Subgroups — #16836252

Order 3712.

492 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14272231	C₂	1	1	yes	maximal, characteristic, complemented
2	#14272230	C₂	1	1	yes	maximal, characteristic
3	#14272202	C₂	1	4	yes	maximal, complemented
4	SD₁₆×#1181	C₂	1	4	yes	maximal, complemented
5	#14272173	C₂	1	1	yes	maximal, characteristic, complemented
6	#14272156	C₂	1	1	yes	maximal, characteristic, complemented
7	#14272155	C₂	1	1	yes	maximal, characteristic
8	#14272126	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×SD₁₆×D₂₉	C₂	1	1	yes	maximal, characteristic
10	C₂×Q₈×D₂₉	C₂²	1	1	yes	characteristic
11	C₂×D₄×D₂₉	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊#1181	—	2	2	no	—
13	Q₈×#1181	—	2	2	no	—
14	Q₈×#1181	C₂²	1	4	yes	complemented
15	D₄×#1181	—	2	2	no	—
16	D₄×#1181	C₂²	1	4	yes	—
17	#11745345	—	2	2	no	—
18	#11745345	C₂²	1	1	yes	characteristic
19	#11745345	C₂²	1	1	yes	characteristic
20	#11745345	C₂²	1	1	yes	characteristic
21	D₂₉⋊M₁₆	C₂²	1	1	yes	characteristic
22	C₂×SD₁₆×C₂₉	C₄	1	1	yes	characteristic, complemented
23	C₂×#90784	C₄	1	1	yes	characteristic, complemented
24	C₂×#90783	C₄	1	1	yes	characteristic, complemented
25	SD₁₆×D₂₉	C₄	1	2	yes	complemented
26	SD₁₆×D₂₉	C₄	1	2	yes	complemented
27	SD₁₆×D₂₉	C₂²	1	2	yes	—
28	SD₁₆×D₂₉	C₂²	1	2	yes	—
29	C₂×#90773	C₄	1	1	yes	characteristic, complemented
30	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
31	#11745225	C₂²	1	4	yes	complemented
32	#11745223	C₂²	1	4	yes	—
33	#11745210	C₂²	1	2	yes	complemented
34	#11745209	C₂²	1	2	yes	complemented
35	#11745208	C₂²	1	2	yes	complemented
36	C₈×#1181	C₂²	1	2	yes	complemented
37	C₂³×D₂₉	—	2	1	no	—
38	C₂²×#1181	—	2	4	no	—
39	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic, complemented
40	C₂×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊D₂₉	—	2	1	no	—
42	Q₈×D₂₉	—	2	1	no	—
43	Q₈×D₂₉	—	2	1	no	—
44	Q₈×D₂₉	C₂×C₄	1	2	yes	complemented
45	Q₈×D₂₉	C₂³	1	2	yes	—
46	D₄×D₂₉	—	2	1	no	—
47	D₄×D₂₉	—	2	1	no	—
48	D₄×D₂₉	C₂×C₄	1	2	yes	—
49	D₄×D₂₉	C₂³	1	2	yes	—
50	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₂₉	—	2	1	no	—
52	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
53	C₂×Dic₅₈	—	2	1	no	—
54	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic, complemented
55	C₂²⋊#1181	—	2	1	no	—
56	C₂²⋊#1181	—	2	1	no	—
57	C₂²⋊#1181	—	2	4	no	—
58	C₂²⋊#1181	D₄	1	2	yes	—
59	#90800	—	2	1	no	—
60	C₄⋊#1181	—	2	4	no	—
61	C₄⋊#1181	D₄	1	2	yes	—
62	C₄⋊#1181	C₂³	1	2	yes	—
63	C₄⋊#1181	C₂³	1	2	yes	—
64	C₄×#1181	—	2	4	no	—
65	C₄×#1181	—	2	1	no	—
66	C₄×#1181	—	2	1	no	—
67	C₄×#1181	C₂³	1	2	yes	—
68	#90796	—	2	1	no	—
69	D₂₉⋊C₈	C₂³	1	2	yes	—
70	SD₁₆×C₂₉	C₂×C₄	1	2	yes	complemented
71	SD₁₆×C₂₉	C₂×C₄	1	2	yes	complemented
72	C₂×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic, complemented
73	#90784	C₂×C₄	1	2	yes	complemented
74	#90784	C₂×C₄	1	2	yes	complemented
75	#90783	C₂×C₄	1	2	yes	complemented
76	#90783	C₂×C₄	1	2	yes	complemented
77	C₂×#6597	C₂×C₄	1	1	yes	characteristic, complemented
78	#90773	C₂×C₄	1	2	yes	complemented
79	#90773	C₂×C₄	1	2	yes	complemented
80	C₈×D₂₉	C₂×C₄	1	1	yes	characteristic, complemented
81	C₈×D₂₉	C₂×C₄	1	1	yes	characteristic, complemented
82	C₈×D₂₉	C₂³	1	1	yes	characteristic
83	C₈×D₂₉	C₂³	1	1	yes	characteristic
84	C₂³×C₂₉	—	2	1	no	—
85	C₂²×D₂₉	—	2	1	no	—
86	C₂²×D₂₉	—	2	2	no	—
87	C₂²×D₂₉	—	2	2	no	—
88	C₂²×D₂₉	—	8	2	no	—
89	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
90	C₂×#1181	—	2	2	no	—
91	C₂×#1181	—	2	2	no	—
92	C₂×#1181	—	8	4	no	—
93	C₂×#1181	C₂×D₄	1	4	yes	—
94	Q₈×C₂₉	—	2	1	no	—
95	Q₈×C₂₉	C₂²×C₄	1	2	yes	—
96	D₄×C₂₉	—	2	1	no	—
97	D₄×C₂₉	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₂₉	—	2	1	no	—
99	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
100	C₂²⋊D₂₉	—	2	2	no	—
101	C₂²⋊D₂₉	—	2	2	no	—
102	C₂×Dic₂₉	—	2	1	no	—
103	C₂×Dic₂₉	#41	1	1	yes	characteristic
104	D₁₁₆	—	2	1	no	—
105	D₁₁₆	C₂²×C₄	1	2	yes	—
106	C₄×D₂₉	—	2	2	no	—
107	C₄×D₂₉	—	2	2	no	—
108	C₄×D₂₉	C₂²×C₄	1	1	yes	characteristic
109	C₄×D₂₉	C₂×D₄	1	1	yes	characteristic
110	C₄×D₂₉	#41	1	1	yes	characteristic
111	C₄×D₂₉	C₂⁴	1	1	yes	characteristic
112	Dic₅₈	—	2	1	no	—
113	Dic₅₈	—	2	2	no	—
114	Dic₅₈	—	2	2	no	—
115	Dic₅₈	C₂²×C₄	1	2	yes	—
116	#6599	—	2	2	no	—
117	C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
118	C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
119	#6597	C₂²×C₄	1	1	yes	characteristic
120	#6597	C₂²×C₄	1	1	yes	characteristic
121	#2949	—	29	1	no	maximal, Sylow
122	C₂²×C₂₉	—	2	2	no	—
123	C₂²×C₂₉	—	8	1	no	—
124	C₂²×C₂₉	C₄×D₄	1	1	yes	characteristic
125	C₂×D₂₉	—	2	2	no	—
126	C₂×D₂₉	—	2	2	no	—
127	C₂×D₂₉	—	4	4	no	—
128	C₂×D₂₉	—	8	2	no	—
129	C₂×D₂₉	—	8	1	no	—
130	C₂×D₂₉	C₄×D₄	1	1	yes	characteristic
131	C₂×D₂₉	C₂²×D₄	1	1	yes	characteristic
132	#1181	—	2	4	no	—
133	#1181	—	4	2	no	—
134	#1181	—	4	2	no	—
135	C₄×C₂₉	—	2	2	no	—
136	C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
137	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
138	Dic₂₉	—	2	2	no	—
139	Dic₂₉	C₄×D₄	1	1	yes	characteristic
140	Dic₂₉	C₂×#41	1	1	yes	characteristic
141	C₂²×SD₁₆	—	29	1	no	—
142	#519	—	29	1	no	—
143	#518	—	29	1	no	—
144	#440	—	29	4	no	—
145	C₄×SD₁₆	—	29	4	no	—
146	#427	—	29	1	no	—
147	#419	—	29	1	no	—
148	#418	—	29	1	no	—
149	#405	—	29	1	no	—
150	C₂×C₂₉	—	2	1	no	—
151	C₂×C₂₉	—	4	2	no	—
152	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
153	D₂₉	—	2	1	no	—
154	D₂₉	—	4	2	no	—
155	D₂₉	#577	1	2	yes	—
156	C₂²×Q₈	—	29	1	no	—
157	C₂²×D₄	—	29	1	no	—
158	C₂×SD₁₆	—	29	1	no	—
159	C₂×SD₁₆	—	29	1	no	—
160	C₂×SD₁₆	—	29	2	no	—
161	C₂×SD₁₆	—	29	2	no	—
162	C₂×SD₁₆	—	29	2	no	—
163	C₂×SD₁₆	—	29	2	no	—
164	C₂×SD₁₆	—	29	1	no	—
165	C₂×SD₁₆	#1181	1	1	yes	characteristic, complemented
166	C₂×M₁₆	—	29	1	no	—
167	C₂²×C₈	—	29	1	no	—
168	C₄×Q₈	—	29	4	no	—
169	C₄×Q₈	—	58	2	no	—
170	C₄×D₄	—	29	4	no	—
171	C₄×D₄	—	58	2	no	—
172	#117	—	29	1	no	—
173	#117	—	29	1	no	—
174	#117	—	29	1	no	—
175	#117	—	58	2	no	—
176	C₂×C₂²⋊C₄	—	58	2	no	—
177	#107	—	29	2	no	—
178	#106	—	29	2	no	—
179	#103	—	29	4	no	—
180	#102	—	29	4	no	—
181	#97	—	29	2	no	—
182	C₄×C₈	—	29	2	no	—
183	C₂₉	#2949	1	1	yes	characteristic, complemented, Sylow
184	C₂⁴	—	58	1	no	—
185	C₂×Q₈	—	29	2	no	—
186	C₂×Q₈	—	29	2	no	—
187	C₂×Q₈	—	29	1	no	—
188	C₂×Q₈	—	58	1	no	—
189	C₂×Q₈	—	58	1	no	—
190	C₂×Q₈	—	58	1	no	—
191	C₂×Q₈	C₂×#1181	1	1	yes	characteristic, complemented
192	C₂×D₄	—	29	2	no	—
193	C₂×D₄	—	29	2	no	—
194	C₂×D₄	—	29	1	no	—
195	C₂×D₄	—	58	1	no	—
196	C₂×D₄	—	58	1	no	—
197	C₂×D₄	—	58	1	no	—
198	C₂×D₄	C₂×#1181	1	1	yes	characteristic
199	C₂²×C₄	—	29	1	no	—
200	C₂²×C₄	—	58	1	no	—
201	C₂²×C₄	—	58	4	no	—
202	SD₁₆	—	29	2	no	—
203	SD₁₆	—	29	2	no	—
204	SD₁₆	—	29	2	no	—
205	SD₁₆	—	29	2	no	—
206	SD₁₆	—	29	2	no	—
207	SD₁₆	—	29	2	no	—
208	SD₁₆	C₂×#1181	1	2	yes	complemented
209	SD₁₆	C₂×#1181	1	2	yes	complemented
210	M₁₆	—	58	1	no	—
211	M₁₆	—	58	1	no	—
212	C₂×C₈	—	29	1	no	—
213	C₂×C₈	—	29	1	no	—
214	C₂×C₈	—	29	1	no	—
215	C₂×C₈	—	29	1	no	—
216	C₂×C₈	—	29	1	no	—
217	C₂×C₈	—	29	2	no	—
218	C₂×C₈	C₂×#1181	1	1	yes	characteristic, complemented
219	C₄⋊C₄	—	29	2	no	—
220	C₄⋊C₄	—	29	2	no	—
221	C₄⋊C₄	—	29	2	no	—
222	C₄⋊C₄	—	58	4	no	—
223	C₂²⋊C₄	—	29	2	no	—
224	C₂²⋊C₄	—	58	4	no	—
225	C₂²⋊C₄	—	58	1	no	—
226	C₂²⋊C₄	—	58	1	no	—
227	C₄²	—	29	2	no	—
228	C₄²	—	58	1	no	—
229	C₄²	—	58	1	no	—
230	C₄²	—	58	4	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	29	1	no	—
233	C₂³	—	58	2	no	—
234	C₂³	—	58	2	no	—
235	C₂³	—	58	1	no	—
236	C₂³	—	232	2	no	—
237	Q₈	—	2	1	no	—
238	Q₈	—	29	2	no	—
239	Q₈	—	58	2	no	—
240	Q₈	—	58	2	no	—
241	Q₈	—	58	1	no	—
242	Q₈	C₂²×#1181	1	2	yes	—
243	D₄	—	2	1	no	—
244	D₄	—	29	2	no	—
245	D₄	—	58	2	no	—
246	D₄	—	58	2	no	—
247	D₄	—	58	1	no	—
248	D₄	C₂²×#1181	1	2	yes	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	29	1	no	—
251	C₂×C₄	—	29	1	no	—
252	C₂×C₄	—	29	1	no	—
253	C₂×C₄	—	29	1	no	—
254	C₂×C₄	—	29	1	no	—
255	C₂×C₄	—	29	4	no	—
256	C₂×C₄	—	58	2	no	—
257	C₂×C₄	—	58	2	no	—
258	C₂×C₄	—	58	1	no	—
259	C₂×C₄	—	58	2	no	—
260	C₂×C₄	—	58	2	no	—
261	C₂×C₄	—	232	4	no	—
262	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
263	C₈	—	29	1	no	—
264	C₈	—	29	1	no	—
265	C₈	—	58	2	no	—
266	C₈	C₂²×#1181	1	1	yes	characteristic
267	C₈	C₂²×#1181	1	1	yes	characteristic
268	C₂²	—	2	2	no	—
269	C₂²	—	8	1	no	—
270	C₂²	—	29	1	no	—
271	C₂²	—	29	1	no	—
272	C₂²	—	58	2	no	—
273	C₂²	—	58	2	no	—
274	C₂²	—	116	4	no	—
275	C₂²	—	232	1	no	—
276	C₂²	—	232	2	no	—
277	C₂²	D₄×#1181	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	29	1	no	—
280	C₄	—	29	1	no	—
281	C₄	—	58	4	no	—
282	C₄	—	58	2	no	—
283	C₄	—	116	2	no	—
284	C₄	—	116	2	no	—
285	C₄	D₄×#1181	1	1	yes	characteristic
286	C₄	C₂³×#1181	1	1	yes	characteristic
287	C₂	—	2	1	no	—
288	C₂	—	4	2	no	—
289	C₂	—	29	2	no	—
290	C₂	—	58	1	no	—
291	C₂	—	116	2	no	—
292	C₂	C₂×D₄×#1181	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#1181
commutator	C₄×C₂₉	C₂³×C₄
Frattini	C₄	C₂³×#1181
Fitting	C₂×SD₁₆×C₂₉	C₄
socle	C₂×C₂₉	C₂×C₄×D₄
prime radical	C₂×D₄×D₂₉	C₂²
cosocle	C₄×D₂₉	C₂⁴
hypercenter	C₂×SD₁₆	#1181
hypocenter	C₂₉	#2949
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #14272231	C₂	semidirect
3. #14272202	C₂	semidirect
4. SD₁₆×#1181	C₂	semidirect
5. #14272173	C₂	semidirect
6. #14272156	C₂	semidirect
8. #14272126	C₂	semidirect
14. Q₈×#1181	C₂²	semidirect
22. C₂×SD₁₆×C₂₉	C₄	semidirect
23. C₂×#90784	C₄	semidirect
24. C₂×#90783	C₄	semidirect
25. SD₁₆×D₂₉	C₄	semidirect
26. SD₁₆×D₂₉	C₄	semidirect
29. C₂×#90773	C₄	semidirect
31. #11745225	C₂²	semidirect
33. #11745210	C₂²	semidirect
34. #11745209	C₂²	semidirect
35. #11745208	C₂²	semidirect
36. C₈×#1181	C₂²	semidirect
39. C₂×Q₈×C₂₉	C₂×C₄	semidirect
44. Q₈×D₂₉	C₂×C₄	semidirect
54. C₂×Dic₅₈	C₂×C₄	semidirect
70. SD₁₆×C₂₉	C₂×C₄	semidirect
71. SD₁₆×C₂₉	C₂×C₄	semidirect
72. C₂×C₈×C₂₉	C₂×C₄	semidirect
73. #90784	C₂×C₄	semidirect
74. #90784	C₂×C₄	semidirect
75. #90783	C₂×C₄	semidirect
76. #90783	C₂×C₄	semidirect
77. C₂×#6597	C₂×C₄	semidirect
78. #90773	C₂×C₄	semidirect
79. #90773	C₂×C₄	semidirect
80. C₈×D₂₉	C₂×C₄	semidirect
81. C₈×D₂₉	C₂×C₄	semidirect
165. C₂×SD₁₆	#1181	semidirect
183. C₂₉	#2949	semidirect
191. C₂×Q₈	C₂×#1181	semidirect
208. SD₁₆	C₂×#1181	semidirect
209. SD₁₆	C₂×#1181	semidirect
218. C₂×C₈	C₂×#1181	semidirect

Maximal subgroups

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

subgroup	index
#14272231	2
#14272230	2
#14272202	2
SD₁₆×#1181	2
#14272173	2
#14272156	2
#14272155	2
#14272126	2
C₂×SD₁₆×D₂₉	2
#2949	29