Subgroups — #16836409

Order 3712.

432 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14272212	C₂	1	2	yes	maximal, complemented
2	#14272191	C₂	1	2	yes	maximal, complemented
3	#14272190	C₂	1	4	yes	maximal, complemented
4	#14272182	C₂	1	2	yes	maximal, complemented
5	#14272179	C₂	1	1	yes	maximal, characteristic, complemented
6	D₂₉⋊(C₄⋊C₈)	C₂	1	2	yes	maximal, complemented
7	C₂×C₄⋊#6599	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271631	C₂	1	1	yes	maximal, characteristic
9	C₂×#90805	C₄	1	1	yes	characteristic
10	C₂²×#6599	C₂²	1	2	yes	—
11	C₂×D₂₉⋊C₈	C₂²	1	4	yes	—
12	C₂₉×#117	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
14	#11745255	C₄	1	1	yes	characteristic
15	#11745255	C₂²	1	1	yes	characteristic
16	#11745249	C₄	1	1	yes	characteristic
17	#11745249	C₂²	1	1	yes	characteristic
18	#11745243	C₄	1	2	yes	—
19	#11745243	C₂²	1	2	yes	—
20	#11745234	C₄	1	2	yes	—
21	C₄²×D₂₉	C₂²	1	2	yes	—
22	C₂²⋊#6599	C₂²	1	2	yes	—
23	#11745220	C₂²	1	4	yes	—
24	#11745219	C₂²	1	4	yes	—
25	C₄⋊#6599	C₂²	1	2	yes	—
26	C₄⋊#6599	C₂²	1	2	yes	—
27	C₄⋊#6599	C₂²	1	1	yes	characteristic
28	C₄⋊#6599	C₂²	1	1	yes	characteristic
29	C₄×#6599	C₂²	1	2	yes	—
30	C₄×#6599	C₂²	1	4	yes	—
31	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₂²⋊D₂₉	C₂×C₄	1	2	yes	—
33	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
35	#90805	C₈	1	8	yes	complemented
36	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₂₉	C₂³	1	2	yes	—
40	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
41	C₂×#6599	—	4	2	no	—
42	C₂×#6599	C₂³	1	2	yes	—
43	C₂×#6599	C₂³	1	4	yes	—
44	C₂×#6599	C₂³	1	2	yes	—
45	D₂₉⋊C₈	—	4	4	no	—
46	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
47	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
48	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
49	C₄²×C₂₉	C₂×C₄	1	2	yes	—
50	#90781	C₂×C₄	1	2	yes	—
51	#90781	C₂×C₄	1	2	yes	—
52	#90779	C₂×C₄	1	2	yes	—
53	#90779	C₂×C₄	1	2	yes	—
54	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
57	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
58	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
59	C₄×Dic₂₉	C₂³	1	2	yes	—
60	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₂₉	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₂₉	C₂×C₈	1	4	yes	complemented
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	1	yes	characteristic
66	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²⋊D₂₉	C₂×C₈	1	8	yes	complemented
68	C₂×Dic₂₉	—	2	1	no	—
69	C₂×Dic₂₉	—	2	1	no	—
70	C₂×Dic₂₉	—	4	1	no	—
71	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
75	C₂×Dic₂₉	C₂⁴	1	1	yes	characteristic
76	D₁₁₆	C₂×C₈	1	4	yes	complemented
77	C₄×D₂₉	—	2	2	no	—
78	C₄×D₂₉	—	2	2	no	—
79	C₄×D₂₉	—	4	2	no	—
80	C₄×D₂₉	C₂×C₈	1	8	yes	complemented
81	Dic₅₈	C₂×C₈	1	4	yes	complemented
82	#6599	—	2	2	no	—
83	#6599	—	2	4	no	—
84	#6599	—	2	2	no	—
85	#2988	—	29	1	no	maximal, Sylow
86	C₂²×C₂₉	—	2	1	no	—
87	C₂²×C₂₉	—	2	1	no	—
88	C₂²×C₂₉	C₂²×C₈	1	2	yes	—
89	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
90	C₂×D₂₉	—	2	2	no	—
91	C₂×D₂₉	—	2	2	no	—
92	C₂×D₂₉	C₂²×C₈	1	4	yes	—
93	C₄×C₂₉	—	2	2	no	—
94	C₄×C₂₉	—	2	2	no	—
95	C₄×C₂₉	C₂²×C₈	1	2	yes	—
96	C₄×C₂₉	C₂²×C₈	1	2	yes	—
97	Dic₂₉	—	2	1	no	—
98	Dic₂₉	—	2	1	no	—
99	Dic₂₉	—	2	1	no	—
100	Dic₂₉	C₂²×C₈	1	4	yes	—
101	Dic₂₉	H(5,𝔽₂)	1	1	yes	characteristic
102	Dic₂₉	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×#41	—	29	1	no	—
104	C₈×Q₈	—	29	2	no	—
105	C₈×D₄	—	29	4	no	—
106	C₈×D₄	—	29	2	no	—
107	#431	—	29	1	no	—
108	#431	—	29	2	no	—
109	C₂×C₄⋊C₈	—	29	1	no	—
110	C₂×C₄⋊C₈	—	29	2	no	—
111	C₂×C₂₉	—	2	2	no	—
112	C₂×C₂₉	#520	1	1	yes	characteristic
113	C₂×C₂₉	C₂³×C₈	1	1	yes	characteristic
114	C₂×C₂₉	#568	1	1	yes	characteristic
115	D₂₉	—	2	4	no	—
116	C₂×#41	—	29	1	no	—
117	C₂²×C₈	—	29	4	no	—
118	C₂²×C₈	—	29	2	no	—
119	C₄×Q₈	—	29	1	no	—
120	C₄×Q₈	—	29	1	no	—
121	C₄×D₄	—	29	1	no	—
122	C₄×D₄	—	29	1	no	—
123	C₄×D₄	—	29	2	no	—
124	C₄×D₄	—	29	2	no	—
125	#117	—	29	2	no	—
126	#117	#1181	1	1	yes	characteristic
127	C₂×C₄²	—	29	1	no	—
128	C₂×C₄²	—	29	2	no	—
129	C₄⋊C₈	—	29	2	no	—
130	C₄⋊C₈	—	29	1	no	—
131	C₄⋊C₈	—	29	2	no	—
132	C₄⋊C₈	—	29	4	no	—
133	C₄⋊C₈	—	29	1	no	—
134	C₂²⋊C₈	—	29	4	no	—
135	C₂²⋊C₈	—	29	2	no	—
136	C₄×C₈	—	29	2	no	—
137	C₄×C₈	—	29	4	no	—
138	C₂₉	#2988	1	1	yes	characteristic, complemented, Sylow
139	#41	—	29	8	no	—
140	C₂×Q₈	—	29	1	no	—
141	C₂×D₄	—	29	2	no	—
142	C₂×D₄	—	29	1	no	—
143	C₂²×C₄	—	29	2	no	—
144	C₂²×C₄	—	29	1	no	—
145	C₂²×C₄	—	29	1	no	—
146	C₂²×C₄	—	29	2	no	—
147	C₂²×C₄	—	29	2	no	—
148	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
149	C₂×C₈	—	29	2	no	—
150	C₂×C₈	—	29	4	no	—
151	C₂×C₈	—	29	2	no	—
152	C₂×C₈	—	116	4	no	—
153	C₂×C₈	—	116	2	no	—
154	C₄⋊C₄	—	29	2	no	—
155	C₄⋊C₄	—	29	2	no	—
156	C₄⋊C₄	C₂×#1181	1	1	yes	characteristic
157	C₄⋊C₄	C₂×#1181	1	1	yes	characteristic
158	C₂²⋊C₄	—	29	2	no	—
159	C₂²⋊C₄	—	29	2	no	—
160	C₂²⋊C₄	C₂×#1181	1	2	yes	—
161	C₄²	—	29	2	no	—
162	C₄²	—	29	1	no	—
163	C₄²	—	29	1	no	—
164	C₄²	—	29	1	no	—
165	C₄²	—	29	2	no	—
166	C₄²	—	29	1	no	—
167	C₄²	C₂×#1181	1	2	yes	—
168	C₂³	—	29	2	no	—
169	C₂³	C₂²×#1181	1	1	yes	characteristic
170	Q₈	—	29	4	no	—
171	D₄	—	29	8	no	—
172	D₄	—	29	4	no	—
173	C₂×C₄	—	29	1	no	—
174	C₂×C₄	—	29	2	no	—
175	C₂×C₄	—	29	8	no	—
176	C₂×C₄	—	29	1	no	—
177	C₂×C₄	—	29	1	no	—
178	C₂×C₄	—	29	1	no	—
179	C₂×C₄	—	58	2	no	—
180	C₂×C₄	—	58	2	no	—
181	C₂×C₄	—	58	1	no	—
182	C₂×C₄	—	58	1	no	—
183	C₂×C₄	—	116	2	no	—
184	C₂×C₄	—	116	1	no	—
185	C₂×C₄	D₂₉⋊C₈	1	4	yes	complemented
186	C₂×C₄	C₂²×#1181	1	2	yes	—
187	C₂×C₄	C₂²×#1181	1	2	yes	—
188	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
189	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
190	C₈	—	58	2	no	—
191	C₈	—	58	4	no	—
192	C₈	—	58	2	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	29	4	no	—
196	C₂²	—	58	2	no	—
197	C₂²	—	58	2	no	—
198	C₂²	C₂×D₂₉⋊C₈	1	2	yes	—
199	C₂²	C₂³×#1181	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	29	1	no	—
203	C₄	—	29	1	no	—
204	C₄	—	29	4	no	—
205	C₄	—	58	1	no	—
206	C₄	—	58	1	no	—
207	C₄	—	58	1	no	—
208	C₄	C₂×D₂₉⋊C₈	1	2	yes	—
209	C₄	C₂×D₂₉⋊C₈	1	2	yes	—
210	C₂	—	2	2	no	—
211	C₂	—	58	4	no	—
212	C₂	#14272232	1	1	yes	characteristic
213	C₂	C₂²×D₂₉⋊C₈	1	1	yes	characteristic
214	C₂	#14272242	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#1181
commutator	C₂×C₂₉	C₂³×C₈
Fitting	C₂₉×#117	C₄
socle	C₂²×C₂₉	C₂³×C₄
prime radical	C₂×#90805	C₄
cosocle	C₂×Dic₂₉	C₂⁴
hypercenter	#117	#1181
hypocenter	C₂₉	#2988
LCS	G ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #14272212	C₂	semidirect
2. #14272191	C₂	semidirect
3. #14272190	C₂	semidirect
4. #14272182	C₂	semidirect
5. #14272179	C₂	semidirect
6. D₂₉⋊(C₄⋊C₈)	C₂	semidirect
7. C₂×C₄⋊#6599	C₂	semidirect
35. #90805	C₈	semidirect
62. C₂×C₄×C₂₉	C₂×C₈	semidirect
67. C₂²⋊D₂₉	C₂×C₈	semidirect
76. D₁₁₆	C₂×C₈	semidirect
80. C₄×D₂₉	C₂×C₈	semidirect
81. Dic₅₈	C₂×C₈	semidirect
138. C₂₉	#2988	semidirect
185. C₂×C₄	D₂₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#14272212	2
#14272191	2
#14272190	2
#14272182	2
#14272179	2
D₂₉⋊(C₄⋊C₈)	2
C₂×C₄⋊#6599	2
#14271631	2
#2988	29