Subgroups — #16828574

Order 3712.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×D₂₉	C₂	1	1	yes	maximal, characteristic
2	#14271294	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271293	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271292	C₂	1	2	yes	maximal, complemented
5	#14271291	C₂	1	2	yes	maximal, complemented
6	#14271290	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271219	C₂	1	2	yes	maximal
8	#14271218	C₂	1	2	yes	maximal
9	#14271215	C₂	1	1	yes	maximal, characteristic
10	#14271214	C₂	1	1	yes	maximal, characteristic
11	C₂₉×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₂₉	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
14	Q₈×Dic₂₉	C₄	1	1	yes	characteristic
15	C₄⋊C₄×D₂₉	C₄	1	2	yes	—
16	C₄⋊C₄×D₂₉	C₄	1	1	yes	characteristic
17	#11745249	C₄	1	2	yes	—
18	#11745249	C₄	1	1	yes	characteristic
19	C₄²×D₂₉	C₂²	1	1	yes	characteristic
20	C₄²×D₂₉	C₂²	1	2	yes	—
21	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
22	C₄×Dic₅₈	C₂²	1	2	yes	—
23	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
25	C₂₉×#97	C₂²	1	2	yes	—
26	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
27	#11745168	C₂²	1	1	yes	characteristic
28	#11745168	C₂²	1	1	yes	characteristic
29	#11745168	C₂²	1	2	yes	—
30	#11745163	C₂²	1	2	yes	—
31	#11745163	C₂²	1	2	yes	—
32	#11745162	C₂²	1	2	yes	—
33	#11745162	C₂²	1	1	yes	characteristic
34	#11745162	C₂²	1	2	yes	—
35	#11745162	C₂²	1	1	yes	characteristic
36	C₈×Dic₂₉	C₂²	1	2	yes	—
37	C₄⋊#6597	C₂²	1	2	yes	—
38	C₄⋊#6597	C₂²	1	1	yes	characteristic
39	#11745151	C₂²	1	2	yes	—
40	C₄×#6597	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
42	Q₈×D₂₉	—	2	4	no	—
43	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
48	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
49	C₂×Dic₅₈	C₂×C₄	1	2	yes	—
50	C₂×C₈×C₂₉	C₂³	1	2	yes	—
51	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
54	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₂₉	C₂³	1	1	yes	characteristic
56	C₄²×C₂₉	C₂³	1	2	yes	—
57	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
59	#90779	C₂×C₄	1	2	yes	—
60	#90779	C₂×C₄	1	2	yes	—
61	#90779	C₂×C₄	1	2	yes	—
62	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
63	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
65	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
67	C₄×Dic₂₉	C₂³	1	2	yes	—
68	C₂×#6597	C₂³	1	1	yes	characteristic
69	C₂×#6597	C₂³	1	1	yes	characteristic
70	C₂×#6597	C₂³	1	2	yes	—
71	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
72	Q₈×C₂₉	—	2	2	no	—
73	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
78	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
81	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
82	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
83	C₄×D₂₉	—	2	4	no	—
84	C₄×D₂₉	—	2	1	no	—
85	C₄×D₂₉	—	2	1	no	—
86	C₄×D₂₉	—	4	1	no	—
87	C₄×D₂₉	—	4	2	no	—
88	C₄×D₂₉	M₁₆	1	4	yes	—
89	Dic₅₈	—	2	2	no	—
90	Dic₅₈	—	2	4	no	—
91	C₈×C₂₉	—	2	1	no	—
92	C₈×C₂₉	—	2	2	no	—
93	C₈×C₂₉	—	2	1	no	—
94	#6597	—	2	1	no	—
95	#6597	—	2	2	no	—
96	#6597	—	2	1	no	—
97	#2983	—	29	1	no	maximal, Sylow
98	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
99	C₂×D₂₉	—	2	1	no	—
100	C₂×D₂₉	—	2	1	no	—
101	C₂×D₂₉	C₂×M₁₆	1	2	yes	—
102	C₄×C₂₉	—	2	2	no	—
103	C₄×C₂₉	—	2	1	no	—
104	C₄×C₂₉	—	2	2	no	—
105	C₄×C₂₉	C₂×M₁₆	1	2	yes	—
106	C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
107	C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
108	Dic₂₉	—	2	1	no	—
109	Dic₂₉	—	2	2	no	—
110	Dic₂₉	—	2	1	no	—
111	Dic₂₉	—	2	2	no	—
112	Dic₂₉	C₂×M₁₆	1	2	yes	—
113	C₂×C₄×Q₈	—	29	1	no	—
114	#446	—	29	1	no	—
115	#446	—	29	1	no	—
116	#446	—	29	2	no	—
117	#446	—	29	2	no	—
118	#446	—	29	1	no	—
119	#446	D₂₉	1	1	yes	characteristic, complemented
120	#432	—	29	2	no	—
121	#432	—	29	2	no	—
122	#431	—	29	1	no	—
123	#431	—	29	1	no	—
124	C₂×C₂₉	#519	1	1	yes	characteristic
125	C₂×C₂₉	C₂²×M₁₆	1	1	yes	characteristic
126	C₂×C₂₉	#568	1	1	yes	characteristic
127	D₂₉	—	2	2	no	—
128	C₂²×Q₈	—	29	1	no	—
129	C₄×Q₈	—	29	1	no	—
130	C₄×Q₈	—	29	2	no	—
131	C₄×Q₈	—	29	1	no	—
132	C₄×Q₈	—	29	2	no	—
133	C₄×Q₈	—	29	1	no	—
134	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	29	1	no	—
136	C₂×C₄⋊C₄	—	29	2	no	—
137	C₂×C₄²	—	29	1	no	—
138	C₂×C₄²	—	29	2	no	—
139	C₄⋊C₈	—	29	1	no	—
140	C₄⋊C₈	—	29	1	no	—
141	C₄⋊C₈	—	29	2	no	—
142	C₄⋊C₈	—	29	2	no	—
143	C₄⋊C₈	—	29	1	no	—
144	C₄⋊C₈	—	29	2	no	—
145	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
146	C₄⋊C₈	C₂×D₂₉	1	2	yes	—
147	C₂²⋊C₈	—	29	2	no	—
148	C₂²⋊C₈	—	29	1	no	—
149	C₂²⋊C₈	—	29	1	no	—
150	#97	—	29	2	no	—
151	#97	—	29	2	no	—
152	#97	—	29	2	no	—
153	#97	C₂×D₂₉	1	2	yes	—
154	C₄×C₈	—	29	2	no	—
155	C₄×C₈	—	29	1	no	—
156	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
157	C₂₉	#2983	1	1	yes	characteristic, complemented, Sylow
158	C₂×Q₈	—	29	2	no	—
159	C₂×Q₈	—	29	1	no	—
160	C₂×Q₈	—	58	4	no	—
161	C₂×Q₈	C₄×D₂₉	1	1	yes	characteristic
162	C₂²×C₄	—	29	1	no	—
163	C₂²×C₄	—	29	2	no	—
164	C₂²×C₄	—	29	1	no	—
165	C₂²×C₄	—	29	1	no	—
166	C₂²×C₄	—	29	2	no	—
167	C₂×C₈	—	29	2	no	—
168	C₂×C₈	—	29	1	no	—
169	C₂×C₈	—	29	1	no	—
170	C₂×C₈	C₂²×D₂₉	1	2	yes	—
171	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
173	C₄⋊C₄	—	29	2	no	—
174	C₄⋊C₄	—	29	2	no	—
175	C₄⋊C₄	—	29	2	no	—
176	C₄⋊C₄	—	29	2	no	—
177	C₄⋊C₄	—	29	1	no	—
178	C₄⋊C₄	C₄×D₂₉	1	2	yes	—
179	C₄⋊C₄	C₄×D₂₉	1	1	yes	characteristic
180	C₄²	—	29	1	no	—
181	C₄²	—	29	2	no	—
182	C₄²	—	29	1	no	—
183	C₄²	—	29	2	no	—
184	C₄²	—	29	1	no	—
185	C₄²	—	29	2	no	—
186	C₄²	C₂²×D₂₉	1	1	yes	characteristic
187	C₄²	C₂²×D₂₉	1	2	yes	—
188	C₂³	—	29	1	no	—
189	Q₈	—	2	2	no	—
190	Q₈	—	58	4	no	—
191	Q₈	—	58	2	no	—
192	C₂×C₄	—	29	1	no	—
193	C₂×C₄	—	29	4	no	—
194	C₂×C₄	—	29	1	no	—
195	C₂×C₄	—	29	1	no	—
196	C₂×C₄	—	29	2	no	—
197	C₂×C₄	—	29	2	no	—
198	C₂×C₄	—	58	4	no	—
199	C₂×C₄	—	58	1	no	—
200	C₂×C₄	—	58	1	no	—
201	C₂×C₄	—	116	1	no	—
202	C₂×C₄	—	116	2	no	—
203	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	2	no	—
210	C₈	—	2	1	no	—
211	C₈	—	58	2	no	—
212	C₈	—	58	1	no	—
213	C₈	—	58	1	no	—
214	C₂²	—	29	2	no	—
215	C₂²	—	58	1	no	—
216	C₂²	—	58	1	no	—
217	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	29	2	no	—
222	C₄	—	58	1	no	—
223	C₄	—	58	2	no	—
224	C₄	—	58	1	no	—
225	C₄	—	58	2	no	—
226	C₄	M₁₆×D₂₉	1	2	yes	—
227	C₄	#11745362	1	1	yes	characteristic
228	C₄	#11745365	1	1	yes	characteristic
229	C₂	—	58	2	no	—
230	C₂	#14271639	1	1	yes	characteristic
231	C₂	C₂×M₁₆×D₂₉	1	1	yes	characteristic
232	C₂	#14271928	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#446	C₂
prime radical	C₂²×D₂₉	C₂²×C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#446	D₂₉
hypocenter	C₂₉	#2983
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #14271294	C₂	semidirect
3. #14271293	C₂	semidirect
4. #14271292	C₂	semidirect
5. #14271291	C₂	semidirect
6. #14271290	C₂	semidirect
11. C₂₉×#446	C₂	semidirect
119. #446	D₂₉	semidirect
157. C₂₉	#2983	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×D₂₉	2
#14271294	2
#14271293	2
#14271292	2
#14271291	2
#14271290	2
#14271219	2
#14271218	2
#14271215	2
#14271214	2
C₂₉×#446	2
#2983	29