Subgroups — #16827875

Order 3712.

408 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	#14271069	C₂	1	1	yes	maximal, characteristic
2	#14271068	C₂	1	1	yes	maximal, characteristic
3	#14271068	C₂	1	1	yes	maximal, characteristic
4	#14271062	C₂	1	1	yes	maximal, characteristic
5	#14271059	C₂	1	1	yes	maximal, characteristic
6	#14271049	C₂	1	2	yes	maximal
7	#14271046	C₂	1	2	yes	maximal
8	#14271044	C₂	1	2	yes	maximal
9	#14271030	C₂	1	2	yes	maximal
10	C₄²×Dic₂₉	C₂	1	1	yes	maximal, characteristic
11	C₂₉×#389	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₅₈	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₂₉	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
19	C₂×#90779	C₂²	1	2	yes	—
20	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
26	C₂₉×#95	C₂²	1	2	yes	—
27	#11745179	C₂²	1	2	yes	—
28	#11745179	C₂²	1	2	yes	—
29	#11745179	C₂²	1	2	yes	—
30	#11745179	C₂²	1	2	yes	—
31	#11745179	C₂²	1	2	yes	—
32	#11745179	C₂²	1	2	yes	—
33	#11745179	C₂²	1	2	yes	—
34	#11745179	C₂²	1	2	yes	—
35	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
36	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
40	C₂²×Dic₂₉	C₂³	1	2	yes	—
41	C₂²×Dic₂₉	C₂³	1	2	yes	—
42	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
43	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
44	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
45	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
46	C₂×Dic₅₈	—	2	2	no	—
47	C₂×Dic₅₈	—	2	2	no	—
48	C₂×Dic₅₈	—	2	2	no	—
49	C₄⋊C₄×C₂₉	—	2	2	no	—
50	C₄⋊C₄×C₂₉	—	2	4	no	—
51	C₄⋊C₄×C₂₉	—	2	2	no	—
52	C₄²×C₂₉	—	2	2	no	—
53	C₄⋊Dic₂₉	—	2	2	no	—
54	C₄⋊Dic₂₉	—	2	4	no	—
55	#90779	—	2	4	no	—
56	C₄×Dic₂₉	—	2	2	no	—
57	C₄×Dic₂₉	—	2	2	no	—
58	C₄×Dic₂₉	—	2	2	no	—
59	C₄×Dic₂₉	—	2	2	no	—
60	C₄×Dic₂₉	—	2	2	no	—
61	C₄×Dic₂₉	Q₈	1	4	yes	complemented
62	C₂³×C₂₉	C₂⁴	1	1	yes	characteristic
63	C₂×C₄×C₂₉	—	2	1	no	—
64	C₂×C₄×C₂₉	—	2	1	no	—
65	C₂×C₄×C₂₉	—	2	1	no	—
66	C₂×C₄×C₂₉	—	2	1	no	—
67	C₂×C₄×C₂₉	—	2	2	no	—
68	C₂×C₄×C₂₉	—	2	2	no	—
69	C₂×C₄×C₂₉	—	2	2	no	—
70	C₂×C₄×C₂₉	—	2	2	no	—
71	C₂×C₄×C₂₉	—	2	2	no	—
72	C₂×C₄×C₂₉	—	2	2	no	—
73	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
74	C₂×C₄×C₂₉	#41	1	2	yes	—
75	C₂×C₄×C₂₉	#41	1	2	yes	—
76	C₂×C₄×C₂₉	#41	1	2	yes	—
77	C₂×C₄×C₂₉	#41	1	2	yes	—
78	C₂×Dic₂₉	—	2	1	no	—
79	C₂×Dic₂₉	—	2	1	no	—
80	C₂×Dic₂₉	—	2	1	no	—
81	C₂×Dic₂₉	—	2	1	no	—
82	C₂×Dic₂₉	—	2	1	no	—
83	C₂×Dic₂₉	—	2	1	no	—
84	C₂×Dic₂₉	—	2	1	no	—
85	C₂×Dic₂₉	—	2	1	no	—
86	C₂×Dic₂₉	—	2	2	no	—
87	C₂×Dic₂₉	—	2	2	no	—
88	C₂×Dic₂₉	—	2	2	no	—
89	C₂×Dic₂₉	—	2	2	no	—
90	C₂×Dic₂₉	—	2	2	no	—
91	C₂×Dic₂₉	—	2	2	no	—
92	C₂×Dic₂₉	C₂×Q₈	1	2	yes	—
93	C₂×Dic₂₉	C₂×Q₈	1	2	yes	—
94	C₂×Dic₂₉	#41	1	2	yes	—
95	C₂×Dic₂₉	#41	1	2	yes	—
96	Dic₅₈	—	4	4	no	—
97	#2863	—	29	1	no	maximal, Sylow
98	C₂²×C₂₉	C₂²×Q₈	1	1	yes	characteristic
99	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
100	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
101	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
102	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
103	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
104	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
105	C₄×C₂₉	—	2	2	no	—
106	C₄×C₂₉	—	2	2	no	—
107	C₄×C₂₉	—	4	2	no	—
108	C₄×C₂₉	—	4	2	no	—
109	C₄×C₂₉	#126	1	4	yes	—
110	Dic₂₉	—	2	2	no	—
111	Dic₂₉	—	2	2	no	—
112	Dic₂₉	—	2	2	no	—
113	Dic₂₉	—	2	2	no	—
114	Dic₂₉	—	4	2	no	—
115	Dic₂₉	—	4	2	no	—
116	#391	—	29	1	no	—
117	#391	—	29	1	no	—
118	#391	—	29	1	no	—
119	#389	—	29	1	no	—
120	#389	—	29	1	no	—
121	#389	D₂₉	1	1	yes	characteristic
122	#387	—	29	2	no	—
123	#387	—	29	2	no	—
124	#387	—	29	2	no	—
125	#383	—	29	2	no	—
126	C₄³	—	29	1	no	—
127	C₂×C₂₉	C₂×#126	1	1	yes	characteristic
128	C₂×C₂₉	#529	1	1	yes	characteristic
129	C₂×C₂₉	#529	1	1	yes	characteristic
130	C₂×C₂₉	#529	1	1	yes	characteristic
131	C₂×C₂₉	#533	1	1	yes	characteristic
132	C₂×C₂₉	#533	1	1	yes	characteristic
133	C₂×C₂₉	#533	1	1	yes	characteristic
134	C₂²×Q₈	—	29	1	no	—
135	C₂×C₄⋊C₄	—	29	2	no	—
136	C₂×C₄⋊C₄	—	29	2	no	—
137	C₂×C₄⋊C₄	—	29	1	no	—
138	C₂×C₄⋊C₄	C₂×D₂₉	1	2	yes	—
139	C₂×C₄⋊C₄	C₂×D₂₉	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂×D₂₉	1	1	yes	characteristic
141	C₂×C₄²	—	29	1	no	—
142	C₂×C₄²	—	29	1	no	—
143	C₂×C₄²	—	29	1	no	—
144	C₂×C₄²	—	29	1	no	—
145	C₂×C₄²	—	29	1	no	—
146	C₂×C₄²	—	29	1	no	—
147	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
148	#95	—	29	2	no	—
149	#95	—	29	2	no	—
150	#95	—	29	2	no	—
151	#95	—	29	2	no	—
152	#95	—	29	2	no	—
153	#95	—	29	2	no	—
154	#95	—	29	2	no	—
155	#95	—	29	2	no	—
156	#95	C₂×D₂₉	1	2	yes	—
157	C₂₉	#2863	1	1	yes	characteristic, complemented, Sylow
158	C₂×Q₈	—	58	2	no	—
159	C₂×Q₈	—	58	2	no	—
160	C₂×Q₈	—	58	2	no	—
161	C₂²×C₄	—	29	1	no	—
162	C₂²×C₄	—	29	1	no	—
163	C₂²×C₄	—	29	1	no	—
164	C₂²×C₄	—	29	1	no	—
165	C₂²×C₄	—	29	2	no	—
166	C₂²×C₄	—	29	2	no	—
167	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
168	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
169	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	58	4	no	—
176	C₄⋊C₄	—	58	4	no	—
177	C₄⋊C₄	—	58	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	29	4	no	—
180	C₄²	—	58	2	no	—
181	C₄²	—	58	2	no	—
182	C₄²	—	58	2	no	—
183	C₄²	—	58	2	no	—
184	C₄²	—	58	2	no	—
185	C₂³	C₂³×D₂₉	1	1	yes	characteristic
186	Q₈	—	116	4	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	29	2	no	—
198	C₂×C₄	—	29	2	no	—
199	C₂×C₄	—	29	2	no	—
200	C₂×C₄	—	29	2	no	—
201	C₂×C₄	—	58	1	no	—
202	C₂×C₄	—	58	1	no	—
203	C₂×C₄	—	58	1	no	—
204	C₂×C₄	—	58	1	no	—
205	C₂×C₄	—	58	1	no	—
206	C₂×C₄	—	58	1	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	—	58	2	no	—
214	C₂×C₄	—	58	2	no	—
215	C₂×C₄	#90805	1	2	yes	—
216	C₂×C₄	#90807	1	2	yes	—
217	C₂×C₄	#90807	1	2	yes	—
218	C₂×C₄	Q₈×D₂₉	1	2	yes	—
219	C₂×C₄	#90809	1	2	yes	—
220	C₂²	C₂×#90805	1	1	yes	characteristic
221	C₂²	C₂×#90807	1	1	yes	characteristic
222	C₂²	C₂×#90807	1	1	yes	characteristic
223	C₂²	C₂×Q₈×D₂₉	1	1	yes	characteristic
224	C₂²	C₂×#90809	1	1	yes	characteristic
225	C₂²	#41×D₂₉	1	1	yes	characteristic
226	C₂²	#41×D₂₉	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	2	no	—
231	C₄	—	58	2	no	—
232	C₄	—	58	2	no	—
233	C₄	—	58	2	no	—
234	C₄	—	58	2	no	—
235	C₄	—	116	2	no	—
236	C₄	—	116	2	no	—
237	C₄	#11745260	1	4	yes	—
238	C₂	C₂×#11745260	1	1	yes	characteristic
239	C₂	#14271694	1	1	yes	characteristic
240	C₂	#14271695	1	1	yes	characteristic
241	C₂	#14271698	1	1	yes	characteristic
242	C₂	#14271715	1	1	yes	characteristic
243	C₂	#14271717	1	1	yes	characteristic
244	C₂	#14271718	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₉
commutator, socle, prime radical, cosocle	C₂³×C₂₉	C₂⁴
Fitting	C₂₉×#389	C₂
hypercenter	#389	D₂₉
hypocenter	C₂₉	#2863
LCS	G ▷ C₂³×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
61. C₄×Dic₂₉	Q₈	semidirect
157. C₂₉	#2863	semidirect

Maximal subgroups

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

subgroup	index
#14271069	2
#14271068	2
#14271062	2
#14271059	2
#14271049	2
#14271046	2
#14271044	2
#14271030	2
C₄²×Dic₂₉	2
C₂₉×#389	2
#2863	29