Subgroups — #16828557

Order 3712.

410 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271710	C₂	1	1	yes	maximal, characteristic
2	#14271240	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271239	C₂	1	2	yes	maximal, complemented
4	C₈×D₁₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×#6597	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271226	C₂	1	2	yes	maximal, complemented
7	#14271215	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271214	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271131	C₂	1	2	yes	maximal, complemented
10	#14271128	C₂	1	2	yes	maximal, complemented
11	C₂₉×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#90805	C₂²	1	2	yes	—
13	C₂×M₁₆×C₂₉	C₂²	1	2	yes	—
14	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
15	#11745302	C₄	1	1	yes	characteristic
16	#11745300	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₂₉	C₂²	1	2	yes	—
18	C₂²×#6597	C₂²	1	2	yes	—
19	C₂×#90772	C₂²	1	2	yes	—
20	C₂×C₈×D₂₉	C₂²	1	2	yes	—
21	#11745257	C₄	1	1	yes	characteristic
22	#11745250	C₄	1	1	yes	characteristic
23	#11745247	C₄	1	2	yes	—
24	#11745246	C₄	1	2	yes	—
25	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
26	C₄²×D₂₉	C₂²	1	1	yes	characteristic
27	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂₉×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
30	C₂²⋊#6597	C₂²	1	2	yes	—
31	#11745168	C₂²	1	2	yes	—
32	#11745168	C₂²	1	2	yes	—
33	#11745168	C₂²	1	1	yes	characteristic
34	#11745168	C₂²	1	2	yes	—
35	#11745168	C₂²	1	1	yes	characteristic
36	#11745162	C₂²	1	1	yes	characteristic
37	#11745162	C₂²	1	1	yes	characteristic
38	C₈×Dic₂₉	C₂²	1	2	yes	—
39	C₄⋊#6597	C₂²	1	1	yes	characteristic
40	C₄×#6597	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₂₉	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊D₂₉	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊D₂₉	C₂×C₄	1	2	yes	—
46	#90805	—	4	2	no	—
47	#90805	—	4	2	no	—
48	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
49	C₂×D₁₁₆	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₂₉	C₂³	1	2	yes	—
53	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
54	C₂×Dic₅₈	C₂×C₄	1	2	yes	—
55	M₁₆×C₂₉	—	4	2	no	—
56	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₂₉	C₂³	1	2	yes	—
58	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
60	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
61	C₄²×C₂₉	C₂³	1	1	yes	characteristic
62	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
63	#90781	C₂×C₄	1	2	yes	—
64	#90781	C₂×C₄	1	2	yes	—
65	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
66	#90779	C₂×C₄	1	2	yes	—
67	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
68	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
70	C₂×#6597	—	4	2	no	—
71	C₂×#6597	C₂³	1	1	yes	characteristic
72	C₂×#6597	C₂³	1	2	yes	—
73	C₂×#6597	C₂³	1	1	yes	characteristic
74	#90772	—	4	2	no	—
75	C₈×D₂₉	—	4	2	no	—
76	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
77	C₂²×D₂₉	C₂²×C₄	1	2	yes	—
78	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
79	D₄×C₂₉	—	2	2	no	—
80	C₂×C₄×C₂₉	—	2	2	no	—
81	C₂×C₄×C₂₉	—	2	2	no	—
82	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
86	C₂²⋊D₂₉	—	2	4	no	—
87	C₂²⋊D₂₉	—	4	2	no	—
88	C₂²⋊D₂₉	—	4	2	no	—
89	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
92	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
93	D₁₁₆	—	2	2	no	—
94	D₁₁₆	—	4	2	no	—
95	C₄×D₂₉	—	2	2	no	—
96	C₄×D₂₉	—	2	2	no	—
97	C₄×D₂₉	—	2	2	no	—
98	C₄×D₂₉	—	2	1	no	—
99	C₄×D₂₉	—	2	1	no	—
100	C₄×D₂₉	—	2	2	no	—
101	Dic₅₈	—	4	2	no	—
102	C₈×C₂₉	—	2	1	no	—
103	C₈×C₂₉	—	2	2	no	—
104	C₈×C₂₉	—	2	1	no	—
105	#6597	—	2	2	no	—
106	#6597	—	2	1	no	—
107	#6597	—	2	1	no	—
108	#2979	—	29	1	no	maximal, Sylow
109	C₂²×C₂₉	—	2	2	no	—
110	C₂²×C₂₉	—	2	2	no	—
111	C₂²×C₂₉	—	2	2	no	—
112	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
113	C₂×D₂₉	—	2	1	no	—
114	C₂×D₂₉	—	2	2	no	—
115	C₂×D₂₉	—	2	1	no	—
116	C₂×D₂₉	—	2	1	no	—
117	C₂×D₂₉	—	2	2	no	—
118	C₂×D₂₉	—	2	2	no	—
119	C₄×C₂₉	—	2	2	no	—
120	C₄×C₂₉	—	2	1	no	—
121	C₄×C₂₉	#131	1	2	yes	—
122	C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
123	C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
124	Dic₂₉	—	2	1	no	—
125	Dic₂₉	—	2	2	no	—
126	Dic₂₉	—	2	1	no	—
127	Dic₂₉	#131	1	2	yes	—
128	#532	—	29	1	no	—
129	#436	—	29	2	no	—
130	#436	—	29	1	no	—
131	#436	D₂₉	1	1	yes	characteristic, complemented
132	C₈×D₄	—	29	1	no	—
133	C₈×D₄	—	29	2	no	—
134	C₈×D₄	—	29	1	no	—
135	#431	—	29	1	no	—
136	#431	—	29	1	no	—
137	#408	—	29	2	no	—
138	#408	—	29	2	no	—
139	C₂×C₂₉	—	4	2	no	—
140	C₂×C₂₉	#518	1	1	yes	characteristic
141	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
142	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
143	D₂₉	—	4	1	no	—
144	D₂₉	—	4	2	no	—
145	C₂×#41	—	29	2	no	—
146	C₂×M₁₆	—	29	2	no	—
147	C₂×M₁₆	C₂×D₂₉	1	2	yes	—
148	C₂²×C₈	—	29	2	no	—
149	C₂²×C₈	—	29	2	no	—
150	C₄⋊Q₈	—	29	1	no	—
151	#127	—	29	1	no	—
152	#124	—	29	2	no	—
153	#121	—	29	1	no	—
154	#121	—	29	1	no	—
155	#121	—	29	2	no	—
156	C₄×D₄	—	29	2	no	—
157	C₄×D₄	—	29	1	no	—
158	C₄×D₄	C₂×D₂₉	1	1	yes	characteristic
159	C₂×C₄²	—	29	1	no	—
160	C₄⋊C₈	—	29	1	no	—
161	C₄⋊C₈	—	29	1	no	—
162	C₄⋊C₈	—	29	1	no	—
163	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
164	C₂²⋊C₈	—	29	2	no	—
165	C₂²⋊C₈	—	29	2	no	—
166	C₂²⋊C₈	—	29	1	no	—
167	C₂²⋊C₈	—	29	1	no	—
168	C₂²⋊C₈	—	29	2	no	—
169	C₂²⋊C₈	—	29	2	no	—
170	C₂²⋊C₈	C₂×D₂₉	1	2	yes	—
171	C₄×C₈	—	29	2	no	—
172	C₄×C₈	—	29	1	no	—
173	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
174	C₂₉	#2979	1	1	yes	characteristic, complemented, Sylow
175	#41	—	116	2	no	—
176	#41	—	116	2	no	—
177	C₂×Q₈	—	29	2	no	—
178	C₂×D₄	—	29	2	no	—
179	C₂×D₄	—	29	2	no	—
180	C₂×D₄	—	29	2	no	—
181	C₂×D₄	—	29	1	no	—
182	C₂×D₄	—	29	2	no	—
183	C₂×D₄	C₄×D₂₉	1	1	yes	characteristic
184	C₂²×C₄	—	29	1	no	—
185	C₂²×C₄	—	29	1	no	—
186	C₂²×C₄	—	29	1	no	—
187	C₂²×C₄	—	29	2	no	—
188	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
189	M₁₆	—	4	2	no	—
190	M₁₆	—	116	2	no	—
191	C₂×C₈	—	29	2	no	—
192	C₂×C₈	—	29	1	no	—
193	C₂×C₈	—	29	1	no	—
194	C₂×C₈	—	116	2	no	—
195	C₂×C₈	—	116	2	no	—
196	C₂×C₈	C₂²×D₂₉	1	2	yes	—
197	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
198	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
199	C₄⋊C₄	—	29	2	no	—
200	C₄⋊C₄	—	29	1	no	—
201	C₄⋊C₄	C₄×D₂₉	1	1	yes	characteristic
202	C₂²⋊C₄	—	29	2	no	—
203	C₂²⋊C₄	—	29	2	no	—
204	C₂²⋊C₄	—	29	2	no	—
205	C₂²⋊C₄	C₄×D₂₉	1	2	yes	—
206	C₄²	—	29	1	no	—
207	C₄²	—	29	1	no	—
208	C₄²	—	29	1	no	—
209	C₄²	C₂²×D₂₉	1	1	yes	characteristic
210	C₂³	—	29	2	no	—
211	C₂³	—	29	1	no	—
212	C₂³	C₂×C₄×D₂₉	1	2	yes	—
213	Q₈	—	116	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	58	4	no	—
216	D₄	—	58	2	no	—
217	D₄	—	116	2	no	—
218	D₄	—	116	2	no	—
219	D₄	—	116	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	29	1	no	—
223	C₂×C₄	—	29	1	no	—
224	C₂×C₄	—	29	1	no	—
225	C₂×C₄	—	29	2	no	—
226	C₂×C₄	—	58	2	no	—
227	C₂×C₄	—	58	2	no	—
228	C₂×C₄	—	58	1	no	—
229	C₂×C₄	—	58	2	no	—
230	C₂×C₄	—	58	1	no	—
231	C₂×C₄	—	58	2	no	—
232	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
234	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	2	1	no	—
239	C₈	—	58	1	no	—
240	C₈	—	58	2	no	—
241	C₈	—	58	1	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	58	1	no	—
246	C₂²	—	58	2	no	—
247	C₂²	—	58	1	no	—
248	C₂²	—	58	2	no	—
249	C₂²	—	58	1	no	—
250	C₂²	—	58	2	no	—
251	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	29	2	no	—
255	C₄	—	58	1	no	—
256	C₄	—	58	2	no	—
257	C₄	—	58	1	no	—
258	C₄	#11745269	1	2	yes	—
259	C₄	#11745359	1	1	yes	characteristic
260	C₄	#11745364	1	1	yes	characteristic
261	C₂	—	4	2	no	—
262	C₂	—	116	1	no	—
263	C₂	—	116	2	no	—
264	C₂	#14271635	1	1	yes	characteristic
265	C₂	#131×D₂₉	1	1	yes	characteristic
266	C₂	C₂×#11745269	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#436	C₂
prime radical	#14271710	C₂
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#436	D₂₉
hypocenter	C₂₉	#2979
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. #14271240	C₂	semidirect
3. #14271239	C₂	semidirect
4. C₈×D₁₁₆	C₂	semidirect
5. D₄×#6597	C₂	semidirect
6. #14271226	C₂	semidirect
7. #14271215	C₂	semidirect
8. #14271214	C₂	semidirect
9. #14271131	C₂	semidirect
10. #14271128	C₂	semidirect
11. C₂₉×#436	C₂	semidirect
131. #436	D₂₉	semidirect
174. C₂₉	#2979	semidirect

Maximal subgroups

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

subgroup	index
#14271710	2
#14271240	2
#14271239	2
C₈×D₁₁₆	2
D₄×#6597	2
#14271226	2
#14271215	2
#14271214	2
#14271131	2
#14271128	2
C₂₉×#436	2
#2979	29