Subgroups — #16819185

Order 3712.

480 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#90777	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₂₉	C₂	1	1	yes	maximal, characteristic
3	#14270917	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
6	C₂³×Dic₂₉	C₄	1	2	yes	—
7	C₂×M₁₆×C₂₉	—	2	2	no	—
8	C₂×M₁₆×C₂₉	—	2	2	no	—
9	C₂×M₁₆×C₂₉	C₄	1	4	yes	complemented
10	C₂²×C₈×C₂₉	C₂²	1	2	yes	—
11	C₂×C₄×Dic₂₉	—	2	2	no	—
12	C₂×C₄×Dic₂₉	—	2	2	no	—
13	C₂×C₄×Dic₂₉	C₄	1	2	yes	—
14	C₂×C₄×Dic₂₉	C₂²	1	2	yes	—
15	C₂×#90777	—	2	2	no	—
16	C₂×#90777	—	2	2	no	—
17	C₂×#90777	C₄	1	4	yes	complemented
18	C₂²×#6597	C₂²	1	2	yes	—
19	C₂⁴×C₂₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₉	D₄	1	2	yes	—
24	C₂²×C₄×C₂₉	D₄	1	2	yes	—
25	C₂²×C₄×C₂₉	D₄	1	2	yes	—
26	C₂²×C₄×C₂₉	Q₈	1	2	yes	—
27	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
28	C₂²×Dic₂₉	—	2	4	no	—
29	C₂²×Dic₂₉	—	2	8	no	—
30	C₂²×Dic₂₉	C₂×C₄	1	4	yes	—
31	M₁₆×C₂₉	—	2	8	no	—
32	C₂×C₈×C₂₉	—	2	2	no	—
33	C₂×C₈×C₂₉	—	2	2	no	—
34	C₂×C₈×C₂₉	C₂×C₄	1	4	yes	complemented
35	C₄×Dic₂₉	—	2	4	no	—
36	C₄×Dic₂₉	—	2	2	no	—
37	C₄×Dic₂₉	—	2	2	no	—
38	#90777	—	2	8	no	—
39	C₂×#6597	—	2	2	no	—
40	C₂×#6597	—	2	2	no	—
41	C₂×#6597	C₂×C₄	1	4	yes	complemented
42	C₂³×C₂₉	—	2	4	no	—
43	C₂³×C₂₉	C₄²	1	2	yes	—
44	C₂³×C₂₉	C₂²⋊C₄	1	2	yes	—
45	C₂³×C₂₉	C₄⋊C₄	1	2	yes	—
46	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₉	C₄²	1	2	yes	—
48	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₂₉	C₂²⋊C₄	1	4	yes	—
50	C₂×C₄×C₂₉	C₂²⋊C₄	1	4	yes	—
51	C₂×C₄×C₂₉	C₄⋊C₄	1	2	yes	—
52	C₂×C₄×C₂₉	C₄⋊C₄	1	4	yes	—
53	C₂×C₄×C₂₉	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×Dic₂₉	—	2	16	no	—
61	C₂×Dic₂₉	—	2	4	no	—
62	C₂×Dic₂₉	—	2	4	no	—
63	C₂×Dic₂₉	M₁₆	1	8	yes	complemented
64	C₈×C₂₉	—	2	4	no	—
65	#6597	—	2	4	no	—
66	#1728	—	29	1	no	maximal, Sylow
67	C₂²×C₂₉	—	2	4	no	—
68	C₂²×C₂₉	—	2	4	no	—
69	C₂²×C₂₉	—	2	4	no	—
70	C₂²×C₂₉	#95	1	4	yes	—
71	C₂²×C₂₉	C₂×C₄²	1	1	yes	characteristic
72	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₂₉	C₂×M₁₆	1	4	yes	—
75	C₄×C₂₉	#95	1	4	yes	—
76	C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₄×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₄×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
80	Dic₂₉	—	2	8	no	—
81	C₂²×M₁₆	—	29	1	no	—
82	C₂²×M₁₆	D₂₉	1	1	yes	characteristic
83	C₂²×C₄²	—	29	1	no	—
84	#336	—	29	4	no	—
85	C₂×C₂₉	—	2	4	no	—
86	C₂×C₂₉	C₂×#95	1	1	yes	characteristic
87	C₂×C₂₉	C₄×M₁₆	1	2	yes	—
88	C₂×C₂₉	#407	1	2	yes	—
89	C₂×C₂₉	#423	1	2	yes	—
90	C₂³×C₄	—	29	2	no	—
91	C₂³×C₄	C₂×D₂₉	1	1	yes	characteristic
92	C₂×M₁₆	—	2	2	no	—
93	C₂×M₁₆	—	2	2	no	—
94	C₂×M₁₆	—	29	4	no	—
95	C₂×M₁₆	—	58	2	no	—
96	C₂×M₁₆	—	58	2	no	—
97	C₂×M₁₆	Dic₂₉	1	4	yes	complemented
98	C₂²×C₈	—	29	2	no	—
99	C₂²×C₈	C₂×D₂₉	1	2	yes	—
100	C₂×C₄²	—	29	2	no	—
101	C₂×C₄²	—	29	2	no	—
102	C₂×C₄²	—	58	2	no	—
103	C₂×C₄²	—	58	2	no	—
104	C₂₉	#1728	1	1	yes	characteristic, complemented, Sylow
105	C₂⁴	C₄×D₂₉	1	1	yes	characteristic
106	C₂²×C₄	—	29	4	no	—
107	C₂²×C₄	—	58	8	no	—
108	C₂²×C₄	—	58	4	no	—
109	C₂²×C₄	Dic₅₈	1	2	yes	—
110	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
111	C₂²×C₄	C₄×D₂₉	1	2	yes	—
112	C₂²×C₄	D₁₁₆	1	2	yes	—
113	C₂²×C₄	C₂×Dic₂₉	1	2	yes	—
114	C₂²×C₄	C₂²⋊D₂₉	1	2	yes	—
115	C₂²×C₄	C₂²⋊D₂₉	1	2	yes	—
116	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
117	M₁₆	—	2	8	no	—
118	M₁₆	—	58	8	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	29	4	no	—
122	C₂×C₈	—	58	2	no	—
123	C₂×C₈	—	58	2	no	—
124	C₂×C₈	C₂×Dic₂₉	1	4	yes	complemented
125	C₄²	—	58	4	no	—
126	C₄²	—	58	2	no	—
127	C₄²	—	58	2	no	—
128	C₂³	—	2	4	no	—
129	C₂³	C₄×Dic₂₉	1	2	yes	—
130	C₂³	#90779	1	2	yes	—
131	C₂³	#90781	1	2	yes	—
132	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
133	C₂×C₄	—	29	8	no	—
134	C₂×C₄	—	58	16	no	—
135	C₂×C₄	—	58	4	no	—
136	C₂×C₄	—	58	4	no	—
137	C₂×C₄	C₄×Dic₂₉	1	2	yes	—
138	C₂×C₄	#90779	1	4	yes	—
139	C₂×C₄	#90779	1	2	yes	—
140	C₂×C₄	C₄⋊Dic₂₉	1	4	yes	—
141	C₂×C₄	#90781	1	4	yes	—
142	C₂×C₄	#90781	1	2	yes	—
143	C₂×C₄	C₂²⋊Dic₂₉	1	4	yes	—
144	C₂×C₄	C₂×Dic₅₈	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
146	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
147	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
149	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
150	C₈	—	2	4	no	—
151	C₈	—	58	4	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	#11745179	1	4	yes	—
156	C₂²	M₁₆×D₂₉	1	4	yes	—
157	C₂²	C₂×C₄×Dic₂₉	1	1	yes	characteristic
158	C₂²	C₂×#90779	1	1	yes	characteristic
159	C₂²	C₂×#90781	1	1	yes	characteristic
160	C₄	—	58	8	no	—
161	C₄	#11745179	1	4	yes	—
162	C₄	C₂×#90779	1	1	yes	characteristic
163	C₄	C₂×C₄⋊Dic₂₉	1	1	yes	characteristic
164	C₄	C₂×#90781	1	1	yes	characteristic
165	C₄	C₂×C₂²⋊Dic₂₉	1	1	yes	characteristic
166	C₂	—	2	4	no	—
167	C₂	C₂×#11745179	1	1	yes	characteristic
168	C₂	M₁₆×Dic₂₉	1	2	yes	—
169	C₂	#14271126	1	2	yes	—
170	C₂	#14271186	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₉
commutator	C₂²×C₂₉	C₂×C₄²
Fitting	C₂²×M₁₆×C₂₉	C₂
socle	C₂³×C₂₉	C₂²×C₄
prime radical	C₂⁴×C₂₉	C₂×C₄
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	C₂²×M₁₆	D₂₉
hypocenter	C₂₉	#1728
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14270917	C₂	semidirect
9. C₂×M₁₆×C₂₉	C₄	semidirect
17. C₂×#90777	C₄	semidirect
34. C₂×C₈×C₂₉	C₂×C₄	semidirect
41. C₂×#6597	C₂×C₄	semidirect
63. C₂×Dic₂₉	M₁₆	semidirect
97. C₂×M₁₆	Dic₂₉	semidirect
104. C₂₉	#1728	semidirect
124. C₂×C₈	C₂×Dic₂₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#90777	2
C₂²×C₄×Dic₂₉	2
#14270917	2
C₂²×M₁₆×C₂₉	2
#1728	29