Subgroups — #16828439

Order 3712.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 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	#14271291	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₅₈	C₂	1	2	yes	maximal, complemented
5	#14271191	C₂	1	2	yes	maximal
6	C₂×C₄⋊#6597	C₂	1	1	yes	maximal, characteristic
7	#14271121	C₂	1	2	yes	maximal
8	C₄×M₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic
9	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
12	#11745288	C₂²	1	2	yes	—
13	#11745286	C₄	1	2	yes	—
14	#11745286	C₄	1	2	yes	—
15	C₂²×#6597	C₂²	1	2	yes	—
16	#11745232	C₄	1	2	yes	—
17	#11745231	C₄	1	1	yes	characteristic
18	#11745231	C₄	1	1	yes	characteristic
19	C₄×Dic₅₈	C₂²	1	2	yes	—
20	C₄×Dic₅₈	C₂²	1	2	yes	—
21	C₂₉×#97	C₂²	1	2	yes	—
22	C₄×C₈×C₂₉	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	2	yes	—
29	C₄⋊#6597	C₂²	1	1	yes	characteristic
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	M₁₆×C₂₉	—	2	2	no	—
36	M₁₆×C₂₉	Q₈	1	4	yes	complemented
37	C₂×C₈×C₂₉	C₂³	1	2	yes	—
38	C₂×C₈×C₂₉	C₂³	1	2	yes	—
39	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₂₉	C₂³	1	1	yes	characteristic
42	C₄²×C₂₉	C₂³	1	1	yes	characteristic
43	C₂²⋊Dic₂₉	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
45	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
47	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
48	#90779	C₂×C₄	1	4	yes	—
49	#90779	C₂×C₄	1	4	yes	—
50	C₄×Dic₂₉	C₂³	1	4	yes	—
51	C₂×#6597	—	2	4	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	—	2	2	no	—
56	C₂×C₄×C₂₉	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₂₉	#41	1	2	yes	—
64	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₂₉	C₂²×C₄	1	4	yes	—
66	C₂×Dic₂₉	C₂²×C₄	1	4	yes	—
67	Dic₅₈	—	2	4	no	—
68	Dic₅₈	—	4	2	no	—
69	C₈×C₂₉	—	2	2	no	—
70	C₈×C₂₉	C₂×Q₈	1	4	yes	—
71	#6597	—	2	2	no	—
72	#6597	—	2	2	no	—
73	#2962	—	29	1	no	maximal, Sylow
74	C₂²×C₂₉	—	2	1	no	—
75	C₂²×C₂₉	—	2	1	no	—
76	C₂²×C₂₉	C₄×Q₈	1	2	yes	—
77	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
78	C₄×C₂₉	—	2	1	no	—
79	C₄×C₂₉	—	2	1	no	—
80	C₄×C₂₉	—	2	1	no	—
81	C₄×C₂₉	C₄×Q₈	1	2	yes	—
82	C₄×C₂₉	#131	1	2	yes	—
83	C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
84	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
85	Dic₂₉	—	2	4	no	—
86	Dic₂₉	—	2	4	no	—
87	#533	—	29	1	no	—
88	#446	—	29	4	no	—
89	#446	—	29	2	no	—
90	C₈×Q₈	—	29	2	no	—
91	#424	—	29	2	no	—
92	C₂×C₄⋊C₈	—	29	1	no	—
93	#405	—	29	2	no	—
94	C₄×M₁₆	D₂₉	1	1	yes	characteristic
95	C₂×C₂₉	—	2	2	no	—
96	C₂×C₂₉	C₂×C₄×Q₈	1	1	yes	characteristic
97	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
98	C₂×C₂₉	#568	1	1	yes	characteristic
99	C₂×M₁₆	C₂×D₂₉	1	1	yes	characteristic
100	C₂×M₁₆	C₂×D₂₉	1	1	yes	characteristic
101	C₂²×C₈	—	29	2	no	—
102	C₄⋊Q₈	—	29	1	no	—
103	C₄⋊Q₈	—	29	1	no	—
104	#125	—	29	2	no	—
105	C₂²⋊Q₈	—	29	2	no	—
106	C₂²⋊Q₈	—	29	2	no	—
107	C₄×Q₈	—	29	2	no	—
108	C₄×Q₈	—	29	2	no	—
109	#117	—	29	2	no	—
110	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
111	C₄⋊C₈	—	29	4	no	—
112	C₄⋊C₈	—	29	4	no	—
113	C₄⋊C₈	—	29	1	no	—
114	C₄⋊C₈	—	29	1	no	—
115	C₄⋊C₈	—	29	2	no	—
116	#97	—	29	4	no	—
117	#97	C₂×D₂₉	1	2	yes	—
118	C₄×C₈	—	29	4	no	—
119	C₄×C₈	C₂×D₂₉	1	2	yes	—
120	C₂₉	#2962	1	1	yes	characteristic, complemented, Sylow
121	C₂×Q₈	—	29	2	no	—
122	C₂×Q₈	—	29	2	no	—
123	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
124	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
126	M₁₆	—	2	2	no	—
127	M₁₆	Dic₅₈	1	4	yes	complemented
128	C₂×C₈	—	29	2	no	—
129	C₂×C₈	—	29	2	no	—
130	C₂×C₈	—	58	4	no	—
131	C₂×C₈	C₂²×D₂₉	1	2	yes	—
132	C₂×C₈	C₂²×D₂₉	1	2	yes	—
133	C₄⋊C₄	—	29	2	no	—
134	C₄⋊C₄	—	29	2	no	—
135	C₄⋊C₄	—	29	4	no	—
136	C₄⋊C₄	—	29	4	no	—
137	C₄⋊C₄	—	29	2	no	—
138	C₄⋊C₄	—	29	2	no	—
139	C₂²⋊C₄	—	29	4	no	—
140	C₄²	—	29	4	no	—
141	C₄²	C₄×D₂₉	1	1	yes	characteristic
142	C₄²	C₄×D₂₉	1	1	yes	characteristic
143	C₄²	C₂²×D₂₉	1	1	yes	characteristic
144	C₄²	C₂²×D₂₉	1	1	yes	characteristic
145	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
146	Q₈	—	58	4	no	—
147	Q₈	—	116	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	29	4	no	—
151	C₂×C₄	—	29	4	no	—
152	C₂×C₄	C₂×Dic₅₈	1	2	yes	—
153	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
158	C₂×C₄	#90805	1	2	yes	—
159	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	58	2	no	—
162	C₈	—	58	2	no	—
163	C₈	C₂×Dic₅₈	1	4	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₄×Dic₅₈	1	2	yes	—
167	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	58	4	no	—
172	C₄	—	58	4	no	—
173	C₄	C₄×Dic₅₈	1	2	yes	—
174	C₄	#11745269	1	2	yes	—
175	C₄	C₂²×Dic₅₈	1	1	yes	characteristic
176	C₄	C₂×#90805	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×C₄×Dic₅₈	1	1	yes	characteristic
179	C₂	C₂×#11745269	1	1	yes	characteristic
180	C₂	#14271929	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₄×M₁₆×C₂₉	C₂
prime radical	C₂³×C₂₉	C₂²×C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₄×M₁₆	D₂₉
hypocenter	C₂₉	#2962
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #14271293	C₂	semidirect
3. #14271291	C₂	semidirect
4. C₈×Dic₅₈	C₂	semidirect
36. M₁₆×C₂₉	Q₈	semidirect
120. C₂₉	#2962	semidirect
127. M₁₆	Dic₅₈	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
#14271291	2
C₈×Dic₅₈	2
#14271191	2
C₂×C₄⋊#6597	2
#14271121	2
C₄×M₁₆×C₂₉	2
#2962	29