Subgroups — #16828428

Order 3712.

348 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271717	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	1	yes	maximal, characteristic, complemented
5	#14271292	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271291	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271291	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271290	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271188	C₂	1	1	yes	maximal, characteristic
10	#14271188	C₂	1	1	yes	maximal, characteristic
11	#14271185	C₂	1	1	yes	maximal, characteristic
12	#14271120	C₂	1	1	yes	maximal, characteristic
13	#14271117	C₂	1	1	yes	maximal, characteristic
14	#97×D₂₉	C₂	1	1	yes	maximal, characteristic
15	C₂₉×#446	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
17	#11745308	C₄	1	1	yes	characteristic
18	#11745306	C₄	1	1	yes	characteristic
19	C₂×#90772	C₂²	1	1	yes	characteristic
20	C₂×#90772	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
23	#11745259	C₄	1	1	yes	characteristic
24	#11745258	C₄	1	1	yes	characteristic
25	#11745258	C₄	1	1	yes	characteristic
26	#11745251	C₄	1	1	yes	characteristic
27	#11745251	C₄	1	1	yes	characteristic
28	#11745250	C₄	1	1	yes	characteristic
29	#11745234	C₂²	1	1	yes	characteristic
30	#11745234	C₂²	1	1	yes	characteristic
31	C₄²×D₂₉	C₂²	1	1	yes	characteristic
32	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
33	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
34	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
35	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₂₉×#97	C₂²	1	1	yes	characteristic
39	C₂₉×#97	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
41	#11745163	C₂²	1	1	yes	characteristic
42	#11745163	C₂²	1	1	yes	characteristic
43	#11745163	C₂²	1	1	yes	characteristic
44	#11745163	C₂²	1	1	yes	characteristic
45	#11745162	C₂²	1	1	yes	characteristic
46	#11745162	C₂²	1	1	yes	characteristic
47	#11745162	C₂²	1	1	yes	characteristic
48	#11745162	C₂²	1	1	yes	characteristic
49	#11745162	C₂²	1	1	yes	characteristic
50	#11745162	C₂²	1	1	yes	characteristic
51	C₈×Dic₂₉	C₂²	1	1	yes	characteristic
52	C₈×Dic₂₉	C₂²	1	1	yes	characteristic
53	C₄⋊#6597	C₂²	1	1	yes	characteristic
54	C₄⋊#6597	C₂²	1	1	yes	characteristic
55	C₄⋊#6597	C₂²	1	1	yes	characteristic
56	#11745151	C₂²	1	1	yes	characteristic
57	#11745151	C₂²	1	1	yes	characteristic
58	C₄×#6597	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
63	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
64	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
65	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
72	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
73	C₄²×C₂₉	C₂³	1	1	yes	characteristic
74	C₄²×C₂₉	C₂³	1	1	yes	characteristic
75	C₄²×C₂₉	C₂³	1	1	yes	characteristic
76	#90781	C₂×C₄	1	1	yes	characteristic
77	#90781	C₂×C₄	1	1	yes	characteristic
78	#90781	C₂×C₄	1	1	yes	characteristic
79	#90781	C₂×C₄	1	1	yes	characteristic
80	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
82	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
83	#90779	C₂×C₄	1	1	yes	characteristic
84	#90779	C₂×C₄	1	1	yes	characteristic
85	#90779	C₂×C₄	1	1	yes	characteristic
86	#90779	C₂×C₄	1	1	yes	characteristic
87	#90779	C₂×C₄	1	1	yes	characteristic
88	#90779	C₂×C₄	1	1	yes	characteristic
89	#90779	C₂×C₄	1	1	yes	characteristic
90	#90779	C₂×C₄	1	1	yes	characteristic
91	#90779	C₂×C₄	1	1	yes	characteristic
92	#90779	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
96	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
97	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
98	C₂×#6597	C₂³	1	1	yes	characteristic
99	C₂×#6597	C₂³	1	1	yes	characteristic
100	C₂×#6597	C₂³	1	1	yes	characteristic
101	C₂×#6597	C₂³	1	1	yes	characteristic
102	#90772	—	2	2	no	—
103	#90772	—	2	2	no	—
104	C₈×D₂₉	—	2	2	no	—
105	C₈×D₂₉	Q₈	1	4	yes	complemented
106	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
107	Q₈×C₂₉	—	4	1	no	—
108	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
117	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
122	C₄×D₂₉	—	4	1	no	—
123	C₄×D₂₉	—	4	1	no	—
124	C₄×D₂₉	C₂×Q₈	1	2	yes	—
125	C₄×D₂₉	#41	1	2	yes	—
126	Dic₅₈	—	4	1	no	—
127	Dic₅₈	—	4	1	no	—
128	Dic₅₈	—	4	1	no	—
129	C₈×C₂₉	—	2	1	no	—
130	C₈×C₂₉	—	2	1	no	—
131	C₈×C₂₉	—	2	1	no	—
132	C₈×C₂₉	C₂×Q₈	1	2	yes	—
133	#6597	—	2	1	no	—
134	#6597	—	2	1	no	—
135	#6597	—	2	1	no	—
136	#6597	C₂×Q₈	1	2	yes	—
137	#2961	—	29	1	no	maximal, Sylow
138	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
139	C₂×D₂₉	—	2	1	no	—
140	C₂×D₂₉	—	2	1	no	—
141	C₂×D₂₉	C₄×Q₈	1	2	yes	—
142	C₄×C₂₉	—	2	1	no	—
143	C₄×C₂₉	—	2	1	no	—
144	C₄×C₂₉	—	2	1	no	—
145	C₄×C₂₉	—	2	1	no	—
146	C₄×C₂₉	—	2	1	no	—
147	C₄×C₂₉	—	2	1	no	—
148	C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
149	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
150	Dic₂₉	—	2	1	no	—
151	Dic₂₉	—	2	1	no	—
152	Dic₂₉	—	2	1	no	—
153	Dic₂₉	—	2	1	no	—
154	Dic₂₉	—	2	1	no	—
155	Dic₂₉	—	2	1	no	—
156	Dic₂₉	C₄×Q₈	1	2	yes	—
157	#533	—	29	1	no	—
158	#446	—	29	1	no	—
159	#446	—	29	1	no	—
160	#446	—	29	1	no	—
161	#446	—	29	1	no	—
162	#446	—	29	1	no	—
163	#446	—	29	1	no	—
164	#446	—	29	1	no	—
165	#446	D₂₉	1	1	yes	characteristic, complemented
166	#424	—	29	1	no	—
167	#424	—	29	1	no	—
168	#423	—	29	1	no	—
169	#405	—	29	1	no	—
170	#405	—	29	1	no	—
171	C₂×#97	—	29	1	no	—
172	C₂×C₂₉	C₂×C₄×Q₈	1	1	yes	characteristic
173	C₂×C₂₉	#568	1	1	yes	characteristic
174	C₂×C₂₉	#568	1	1	yes	characteristic
175	D₂₉	—	2	2	no	—
176	C₂×M₁₆	—	29	1	no	—
177	C₂×M₁₆	—	29	1	no	—
178	C₂²×C₈	—	29	1	no	—
179	C₂²×C₈	—	29	1	no	—
180	C₄⋊Q₈	—	29	1	no	—
181	C₄⋊Q₈	—	29	1	no	—
182	#125	—	29	1	no	—
183	#125	—	29	1	no	—
184	C₂²⋊Q₈	—	29	1	no	—
185	C₂²⋊Q₈	—	29	1	no	—
186	C₂²⋊Q₈	—	29	1	no	—
187	C₂²⋊Q₈	—	29	1	no	—
188	C₄×Q₈	—	29	1	no	—
189	C₄×Q₈	—	29	1	no	—
190	C₄×Q₈	—	29	1	no	—
191	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic
192	#117	—	29	1	no	—
193	#117	—	29	1	no	—
194	C₂×C₄²	—	29	1	no	—
195	C₄⋊C₈	—	29	1	no	—
196	C₄⋊C₈	—	29	1	no	—
197	C₄⋊C₈	—	29	1	no	—
198	C₄⋊C₈	—	29	1	no	—
199	C₄⋊C₈	—	29	1	no	—
200	C₄⋊C₈	—	29	1	no	—
201	C₄⋊C₈	—	29	1	no	—
202	C₄⋊C₈	—	29	1	no	—
203	C₄⋊C₈	—	29	1	no	—
204	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
205	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
206	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
207	#97	—	29	1	no	—
208	#97	—	29	1	no	—
209	#97	—	29	1	no	—
210	#97	—	29	1	no	—
211	#97	—	29	1	no	—
212	#97	—	29	1	no	—
213	#97	C₂×D₂₉	1	1	yes	characteristic
214	#97	C₂×D₂₉	1	1	yes	characteristic
215	C₄×C₈	—	29	1	no	—
216	C₄×C₈	—	29	1	no	—
217	C₄×C₈	—	29	1	no	—
218	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
219	C₂₉	#2961	1	1	yes	characteristic, complemented, Sylow
220	C₂×Q₈	—	29	1	no	—
221	C₂×Q₈	—	29	1	no	—
222	C₂×Q₈	—	29	1	no	—
223	C₂×Q₈	C₄×D₂₉	1	1	yes	characteristic
224	C₂²×C₄	—	29	1	no	—
225	C₂²×C₄	—	29	1	no	—
226	C₂²×C₄	—	29	1	no	—
227	M₁₆	—	58	2	no	—
228	M₁₆	—	58	2	no	—
229	C₂×C₈	—	29	1	no	—
230	C₂×C₈	—	29	4	no	—
231	C₂×C₈	—	29	1	no	—
232	C₂×C₈	—	29	1	no	—
233	C₂×C₈	—	29	1	no	—
234	C₂×C₈	—	58	2	no	—
235	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
236	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
237	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
239	C₄⋊C₄	—	29	1	no	—
240	C₄⋊C₄	—	29	1	no	—
241	C₄⋊C₄	—	29	1	no	—
242	C₄⋊C₄	—	29	1	no	—
243	C₄⋊C₄	—	29	1	no	—
244	C₄⋊C₄	—	29	1	no	—
245	C₄⋊C₄	—	29	1	no	—
246	C₄⋊C₄	—	29	1	no	—
247	C₄⋊C₄	—	29	1	no	—
248	C₄⋊C₄	—	29	1	no	—
249	C₄⋊C₄	—	29	1	no	—
250	C₄⋊C₄	—	29	1	no	—
251	C₄⋊C₄	—	29	1	no	—
252	C₄⋊C₄	C₄×D₂₉	1	1	yes	characteristic
253	C₄⋊C₄	C₄×D₂₉	1	1	yes	characteristic
254	C₄⋊C₄	C₄×D₂₉	1	1	yes	characteristic
255	C₂²⋊C₄	—	29	1	no	—
256	C₂²⋊C₄	—	29	1	no	—
257	C₂²⋊C₄	—	29	1	no	—
258	C₂²⋊C₄	—	29	1	no	—
259	C₄²	—	29	1	no	—
260	C₄²	—	29	1	no	—
261	C₄²	—	29	1	no	—
262	C₄²	—	29	1	no	—
263	C₄²	—	29	1	no	—
264	C₄²	C₂²×D₂₉	1	1	yes	characteristic
265	C₄²	C₂²×D₂₉	1	1	yes	characteristic
266	C₄²	C₂²×D₂₉	1	1	yes	characteristic
267	C₂³	—	29	1	no	—
268	Q₈	—	4	1	no	—
269	Q₈	—	116	1	no	—
270	Q₈	—	116	1	no	—
271	Q₈	—	116	1	no	—
272	C₂×C₄	—	29	1	no	—
273	C₂×C₄	—	29	2	no	—
274	C₂×C₄	—	29	2	no	—
275	C₂×C₄	—	29	1	no	—
276	C₂×C₄	—	29	1	no	—
277	C₂×C₄	—	29	1	no	—
278	C₂×C₄	—	29	1	no	—
279	C₂×C₄	—	29	1	no	—
280	C₂×C₄	—	29	1	no	—
281	C₂×C₄	—	116	1	no	—
282	C₂×C₄	—	116	1	no	—
283	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	29	2	no	—
294	C₈	—	58	1	no	—
295	C₈	—	58	1	no	—
296	C₈	—	58	1	no	—
297	C₈	Q₈×D₂₉	1	2	yes	—
298	C₂²	—	29	2	no	—
299	C₂²	—	58	1	no	—
300	C₂²	—	58	1	no	—
301	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	29	2	no	—
309	C₄	—	58	1	no	—
310	C₄	—	58	1	no	—
311	C₄	—	58	1	no	—
312	C₄	—	58	1	no	—
313	C₄	—	58	1	no	—
314	C₄	—	58	1	no	—
315	C₄	C₂×Q₈×D₂₉	1	1	yes	characteristic
316	C₄	#41×D₂₉	1	1	yes	characteristic
317	C₂	—	58	2	no	—
318	C₂	C₄×Q₈×D₂₉	1	1	yes	characteristic
319	C₂	#14271928	1	1	yes	characteristic
320	C₂	#14271929	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	#2961
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. #14271292	C₂	semidirect
6. #14271291	C₂	semidirect
7. #14271291	C₂	semidirect
8. #14271290	C₂	semidirect
15. C₂₉×#446	C₂	semidirect
105. C₈×D₂₉	Q₈	semidirect
165. #446	D₂₉	semidirect
219. C₂₉	#2961	semidirect

Maximal subgroups

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

subgroup	index
#14271717	2
#14271294	2
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#14271291	2
#14271290	2
#14271188	2
#14271185	2
#14271120	2
#14271117	2
#97×D₂₉	2
C₂₉×#446	2
#2961	29