Subgroups — #16836306

Order 3712.

380 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14272214	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#6599	C₂	1	1	yes	maximal, characteristic, complemented
3	#14272198	C₂	1	1	yes	maximal, characteristic, complemented
4	#14272195	C₂	1	2	yes	maximal, complemented
5	#14272190	C₂	1	2	yes	maximal, complemented
6	D₄×#6599	C₂	1	1	yes	maximal, characteristic, complemented
7	#14272187	C₂	1	2	yes	maximal, complemented
8	#14272185	C₂	1	1	yes	maximal, characteristic, complemented
9	#14272127	C₂	1	2	yes	maximal, complemented
10	C₄×D₂₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	#14271692	C₂	1	1	yes	maximal, characteristic
12	C₂×#90800	C₂²	1	2	yes	—
13	C₂²×#6599	C₂²	1	2	yes	—
14	C₂×D₂₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×D₂₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂₉×#124	C₄	1	1	yes	characteristic
17	#11745308	C₄	1	1	yes	characteristic
18	Q₈×Dic₂₉	C₂²	1	1	yes	characteristic
19	#11745300	C₄	1	1	yes	characteristic
20	D₄×Dic₂₉	C₂²	1	1	yes	characteristic
21	#11745245	C₄	1	2	yes	—
22	#11745243	C₂²	1	2	yes	—
23	#11745241	C₄	1	2	yes	—
24	#11745239	C₂²	1	2	yes	—
25	C₄²×D₂₉	C₂²	1	1	yes	characteristic
26	#11745232	C₄	1	1	yes	characteristic
27	C₂²⋊#6599	C₂²	1	2	yes	—
28	C₂²⋊#6599	C₂²	1	2	yes	—
29	#11745220	C₂²	1	2	yes	—
30	#11745219	C₂²	1	2	yes	—
31	#11745218	C₂²	1	2	yes	—
32	#11745218	C₂²	1	2	yes	—
33	C₄⋊#6599	C₂²	1	1	yes	characteristic
34	C₄⋊#6599	C₂²	1	1	yes	characteristic
35	C₄⋊#6599	C₂²	1	2	yes	—
36	C₄×#6599	C₂²	1	1	yes	characteristic
37	C₄×#6599	C₂²	1	1	yes	characteristic
38	C₄×#6599	C₂²	1	1	yes	characteristic
39	C₄×#6599	C₂²	1	1	yes	characteristic
40	C₄×#6599	C₂²	1	2	yes	—
41	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₂₉	C₂×C₄	1	2	yes	—
44	C₂²×Dic₂₉	C₂³	1	2	yes	—
45	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
48	#90800	—	2	4	no	—
49	C₂×#6599	—	2	4	no	—
50	C₂×#6599	C₂³	1	1	yes	characteristic
51	C₂×#6599	C₂³	1	1	yes	characteristic
52	C₂×#6599	C₂³	1	2	yes	—
53	C₂×#6599	C₂³	1	2	yes	—
54	C₂×#6599	C₂³	1	1	yes	characteristic
55	C₂×#6599	C₂³	1	1	yes	characteristic
56	D₂₉⋊C₈	—	2	2	no	—
57	D₂₉⋊C₈	—	2	2	no	—
58	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
59	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
60	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
61	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
62	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
63	#90781	C₂×C₄	1	2	yes	—
64	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
65	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
66	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
67	#90779	C₂×C₄	1	2	yes	—
68	#90779	C₂×C₄	1	2	yes	—
69	#90779	C₂×C₄	1	2	yes	—
70	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
72	C₄×Dic₂₉	C₂³	1	2	yes	—
73	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
74	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
75	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₂₉	—	2	2	no	—
77	D₄×C₂₉	—	2	2	no	—
78	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
82	C₂²⋊D₂₉	—	2	4	no	—
83	C₂×Dic₂₉	—	2	2	no	—
84	C₂×Dic₂₉	—	2	2	no	—
85	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
88	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
89	C₂×Dic₂₉	C₂⁴	1	1	yes	characteristic
90	C₄×D₂₉	—	2	2	no	—
91	C₄×D₂₉	—	2	2	no	—
92	C₄×D₂₉	—	2	1	no	—
93	C₄×D₂₉	—	2	1	no	—
94	#6599	—	2	1	no	—
95	#6599	—	2	1	no	—
96	#6599	—	2	2	no	—
97	#6599	—	2	2	no	—
98	#6599	#41	1	2	yes	—
99	#6599	#41	1	2	yes	—
100	#2966	—	29	1	no	maximal, Sylow
101	C₂²×C₂₉	—	2	2	no	—
102	C₂²×C₂₉	—	2	2	no	—
103	C₂²×C₂₉	—	2	2	no	—
104	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
105	C₂×D₂₉	—	2	1	no	—
106	C₂×D₂₉	—	2	1	no	—
107	C₂×D₂₉	#131	1	2	yes	—
108	C₄×C₂₉	—	2	1	no	—
109	C₄×C₂₉	—	2	1	no	—
110	C₄×C₂₉	—	2	2	no	—
111	C₄×C₂₉	#131	1	2	yes	—
112	Dic₂₉	—	2	2	no	—
113	Dic₂₉	—	2	1	no	—
114	Dic₂₉	—	2	1	no	—
115	Dic₂₉	—	2	2	no	—
116	Dic₂₉	C₂×#41	1	1	yes	characteristic
117	Dic₂₉	C₂×#41	1	1	yes	characteristic
118	#529	—	29	1	no	—
119	#446	—	29	1	no	—
120	C₈×Q₈	—	29	1	no	—
121	#436	—	29	1	no	—
122	#435	—	29	2	no	—
123	C₈×D₄	—	29	1	no	—
124	C₈×D₄	—	29	2	no	—
125	#433	—	29	2	no	—
126	#432	—	29	1	no	—
127	#405	—	29	2	no	—
128	C₂×C₄×C₈	—	29	1	no	—
129	C₂×C₂₉	—	4	2	no	—
130	C₂×C₂₉	C₄×#41	1	1	yes	characteristic
131	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
132	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
133	D₂₉	—	2	2	no	—
134	C₂×M₁₆	—	29	2	no	—
135	C₂²×C₈	—	29	1	no	—
136	C₂²×C₈	—	29	1	no	—
137	C₂²×C₈	—	29	2	no	—
138	#126	—	29	2	no	—
139	#125	—	29	1	no	—
140	#124	#1181	1	1	yes	characteristic
141	#123	—	29	2	no	—
142	C₂²⋊Q₈	—	29	1	no	—
143	#121	—	29	1	no	—
144	C₄×Q₈	—	29	1	no	—
145	C₄×D₄	—	29	1	no	—
146	C₄×D₄	—	29	2	no	—
147	#117	—	29	2	no	—
148	C₂×C₄²	—	29	1	no	—
149	C₄⋊C₈	—	29	1	no	—
150	C₄⋊C₈	—	29	1	no	—
151	C₄⋊C₈	—	29	2	no	—
152	C₄⋊C₈	—	29	2	no	—
153	C₂²⋊C₈	—	29	2	no	—
154	C₂²⋊C₈	—	29	2	no	—
155	C₂²⋊C₈	—	29	2	no	—
156	#97	—	29	2	no	—
157	#97	—	29	2	no	—
158	C₄×C₈	—	29	1	no	—
159	C₄×C₈	—	29	1	no	—
160	C₄×C₈	—	29	1	no	—
161	C₄×C₈	—	29	1	no	—
162	C₄×C₈	—	29	2	no	—
163	C₂₉	#2966	1	1	yes	characteristic, complemented, Sylow
164	C₂×Q₈	C₂×#1181	1	1	yes	characteristic
165	C₂×D₄	—	29	2	no	—
166	C₂×D₄	C₂×#1181	1	1	yes	characteristic
167	C₂²×C₄	—	29	2	no	—
168	C₂²×C₄	—	29	1	no	—
169	C₂²×C₄	—	29	1	no	—
170	C₂²×C₄	—	29	1	no	—
171	M₁₆	—	58	4	no	—
172	C₂×C₈	—	29	1	no	—
173	C₂×C₈	—	29	1	no	—
174	C₂×C₈	—	29	1	no	—
175	C₂×C₈	—	29	1	no	—
176	C₂×C₈	—	29	2	no	—
177	C₂×C₈	—	29	2	no	—
178	C₂×C₈	—	58	2	no	—
179	C₂×C₈	—	58	2	no	—
180	C₂×C₈	—	58	4	no	—
181	C₄⋊C₄	—	29	2	no	—
182	C₄⋊C₄	—	29	2	no	—
183	C₄⋊C₄	—	29	1	no	—
184	C₄⋊C₄	—	29	2	no	—
185	C₄⋊C₄	—	29	1	no	—
186	C₄⋊C₄	—	29	2	no	—
187	C₂²⋊C₄	—	29	2	no	—
188	C₂²⋊C₄	—	29	2	no	—
189	C₂²⋊C₄	—	29	2	no	—
190	C₂²⋊C₄	C₂×#1181	1	2	yes	—
191	C₂²⋊C₄	C₂×#1181	1	2	yes	—
192	C₄²	—	29	1	no	—
193	C₄²	—	29	1	no	—
194	C₄²	—	29	2	no	—
195	C₄²	—	29	1	no	—
196	C₄²	C₂×#1181	1	1	yes	characteristic
197	C₂³	—	29	1	no	—
198	C₂³	C₂²×#1181	1	2	yes	—
199	Q₈	—	2	2	no	—
200	D₄	—	2	2	no	—
201	D₄	—	58	4	no	—
202	C₂×C₄	—	29	1	no	—
203	C₂×C₄	—	29	1	no	—
204	C₂×C₄	—	29	1	no	—
205	C₂×C₄	—	29	2	no	—
206	C₂×C₄	—	29	2	no	—
207	C₂×C₄	—	58	1	no	—
208	C₂×C₄	—	58	1	no	—
209	C₂×C₄	—	58	2	no	—
210	C₂×C₄	—	58	2	no	—
211	C₂×C₄	—	58	2	no	—
212	C₂×C₄	—	58	2	no	—
213	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
214	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
215	C₂×C₄	C₂²×#1181	1	2	yes	—
216	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
217	C₈	—	29	2	no	—
218	C₈	—	29	2	no	—
219	C₈	—	58	1	no	—
220	C₈	—	58	1	no	—
221	C₈	—	58	2	no	—
222	C₈	—	58	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	29	2	no	—
227	C₂²	—	58	1	no	—
228	C₂²	—	58	1	no	—
229	C₂²	C₂³×#1181	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	29	1	no	—
234	C₄	—	29	1	no	—
235	C₄	—	58	2	no	—
236	C₄	—	58	1	no	—
237	C₄	—	58	1	no	—
238	C₄	—	58	2	no	—
239	C₄	#11745346	1	2	yes	—
240	C₂	—	4	2	no	—
241	C₂	—	58	2	no	—
242	C₂	#41×#1181	1	1	yes	characteristic
243	C₂	C₂×#11745346	1	1	yes	characteristic
244	C₂	D₂₉⋊#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#1181
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#124	C₄
prime radical	#11745300	C₄
cosocle	C₂×Dic₂₉	C₂⁴
hypercenter	#124	#1181
hypocenter	C₂₉	#2966
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
1. #14272214	C₂	semidirect
2. Q₈×#6599	C₂	semidirect
3. #14272198	C₂	semidirect
4. #14272195	C₂	semidirect
5. #14272190	C₂	semidirect
6. D₄×#6599	C₂	semidirect
7. #14272187	C₂	semidirect
8. #14272185	C₂	semidirect
9. #14272127	C₂	semidirect
10. C₄×D₂₉⋊C₈	C₂	semidirect
163. C₂₉	#2966	semidirect

Maximal subgroups

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

subgroup	index
#14272214	2
Q₈×#6599	2
#14272198	2
#14272195	2
#14272190	2
D₄×#6599	2
#14272187	2
#14272185	2
#14272127	2
C₄×D₂₉⋊C₈	2
#14271692	2
#2966	29