Subgroups — #16836184

Order 3712.

402 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	#14272188	C₂	1	2	yes	maximal, complemented
2	#14272188	C₂	1	2	yes	maximal, complemented
3	#14272186	C₂	1	1	yes	maximal, characteristic, complemented
4	#14272185	C₂	1	1	yes	maximal, characteristic, complemented
5	#14272182	C₂	1	1	yes	maximal, characteristic, complemented
6	#14272181	C₂	1	1	yes	maximal, characteristic, complemented
7	#14272169	C₂	1	2	yes	maximal, complemented
8	#14272135	C₂	1	1	yes	maximal, characteristic
9	C₂×#11745219	C₂	1	1	yes	maximal, characteristic
10	#14272127	C₂	1	2	yes	maximal, complemented
11	#117×D₂₉	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₄×D₂₉	C₂²	1	1	yes	characteristic
13	C₂×#90800	C₂²	1	2	yes	—
14	C₂²×#6599	C₂²	1	2	yes	—
15	C₂₉×#117	C₄	1	1	yes	characteristic
16	#11745288	C₄	1	1	yes	characteristic
17	#11745254	C₂²	1	1	yes	characteristic
18	#11745254	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×D₂₉	C₄	1	1	yes	characteristic
20	C₄⋊C₄×D₂₉	C₄	1	1	yes	characteristic
21	C₂²⋊C₄×D₂₉	C₄	1	1	yes	characteristic
22	C₂²⋊C₄×D₂₉	C₄	1	1	yes	characteristic
23	#11745239	C₂²	1	1	yes	characteristic
24	#11745239	C₂²	1	1	yes	characteristic
25	#11745234	C₄	1	1	yes	characteristic
26	#11745234	C₄	1	1	yes	characteristic
27	C₄²×D₂₉	C₂²	1	1	yes	characteristic
28	C₄²×D₂₉	C₂²	1	1	yes	characteristic
29	#11745220	C₂²	1	2	yes	complemented
30	#11745220	C₂²	1	2	yes	complemented
31	#11745219	C₂²	1	1	yes	characteristic
32	#11745219	C₂²	1	1	yes	characteristic
33	#11745219	C₂²	1	2	yes	—
34	#11745219	C₂²	1	1	yes	characteristic
35	#11745219	C₂²	1	1	yes	characteristic
36	#11745219	C₂²	1	2	yes	—
37	#11745218	C₂²	1	2	yes	complemented
38	#11745218	C₂²	1	2	yes	complemented
39	C₄⋊#6599	C₂²	1	2	yes	complemented
40	C₄⋊#6599	C₂²	1	2	yes	complemented
41	C₄×#6599	C₂²	1	2	yes	complemented
42	C₄×#6599	C₂²	1	2	yes	complemented
43	C₂³×D₂₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
46	C₂×C₄×D₂₉	—	2	4	no	—
47	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
55	#90800	—	2	4	no	—
56	C₂×#6599	—	2	4	no	—
57	C₂×#6599	C₂³	1	2	yes	—
58	C₂×#6599	C₂³	1	2	yes	—
59	C₂×#6599	C₂³	1	2	yes	—
60	C₂×#6599	C₂³	1	2	yes	—
61	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
63	C₂²⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
64	C₂²⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
66	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
67	C₂²⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
69	#90781	C₂×C₄	1	1	yes	characteristic
70	#90781	C₂×C₄	1	1	yes	characteristic
71	#90781	C₂×C₄	1	1	yes	characteristic
72	#90781	C₂×C₄	1	1	yes	characteristic
73	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
74	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
75	#90779	C₂×C₄	1	1	yes	characteristic
76	#90779	C₂×C₄	1	1	yes	characteristic
77	#90779	C₂×C₄	1	1	yes	characteristic
78	#90779	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
82	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
83	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
84	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
85	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₂₉	—	2	2	no	—
87	C₂²×D₂₉	—	2	2	no	—
88	C₂²×D₂₉	—	4	1	no	—
89	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
90	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₉	—	2	2	no	—
92	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₂₉	#41	1	4	yes	—
104	C₂×Dic₂₉	C₂⁴	1	1	yes	characteristic
105	C₄×D₂₉	—	2	2	no	—
106	C₄×D₂₉	—	2	2	no	—
107	C₄×D₂₉	—	2	2	no	—
108	C₄×D₂₉	—	2	2	no	—
109	C₄×D₂₉	—	2	2	no	—
110	C₄×D₂₉	—	2	2	no	—
111	C₄×D₂₉	—	4	1	no	—
112	C₄×D₂₉	—	4	1	no	—
113	#6599	—	2	2	no	—
114	#6599	—	2	2	no	—
115	#6599	—	2	2	no	—
116	#6599	—	2	2	no	—
117	#2924	—	29	1	no	maximal, Sylow
118	C₂²×C₂₉	—	2	1	no	—
119	C₂²×C₂₉	—	2	1	no	—
120	C₂²×C₂₉	#117	1	2	yes	—
121	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
122	C₂×D₂₉	—	2	1	no	—
123	C₂×D₂₉	—	2	1	no	—
124	C₂×D₂₉	—	2	1	no	—
125	C₂×D₂₉	—	2	1	no	—
126	C₂×D₂₉	—	2	1	no	—
127	C₂×D₂₉	—	4	4	no	—
128	C₂×D₂₉	#131	1	2	yes	—
129	C₄×C₂₉	—	2	1	no	—
130	C₄×C₂₉	—	2	1	no	—
131	C₄×C₂₉	—	2	1	no	—
132	C₄×C₂₉	—	2	1	no	—
133	C₄×C₂₉	—	2	1	no	—
134	C₄×C₂₉	—	2	1	no	—
135	Dic₂₉	—	2	1	no	—
136	Dic₂₉	—	2	1	no	—
137	Dic₂₉	—	2	1	no	—
138	Dic₂₉	—	2	1	no	—
139	Dic₂₉	#117	1	2	yes	—
140	Dic₂₉	C₂×#41	1	2	yes	—
141	C₂×#117	—	29	1	no	—
142	#433	—	29	2	no	—
143	#433	—	29	2	no	—
144	#432	—	29	1	no	—
145	#432	—	29	1	no	—
146	#431	—	29	1	no	—
147	#431	—	29	1	no	—
148	#424	—	29	2	no	—
149	#407	—	29	1	no	—
150	C₂×C₂²⋊C₈	—	29	1	no	—
151	#405	—	29	2	no	—
152	C₂×C₂₉	—	2	2	no	—
153	C₂×C₂₉	C₂×#117	1	1	yes	characteristic
154	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
155	C₂×C₂₉	#568	1	1	yes	characteristic
156	D₂₉	—	2	2	no	—
157	D₂₉	—	4	1	no	—
158	C₂³×C₄	—	29	1	no	—
159	C₂×M₁₆	—	29	2	no	—
160	C₂²×C₈	—	29	2	no	—
161	#117	—	29	1	no	—
162	#117	—	29	1	no	—
163	#117	—	29	1	no	—
164	#117	—	29	1	no	—
165	#117	—	29	1	no	—
166	#117	—	29	1	no	—
167	#117	—	29	1	no	—
168	#117	#1181	1	1	yes	characteristic
169	C₂×C₄⋊C₄	—	29	1	no	—
170	C₂×C₄⋊C₄	—	29	1	no	—
171	C₂×C₂²⋊C₄	—	29	1	no	—
172	C₂×C₂²⋊C₄	—	29	1	no	—
173	C₂×C₄²	—	29	1	no	—
174	C₂×C₄²	—	29	1	no	—
175	C₄⋊C₈	—	29	2	no	—
176	C₄⋊C₈	—	29	2	no	—
177	C₄⋊C₈	—	29	2	no	—
178	C₄⋊C₈	—	29	2	no	—
179	C₂²⋊C₈	—	29	1	no	—
180	C₂²⋊C₈	—	29	1	no	—
181	C₂²⋊C₈	—	29	2	no	—
182	C₂²⋊C₈	—	29	1	no	—
183	C₂²⋊C₈	—	29	1	no	—
184	C₂²⋊C₈	—	29	2	no	—
185	#97	—	29	2	no	—
186	#97	—	29	2	no	—
187	C₄×C₈	—	29	2	no	—
188	C₄×C₈	—	29	2	no	—
189	C₂₉	#2924	1	1	yes	characteristic, complemented, Sylow
190	C₂⁴	—	29	1	no	—
191	C₂²×C₄	—	29	1	no	—
192	C₂²×C₄	—	29	1	no	—
193	C₂²×C₄	—	29	1	no	—
194	C₂²×C₄	—	29	1	no	—
195	C₂²×C₄	—	29	1	no	—
196	C₂²×C₄	—	29	1	no	—
197	C₂²×C₄	—	29	1	no	—
198	C₂²×C₄	—	29	1	no	—
199	C₂²×C₄	—	29	1	no	—
200	C₂²×C₄	—	58	4	no	—
201	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
202	M₁₆	—	58	4	no	—
203	C₂×C₈	—	29	2	no	—
204	C₂×C₈	—	29	2	no	—
205	C₂×C₈	—	29	2	no	—
206	C₂×C₈	—	29	2	no	—
207	C₂×C₈	—	58	4	no	—
208	C₄⋊C₄	—	29	1	no	—
209	C₄⋊C₄	—	29	1	no	—
210	C₄⋊C₄	—	29	1	no	—
211	C₄⋊C₄	—	29	1	no	—
212	C₄⋊C₄	—	29	1	no	—
213	C₄⋊C₄	—	29	1	no	—
214	C₄⋊C₄	C₂×#1181	1	1	yes	characteristic
215	C₄⋊C₄	C₂×#1181	1	1	yes	characteristic
216	C₂²⋊C₄	—	29	1	no	—
217	C₂²⋊C₄	—	29	1	no	—
218	C₂²⋊C₄	—	29	1	no	—
219	C₂²⋊C₄	—	29	1	no	—
220	C₂²⋊C₄	—	29	1	no	—
221	C₂²⋊C₄	—	29	1	no	—
222	C₂²⋊C₄	C₂×#1181	1	1	yes	characteristic
223	C₂²⋊C₄	C₂×#1181	1	1	yes	characteristic
224	C₄²	—	29	1	no	—
225	C₄²	—	29	1	no	—
226	C₄²	—	29	1	no	—
227	C₄²	—	29	1	no	—
228	C₄²	—	29	1	no	—
229	C₄²	—	29	1	no	—
230	C₄²	C₂×#1181	1	1	yes	characteristic
231	C₄²	C₂×#1181	1	1	yes	characteristic
232	C₂³	—	29	1	no	—
233	C₂³	—	29	1	no	—
234	C₂³	—	58	2	no	—
235	C₂³	—	58	2	no	—
236	C₂³	—	116	1	no	—
237	C₂³	C₂²×#1181	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	29	4	no	—
240	C₂×C₄	—	29	1	no	—
241	C₂×C₄	—	29	1	no	—
242	C₂×C₄	—	29	1	no	—
243	C₂×C₄	—	29	1	no	—
244	C₂×C₄	—	29	1	no	—
245	C₂×C₄	—	29	1	no	—
246	C₂×C₄	—	58	2	no	—
247	C₂×C₄	—	58	2	no	—
248	C₂×C₄	—	58	2	no	—
249	C₂×C₄	—	58	2	no	—
250	C₂×C₄	—	58	2	no	—
251	C₂×C₄	—	58	2	no	—
252	C₂×C₄	—	116	1	no	—
253	C₂×C₄	—	116	1	no	—
254	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
255	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
256	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
257	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
258	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
259	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
260	C₈	—	58	2	no	—
261	C₈	—	58	2	no	—
262	C₈	—	58	2	no	—
263	C₈	—	58	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	29	2	no	—
267	C₂²	—	58	1	no	—
268	C₂²	—	58	1	no	—
269	C₂²	—	58	1	no	—
270	C₂²	—	58	1	no	—
271	C₂²	—	58	1	no	—
272	C₂²	—	116	4	no	—
273	C₂²	#11745345	1	2	yes	—
274	C₂²	C₂³×#1181	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	29	2	no	—
282	C₄	—	29	2	no	—
283	C₄	—	58	1	no	—
284	C₄	—	58	1	no	—
285	C₄	—	58	1	no	—
286	C₄	—	58	1	no	—
287	C₂	—	2	2	no	—
288	C₂	—	58	2	no	—
289	C₂	—	116	1	no	—
290	C₂	C₂×#11745345	1	1	yes	characteristic
291	C₂	D₂₉⋊#131	1	1	yes	characteristic
292	C₂	#14272242	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#1181
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#117	C₄
prime radical	C₂³×D₂₉	C₂×C₄
cosocle	C₂×Dic₂₉	C₂⁴
hypercenter	#117	#1181
hypocenter	C₂₉	#2924
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #14272188	C₂	semidirect
2. #14272188	C₂	semidirect
3. #14272186	C₂	semidirect
4. #14272185	C₂	semidirect
5. #14272182	C₂	semidirect
6. #14272181	C₂	semidirect
7. #14272169	C₂	semidirect
10. #14272127	C₂	semidirect
29. #11745220	C₂²	semidirect
30. #11745220	C₂²	semidirect
37. #11745218	C₂²	semidirect
38. #11745218	C₂²	semidirect
39. C₄⋊#6599	C₂²	semidirect
40. C₄⋊#6599	C₂²	semidirect
41. C₄×#6599	C₂²	semidirect
42. C₄×#6599	C₂²	semidirect
189. C₂₉	#2924	semidirect

Maximal subgroups

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

subgroup	index
#14272188	2
#14272186	2
#14272185	2
#14272182	2
#14272181	2
#14272169	2
#14272135	2
C₂×#11745219	2
#14272127	2
#117×D₂₉	2
#2924	29