂²	1	2	yes	—
18	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂₉×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₄×C₈×C₂₉	C₂²	1	2	yes	—
22	C₂²⋊#6597	C₂²	1	2	yes	—
23	#11745163	C₂²	1	4	yes	—
24	#11745162	C₂²	1	4	yes	—
25	#11745162	C₂²	1	4	yes	—
26	C₈×Dic₂₉	C₂²	1	4	yes	—
27	C₄⋊#6597	C₂²	1	1	yes	characteristic
28	C₄⋊#6597	C₂²	1	1	yes	characteristic
29	C₄×#6597	C₂²	1	2	yes	—
30	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
33	C₂×Dic₅₈	C₂×C₄	1	2	yes	—
34	C₂×Dic₅₈	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₂₉	C₂³	1	2	yes	—
36	C₂×C₈×C₂₉	C₂³	1	2	yes	—
37	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₂₉	C₂³	1	1	yes	characteristic
40	C₄²×C₂₉	C₂³	1	1	yes	characteristic
41	C₂²⋊Dic₂₉	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
43	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
44	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
45	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
46	#90779	C₂×C₄	1	4	yes	—
47	#90779	C₂×C₄	1	4	yes	—
48	C₄×Dic₂₉	C₂³	1	4	yes	—
49	C₂×#6597	C₂³	1	2	yes	—
50	C₂×#6597	C₂³	1	2	yes	—
51	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₉	—	2	2	no	—
53	C₂×C₄×C₂₉	—	2	2	no	—
54	C₂×C₄×C₂₉	—	2	1	no	—
55	C₂×C₄×C₂₉	—	2	1	no	—
56	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₂₉	C₂²×C₄	1	4	yes	—
63	C₂×Dic₂₉	C₂²×C₄	1	4	yes	—
64	Dic₅₈	—	2	4	no	—
65	Dic₅₈	—	2	4	no	—
66	C₈×C₂₉	—	2	2	no	—
67	C₈×C₂₉	—	2	2	no	—
68	#6597	—	2	2	no	—
69	#6597	—	2	2	no	—
70	#2986	—	29	1	no	maximal, Sylow
71	C₂²×C₂₉	—	2	1	no	—
72	C₂²×C₂₉	—	2	1	no	—
73	C₂²×C₂₉	—	2	1	no	—
74	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
75	C₄×C₂₉	—	2	1	no	—
76	C₄×C₂₉	—	2	1	no	—
77	C₄×C₂₉	—	2	1	no	—
78	C₄×C₂₉	#131	1	2	yes	—
79	C₄×C₂₉	#131	1	2	yes	—
80	C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
81	C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
82	Dic₂₉	—	2	4	no	—
83	Dic₂₉	—	2	4	no	—
84	#533	—	29	1	no	—
85	#446	—	29	2	no	—
86	#446	—	29	2	no	—
87	C₈×Q₈	—	29	2	no	—
88	C₈×Q₈	—	29	2	no	—
89	#433	—	29	4	no	—
90	#431	—	29	1	no	—
91	#431	D₂₉	1	1	yes	characteristic
92	C₂×C₂₉	—	4	1	no	—
93	C₂×C₂₉	#519	1	1	yes	characteristic
94	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
95	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
96	C₄⋊Q₈	—	29	1	no	—
97	C₄⋊Q₈	—	29	1	no	—
98	#125	—	29	2	no	—
99	C₂²⋊Q₈	—	29	2	no	—
100	C₂²⋊Q₈	—	29	2	no	—
101	C₄×Q₈	—	29	2	no	—
102	C₄×Q₈	—	29	2	no	—
103	#117	—	29	2	no	—
104	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
105	C₄⋊C₈	—	29	4	no	—
106	C₄⋊C₈	—	29	4	no	—
107	C₄⋊C₈	—	29	1	no	—
108	C₄⋊C₈	—	29	1	no	—
109	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
110	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
111	C₂²⋊C₈	—	29	2	no	—
112	C₂²⋊C₈	C₂×D₂₉	1	2	yes	—
113	#97	—	29	4	no	—
114	C₄×C₈	—	29	4	no	—
115	C₄×C₈	—	29	2	no	—
116	C₄×C₈	C₂×D₂₉	1	2	yes	—
117	C₂₉	#2986	1	1	yes	characteristic, complemented, Sylow
118	C₂×Q₈	—	29	2	no	—
119	C₂×Q₈	—	29	2	no	—
120	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
121	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
122	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
123	C₂×C₈	—	29	2	no	—
124	C₂×C₈	—	29	2	no	—
125	C₂×C₈	C₂²×D₂₉	1	2	yes	—
126	C₂×C₈	C₂²×D₂₉	1	2	yes	—
127	C₄⋊C₄	—	29	2	no	—
128	C₄⋊C₄	—	29	2	no	—
129	C₄⋊C₄	—	29	4	no	—
130	C₄⋊C₄	—	29	4	no	—
131	C₄⋊C₄	—	29	2	no	—
132	C₄⋊C₄	—	29	2	no	—
133	C₂²⋊C₄	—	29	4	no	—
134	C₄²	—	29	4	no	—
135	C₄²	C₄×D₂₉	1	1	yes	characteristic
136	C₄²	C₄×D₂₉	1	1	yes	characteristic
137	C₄²	C₂²×D₂₉	1	1	yes	characteristic
138	C₄²	C₂²×D₂₉	1	1	yes	characteristic
139	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
140	Q₈	—	58	4	no	—
141	Q₈	—	58	4	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	29	4	no	—
147	C₂×C₄	—	29	4	no	—
148	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
153	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	2	2	no	—
156	C₈	—	58	2	no	—
157	C₈	—	58	2	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	58	4	no	—
166	C₄	—	58	4	no	—
167	C₄	#11745263	1	2	yes	—
168	C₄	#11745269	1	2	yes	—
169	C₄	#11745365	1	1	yes	characteristic
170	C₄	#11745365	1	1	yes	characteristic
171	C₂	—	4	1	no	—
172	C₂	#14271640	1	1	yes	characteristic
173	C₂	C₂×#11745263	1	1	yes	characteristic
174	C₂	C₂×#11745269	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#431	C₂
prime radical	C₂³×C₂₉	C₂²×C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#431	D₂₉
hypocenter	C₂₉	#2986
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #14271293	C₂	semidirect
3. #14271290	C₂	semidirect
4. C₈×Dic₅₈	C₂	semidirect
5. #14271287	C₂	semidirect
117. C₂₉	#2986	semidirect

Maximal subgroups

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

subgroup	index
#14271720	2
#14271293	2
#14271290	2
C₈×Dic₅₈	2
#14271287	2
#14271221	2
#14271216	2
C₂₉×#431	2
#2986	29