Subgroups — #16827910

Order 3712.

564 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	C₂²×#90777	C₂	1	1	yes	maximal, characteristic
2	C₂×#11745288	C₂	1	1	yes	maximal, characteristic
3	#14271121	C₂	1	4	yes	maximal, complemented
4	#14271118	C₂	1	4	yes	maximal, complemented
5	M₁₆×Dic₂₉	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
8	C₂×M₁₆×C₂₉	C₄	1	8	yes	complemented
9	C₂×M₁₆×C₂₉	C₂²	1	2	yes	—
10	C₂×M₁₆×C₂₉	C₂²	1	2	yes	—
11	C₂²×C₈×C₂₉	C₂²	1	2	yes	—
12	C₂×C₂²⋊Dic₂₉	C₄	1	2	yes	—
13	#11745288	C₄	1	2	yes	—
14	#11745288	C₄	1	2	yes	—
15	#11745288	C₂²	1	2	yes	—
16	#11745288	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₂₉	C₄	1	2	yes	—
18	C₂×C₄×Dic₂₉	C₂²	1	2	yes	—
19	C₂×#90777	C₄	1	8	yes	complemented
20	C₂×#90777	C₂²	1	2	yes	—
21	C₂×#90777	C₂²	1	2	yes	—
22	C₂²×#6597	C₂²	1	2	yes	—
23	#11745163	C₂²	1	8	yes	complemented
24	C₈×Dic₂₉	C₂²	1	8	yes	complemented
25	C₂⁴×C₂₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₉	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₂₉	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
34	C₂²×Dic₂₉	C₂×C₄	1	4	yes	—
35	M₁₆×C₂₉	C₂×C₄	1	8	yes	—
36	M₁₆×C₂₉	C₂×C₄	1	8	yes	complemented
37	C₂×C₈×C₂₉	C₂×C₄	1	8	yes	complemented
38	C₂×C₈×C₂₉	C₂³	1	4	yes	—
39	C₂²⋊Dic₂₉	C₂×C₄	1	4	yes	—
40	C₂²⋊Dic₂₉	C₂×C₄	1	4	yes	—
41	C₄⋊Dic₂₉	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₂₉	C₂×C₄	1	4	yes	—
43	C₄×Dic₂₉	C₂×C₄	1	4	yes	—
44	C₄×Dic₂₉	C₂³	1	4	yes	—
45	#90777	C₂×C₄	1	8	yes	—
46	#90777	C₂×C₄	1	8	yes	complemented
47	C₂×#6597	C₂×C₄	1	8	yes	complemented
48	C₂×#6597	C₂³	1	4	yes	—
49	C₂³×C₂₉	—	4	2	no	—
50	C₂³×C₂₉	C₄²	1	4	yes	—
51	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
52	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
53	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₂₉	C₄²	1	4	yes	—
55	C₂×C₄×C₂₉	C₄²	1	4	yes	—
56	C₂×C₄×C₂₉	C₄²	1	4	yes	—
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	2	yes	—
61	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
67	C₂×Dic₂₉	—	4	4	no	—
68	C₂×Dic₂₉	C₂²×C₄	1	8	yes	—
69	C₈×C₂₉	C₂²×C₄	1	8	yes	—
70	#6597	C₂²×C₄	1	8	yes	—
71	#2873	—	29	1	no	maximal, Sylow
72	C₂²×C₂₉	—	2	2	no	—
73	C₂²×C₂₉	—	2	2	no	—
74	C₂²×C₂₉	—	2	2	no	—
75	C₂²×C₂₉	—	4	4	no	—
76	C₂²×C₂₉	C₂×C₄²	1	2	yes	—
77	C₂²×C₂₉	C₂×C₄²	1	2	yes	—
78	C₂²×C₂₉	C₂×C₄²	1	2	yes	—
79	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
80	C₄×C₂₉	C₂×C₄²	1	2	yes	—
81	C₄×C₂₉	C₂×C₄²	1	2	yes	—
82	C₄×C₂₉	C₂×C₄²	1	2	yes	—
83	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
84	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
85	Dic₂₉	—	2	8	no	—
86	C₂²×M₁₆	—	29	1	no	—
87	C₂²×M₁₆	D₂₉	1	1	yes	characteristic
88	C₂×#117	—	29	1	no	—
89	#405	—	29	4	no	—
90	#405	—	29	4	no	—
91	C₄×M₁₆	—	29	4	no	—
92	C₂×C₂₉	—	2	2	no	—
93	C₂×C₂₉	—	2	2	no	—
94	C₂×C₂₉	—	2	2	no	—
95	C₂×C₂₉	C₂²×C₄²	1	1	yes	characteristic
96	C₂×C₂₉	#568	1	2	yes	—
97	C₂³×C₄	C₂×D₂₉	1	1	yes	characteristic
98	C₂×M₁₆	—	29	8	no	—
99	C₂×M₁₆	—	29	2	no	—
100	C₂×M₁₆	—	29	2	no	—
101	C₂×M₁₆	Dic₂₉	1	8	yes	complemented
102	C₂×M₁₆	C₂×D₂₉	1	2	yes	—
103	C₂×M₁₆	C₂×D₂₉	1	2	yes	—
104	C₂²×C₈	—	29	2	no	—
105	C₂²×C₈	C₂×D₂₉	1	2	yes	—
106	#117	—	29	2	no	—
107	#117	—	29	2	no	—
108	#117	—	29	2	no	—
109	#117	—	29	2	no	—
110	C₂×C₄⋊C₄	—	29	2	no	—
111	C₂×C₂²⋊C₄	—	29	2	no	—
112	C₂×C₄²	—	29	2	no	—
113	#97	—	29	8	no	—
114	C₄×C₈	—	29	8	no	—
115	C₂₉	#2873	1	1	yes	characteristic, complemented, Sylow
116	C₂⁴	C₄×D₂₉	1	1	yes	characteristic
117	C₂²×C₄	—	29	4	no	—
118	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
119	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
120	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
121	C₂²×C₄	C₄×D₂₉	1	4	yes	—
122	C₂²×C₄	C₂×Dic₂₉	1	4	yes	—
123	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
126	M₁₆	—	29	8	no	—
127	M₁₆	—	29	8	no	—
128	M₁₆	C₂×Dic₂₉	1	8	yes	—
129	M₁₆	C₂×Dic₂₉	1	8	yes	complemented
130	C₂×C₈	—	29	8	no	—
131	C₂×C₈	—	29	4	no	—
132	C₂×C₈	C₂×Dic₂₉	1	8	yes	complemented
133	C₂×C₈	C₂²×D₂₉	1	4	yes	—
134	C₄⋊C₄	—	29	4	no	—
135	C₄⋊C₄	—	29	4	no	—
136	C₂²⋊C₄	—	29	4	no	—
137	C₂²⋊C₄	—	29	4	no	—
138	C₄²	—	29	4	no	—
139	C₄²	—	29	4	no	—
140	C₂³	—	4	2	no	—
141	C₂³	C₄×Dic₂₉	1	4	yes	—
142	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
143	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
144	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
145	C₂×C₄	—	29	8	no	—
146	C₂×C₄	—	116	4	no	—
147	C₂×C₄	C₄×Dic₂₉	1	4	yes	—
148	C₂×C₄	C₄×Dic₂₉	1	4	yes	—
149	C₂×C₄	C₄×Dic₂₉	1	4	yes	—
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	C₂×C₄×D₂₉	1	2	yes	—
154	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
155	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
156	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
157	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
158	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
159	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
160	C₈	—	29	8	no	—
161	C₈	C₂²×Dic₂₉	1	8	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	4	4	no	—
166	C₂²	C₂×C₄×Dic₂₉	1	2	yes	—
167	C₂²	C₂×C₄×Dic₂₉	1	2	yes	—
168	C₂²	C₂×C₄×Dic₂₉	1	2	yes	—
169	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
170	C₄	—	58	8	no	—
171	C₄	C₂×C₄×Dic₂₉	1	2	yes	—
172	C₄	C₂×C₄×Dic₂₉	1	2	yes	—
173	C₄	C₂×C₄×Dic₂₉	1	2	yes	—
174	C₄	C₂²×C₄×D₂₉	1	1	yes	characteristic
175	C₄	C₂³×Dic₂₉	1	1	yes	characteristic
176	C₂	—	2	2	no	—
177	C₂	—	2	2	no	—
178	C₂	—	2	2	no	—
179	C₂	C₂²×C₄×Dic₂₉	1	1	yes	characteristic
180	C₂	#14271929	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator	C₂×C₂₉	C₂²×C₄²
Fitting	C₂²×M₁₆×C₂₉	C₂
socle	C₂²×C₂₉	C₂³×C₄
prime radical	C₂⁴×C₂₉	C₂×C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₂²×M₁₆	D₂₉
hypocenter	C₂₉	#2873
LCS	G ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14271121	C₂	semidirect
4. #14271118	C₂	semidirect
5. M₁₆×Dic₂₉	C₂	semidirect
8. C₂×M₁₆×C₂₉	C₄	semidirect
19. C₂×#90777	C₄	semidirect
23. #11745163	C₂²	semidirect
24. C₈×Dic₂₉	C₂²	semidirect
36. M₁₆×C₂₉	C₂×C₄	semidirect
37. C₂×C₈×C₂₉	C₂×C₄	semidirect
46. #90777	C₂×C₄	semidirect
47. C₂×#6597	C₂×C₄	semidirect
101. C₂×M₁₆	Dic₂₉	semidirect
115. C₂₉	#2873	semidirect
129. M₁₆	C₂×Dic₂₉	semidirect
132. 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
C₂×#11745288	2
#14271121	2
#14271118	2
M₁₆×Dic₂₉	2
C₂²×M₁₆×C₂₉	2
#2873	29