Subgroups — #16836267

Order 3712.

458 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14272243	C₂	1	1	yes	maximal, characteristic, complemented
2	#14272242	C₂	1	1	yes	maximal, characteristic, complemented
3	#14272209	C₂	1	2	yes	maximal, complemented
4	#14272208	C₂	1	2	yes	maximal, complemented
5	#14272207	C₂	1	2	yes	maximal, complemented
6	#14272206	C₂	1	2	yes	maximal, complemented
7	#14272178	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745226	C₂	1	1	yes	maximal, characteristic, complemented
9	D₂₉⋊#104	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×#1181	C₂	1	1	yes	maximal, characteristic, complemented
11	#14271962	C₂	1	1	yes	maximal, characteristic
12	#41×D₂₉	C₂²	1	1	yes	characteristic
13	C₂×#90809	C₂²	1	1	yes	characteristic
14	#11745348	—	2	1	no	—
15	#11745348	—	2	1	no	—
16	#11745348	C₂²	1	2	yes	complemented
17	#11745348	C₂²	1	2	yes	complemented
18	#11745348	C₂²	1	2	yes	complemented
19	#11745346	—	2	2	no	—
20	#11745346	C₂²	1	2	yes	complemented
21	C₂×C₄×#1181	C₂²	1	1	yes	characteristic
22	D₂₉⋊M₁₆	—	2	2	no	—
23	D₂₉⋊M₁₆	—	2	1	no	—
24	D₂₉⋊M₁₆	C₂²	1	1	yes	characteristic
25	D₂₉⋊M₁₆	C₂²	1	1	yes	characteristic
26	C₂×#90796	—	2	1	no	—
27	C₂×#90796	C₂²	1	1	yes	characteristic
28	C₂₉×#137	C₄	1	1	yes	characteristic, complemented
29	#11745311	C₄	1	1	yes	characteristic, complemented
30	#11745304	C₄	1	1	yes	characteristic, complemented
31	#11745281	C₂²	1	2	yes	—
32	#11745280	C₄	1	2	yes	complemented
33	#11745278	C₂²	1	2	yes	—
34	#11745276	C₄	1	2	yes	complemented
35	#11745270	C₄	1	1	yes	characteristic, complemented
36	M₁₆×D₂₉	C₂²	1	1	yes	characteristic
37	#11745226	C₂²	1	1	yes	characteristic, complemented
38	#11745226	C₂²	1	1	yes	characteristic, complemented
39	#11745226	C₂²	1	2	yes	complemented
40	#11745226	C₂²	1	2	yes	complemented
41	#11745224	C₂²	1	1	yes	characteristic, complemented
42	#11745224	C₂²	1	1	yes	characteristic, complemented
43	#11745212	C₂²	1	2	yes	complemented
44	#11745211	C₂²	1	2	yes	complemented
45	#11745208	C₂²	1	2	yes	complemented
46	C₈×#1181	C₂²	1	2	yes	complemented
47	C₂²×#1181	—	2	1	no	—
48	#41×C₂₉	C₂×C₄	1	1	yes	characteristic
49	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
50	#90809	—	2	1	no	—
51	#90809	—	2	1	no	—
52	#90809	C₂×C₄	1	2	yes	—
53	#90809	C₂³	1	1	yes	characteristic
54	#90809	C₂³	1	2	yes	—
55	Q₈×D₂₉	C₂×C₄	1	1	yes	characteristic
56	#90807	—	2	1	no	—
57	#90807	C₂³	1	1	yes	characteristic
58	D₄×D₂₉	—	2	1	no	—
59	D₄×D₂₉	C₂×C₄	1	1	yes	characteristic
60	#90805	—	2	1	no	—
61	#90805	C₂×C₄	1	1	yes	characteristic
62	C₂×D₁₁₆	—	2	1	no	—
63	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₂₉	—	2	1	no	—
65	C₂×C₄×D₂₉	—	2	1	no	—
66	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
67	#90800	—	2	1	no	—
68	#90800	—	2	2	no	—
69	#90800	D₄	1	2	yes	—
70	C₂×#6599	—	2	1	no	—
71	C₂×#6599	—	2	2	no	—
72	C₄×#1181	—	2	1	no	—
73	C₄×#1181	C₂³	1	1	yes	characteristic
74	C₄×#1181	C₂³	1	1	yes	characteristic
75	#90796	—	2	1	no	—
76	#90796	—	2	2	no	—
77	#90796	—	2	2	no	—
78	#90796	—	2	2	no	—
79	#90796	D₄	1	1	yes	characteristic
80	#90796	D₄	1	1	yes	characteristic
81	#90796	C₂³	1	2	yes	—
82	#90796	C₂³	1	1	yes	characteristic
83	#90796	C₂³	1	1	yes	characteristic
84	D₂₉⋊C₈	—	2	1	no	—
85	D₂₉⋊C₈	—	2	2	no	—
86	D₂₉⋊C₈	—	2	2	no	—
87	D₂₉⋊C₈	—	2	2	no	—
88	D₂₉⋊C₈	C₂³	1	2	yes	—
89	Q₁₆×C₂₉	C₂×C₄	1	2	yes	complemented
90	SD₁₆×C₂₉	C₂×C₄	1	2	yes	complemented
91	M₁₆×C₂₉	C₂×C₄	1	1	yes	characteristic
92	#90785	C₂×C₄	1	2	yes	complemented
93	#90784	C₂×C₄	1	2	yes	complemented
94	#90784	C₂×C₄	1	2	yes	complemented
95	#90782	C₂×C₄	1	2	yes	complemented
96	#90777	C₂×C₄	1	1	yes	characteristic
97	D₂₃₂	C₂×C₄	1	2	yes	complemented
98	#90773	C₂×C₄	1	2	yes	complemented
99	#90772	C₂×C₄	1	2	yes	—
100	C₈×D₂₉	C₂³	1	2	yes	—
101	C₂²×D₂₉	—	2	1	no	—
102	C₂²×D₂₉	—	2	1	no	—
103	C₂²×D₂₉	#41	1	1	yes	characteristic
104	C₂×#1181	—	2	1	no	—
105	C₂×#1181	—	2	1	no	—
106	C₂×#1181	—	4	2	no	—
107	Q₈×C₂₉	—	2	1	no	—
108	Q₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
109	Q₈×C₂₉	C₂²×C₄	1	2	yes	—
110	D₄×C₂₉	—	2	1	no	—
111	D₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₉	—	2	1	no	—
113	C₂×C₄×C₂₉	—	2	1	no	—
114	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
115	C₂²⋊D₂₉	—	2	1	no	—
116	C₂²⋊D₂₉	—	2	1	no	—
117	C₂²⋊D₂₉	—	2	1	no	—
118	C₂×Dic₂₉	—	2	1	no	—
119	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
120	D₁₁₆	—	2	2	no	—
121	D₁₁₆	—	2	2	no	—
122	D₁₁₆	—	2	1	no	—
123	D₁₁₆	—	2	1	no	—
124	D₁₁₆	C₂²×C₄	1	2	yes	—
125	D₁₁₆	C₂²×C₄	1	1	yes	characteristic
126	C₄×D₂₉	—	2	1	no	—
127	C₄×D₂₉	—	2	1	no	—
128	C₄×D₂₉	—	2	2	no	—
129	C₄×D₂₉	—	2	1	no	—
130	C₄×D₂₉	—	2	2	no	—
131	C₄×D₂₉	C₂²×C₄	1	1	yes	characteristic
132	C₄×D₂₉	C₂×D₄	1	1	yes	characteristic
133	C₄×D₂₉	#41	1	1	yes	characteristic
134	C₄×D₂₉	C₂⁴	1	1	yes	characteristic
135	Dic₅₈	—	2	1	no	—
136	Dic₅₈	C₂²×C₄	1	1	yes	characteristic
137	#6599	—	2	1	no	—
138	#6599	—	2	1	no	—
139	#6599	—	2	2	no	—
140	#6599	C₂×D₄	1	2	yes	—
141	#6599	C₂×D₄	1	2	yes	—
142	C₈×C₂₉	C₂²×C₄	1	2	yes	—
143	#6597	C₂²×C₄	1	2	yes	—
144	#2956	—	29	1	no	maximal, Sylow
145	C₂²×C₂₉	—	2	1	no	—
146	C₂²×C₂₉	C₄×D₄	1	1	yes	characteristic
147	C₂×D₂₉	—	2	2	no	—
148	C₂×D₂₉	—	2	1	no	—
149	C₂×D₂₉	—	4	1	no	—
150	C₂×D₂₉	—	8	1	no	—
151	C₂×D₂₉	—	8	1	no	—
152	C₂×D₂₉	C₄×D₄	1	1	yes	characteristic
153	C₂×D₂₉	C₂×#41	1	1	yes	characteristic
154	#1181	—	2	2	no	—
155	#1181	—	4	1	no	—
156	C₄×C₂₉	—	2	1	no	—
157	C₄×C₂₉	—	2	2	no	—
158	C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
159	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
160	Dic₂₉	—	2	1	no	—
161	Dic₂₉	C₄×D₄	1	1	yes	characteristic
162	Dic₂₉	C₂²×D₄	1	1	yes	characteristic
163	#575	—	29	1	no	—
164	#568	—	29	1	no	—
165	#568	—	29	1	no	—
166	#444	—	29	2	no	—
167	#444	—	29	2	no	—
168	#443	—	29	2	no	—
169	#443	—	29	2	no	—
170	#430	—	29	1	no	—
171	C₂×#104	—	29	1	no	—
172	C₂×#104	—	29	1	no	—
173	C₄×M₁₆	—	29	1	no	—
174	C₂×C₂₉	—	2	1	no	—
175	C₂×C₂₉	—	4	1	no	—
176	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
177	D₂₉	—	2	1	no	—
178	D₂₉	—	2	1	no	—
179	D₂₉	—	4	2	no	—
180	D₂₉	—	4	1	no	—
181	C₂×#41	—	29	1	no	—
182	C₂×#41	—	29	1	no	—
183	#137	—	29	1	no	—
184	#137	—	29	2	no	—
185	#137	#1181	1	1	yes	characteristic, complemented
186	AGL(1,ℤ/8)	—	29	1	no	—
187	AGL(1,ℤ/8)	—	29	2	no	—
188	AGL(1,ℤ/8)	—	29	1	no	—
189	#135	—	29	2	no	—
190	#135	—	29	2	no	—
191	#131	—	29	2	no	—
192	#131	—	29	2	no	—
193	#131	—	29	2	no	—
194	#131	—	29	2	no	—
195	#131	—	58	2	no	—
196	#131	—	58	1	no	—
197	#131	—	58	1	no	—
198	C₂×M₁₆	—	29	1	no	—
199	C₂×M₁₆	—	29	1	no	—
200	C₂×M₁₆	—	29	1	no	—
201	C₂×M₁₆	—	29	1	no	—
202	C₂×M₁₆	—	58	1	no	—
203	C₂×M₁₆	—	58	1	no	—
204	C₂×M₁₆	—	58	2	no	—
205	C₂×C₄²	—	29	1	no	—
206	#108	—	29	2	no	—
207	#108	—	29	2	no	—
208	#104	—	29	1	no	—
209	#104	—	29	1	no	—
210	#104	—	29	2	no	—
211	#104	—	29	1	no	—
212	#104	—	29	1	no	—
213	#104	—	29	2	no	—
214	#97	—	29	2	no	—
215	C₄×C₈	—	29	2	no	—
216	C₂₉	#2956	1	1	yes	characteristic, complemented, Sylow
217	#41	—	29	1	no	—
218	#41	—	29	2	no	—
219	#41	—	29	2	no	—
220	#41	—	29	1	no	—
221	#41	—	29	1	no	—
222	#41	—	58	1	no	—
223	#41	—	58	1	no	—
224	#41	—	58	1	no	—
225	#41	—	58	1	no	—
226	#41	C₂×#1181	1	1	yes	characteristic
227	C₂×Q₈	—	29	1	no	—
228	C₂×Q₈	C₂×#1181	1	1	yes	characteristic
229	C₂×D₄	—	29	1	no	—
230	C₂×D₄	—	29	1	no	—
231	C₂×D₄	—	58	1	no	—
232	C₂×D₄	—	58	1	no	—
233	C₂²×C₄	—	29	1	no	—
234	C₂²×C₄	—	58	1	no	—
235	C₂²×C₄	—	58	1	no	—
236	C₂²×C₄	—	58	1	no	—
237	Q₁₆	—	29	2	no	—
238	Q₁₆	C₂×#1181	1	2	yes	complemented
239	SD₁₆	—	29	2	no	—
240	SD₁₆	—	29	2	no	—
241	SD₁₆	—	29	2	no	—
242	SD₁₆	C₂×#1181	1	2	yes	complemented
243	D₈	—	29	2	no	—
244	D₈	—	29	2	no	—
245	M₁₆	—	29	1	no	—
246	M₁₆	—	29	1	no	—
247	M₁₆	—	29	2	no	—
248	M₁₆	—	29	1	no	—
249	M₁₆	—	29	2	no	—
250	M₁₆	—	29	1	no	—
251	M₁₆	—	29	1	no	—
252	M₁₆	—	29	2	no	—
253	M₁₆	—	58	2	no	—
254	M₁₆	—	58	2	no	—
255	M₁₆	—	58	2	no	—
256	M₁₆	—	58	2	no	—
257	M₁₆	—	58	1	no	—
258	M₁₆	—	58	1	no	—
259	M₁₆	C₂×#1181	1	1	yes	characteristic
260	C₂×C₈	—	29	2	no	—
261	C₂×C₈	—	29	2	no	—
262	C₂×C₈	—	58	2	no	—
263	C₂×C₈	—	58	2	no	—
264	C₂×C₈	—	58	2	no	—
265	C₂×C₈	—	58	2	no	—
266	C₂×C₈	—	58	1	no	—
267	C₂×C₈	—	58	1	no	—
268	C₄²	—	29	1	no	—
269	C₄²	—	29	1	no	—
270	C₄²	—	58	1	no	—
271	C₂³	—	29	1	no	—
272	C₂³	—	58	1	no	—
273	C₂³	—	58	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	29	1	no	—
276	Q₈	—	58	1	no	—
277	Q₈	C₂²×#1181	1	1	yes	characteristic
278	Q₈	C₂²×#1181	1	2	yes	—
279	D₄	—	2	1	no	—
280	D₄	—	29	2	no	—
281	D₄	—	29	1	no	—
282	D₄	—	58	1	no	—
283	D₄	—	58	2	no	—
284	D₄	—	58	2	no	—
285	D₄	—	58	1	no	—
286	D₄	—	58	1	no	—
287	D₄	—	58	1	no	—
288	D₄	—	58	1	no	—
289	D₄	C₂²×#1181	1	1	yes	characteristic
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	29	1	no	—
293	C₂×C₄	—	29	1	no	—
294	C₂×C₄	—	29	1	no	—
295	C₂×C₄	—	29	1	no	—
296	C₂×C₄	—	29	1	no	—
297	C₂×C₄	—	58	1	no	—
298	C₂×C₄	—	58	2	no	—
299	C₂×C₄	—	58	2	no	—
300	C₂×C₄	—	58	1	no	—
301	C₂×C₄	—	58	1	no	—
302	C₂×C₄	—	58	1	no	—
303	C₂×C₄	—	58	1	no	—
304	C₂×C₄	—	58	1	no	—
305	C₂×C₄	—	116	2	no	—
306	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
307	C₈	—	29	2	no	—
308	C₈	—	29	2	no	—
309	C₈	—	29	2	no	—
310	C₈	—	58	1	no	—
311	C₈	—	58	1	no	—
312	C₈	—	58	2	no	—
313	C₈	C₂²×#1181	1	2	yes	—
314	C₂²	—	2	1	no	—
315	C₂²	—	29	1	no	—
316	C₂²	—	29	1	no	—
317	C₂²	—	58	2	no	—
318	C₂²	—	58	1	no	—
319	C₂²	—	116	1	no	—
320	C₂²	—	232	1	no	—
321	C₂²	—	232	1	no	—
322	C₂²	D₄×#1181	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	2	no	—
325	C₄	—	29	1	no	—
326	C₄	—	29	1	no	—
327	C₄	—	58	2	no	—
328	C₄	—	58	1	no	—
329	C₄	—	116	1	no	—
330	C₄	D₄×#1181	1	1	yes	characteristic
331	C₄	C₂³×#1181	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	4	1	no	—
334	C₂	—	58	1	no	—
335	C₂	—	58	1	no	—
336	C₂	—	116	2	no	—
337	C₂	—	116	1	no	—
338	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	C₂₉×#137	C₄
socle	C₂×C₂₉	C₂×C₄×D₄
prime radical	#14271962	C₂
cosocle	C₄×D₂₉	C₂⁴
hypercenter	#137	#1181
hypocenter	C₂₉	#2956
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. #14272243	C₂	semidirect
2. #14272242	C₂	semidirect
3. #14272209	C₂	semidirect
4. #14272208	C₂	semidirect
5. #14272207	C₂	semidirect
6. #14272206	C₂	semidirect
7. #14272178	C₂	semidirect
8. C₂×#11745226	C₂	semidirect
9. D₂₉⋊#104	C₂	semidirect
10. M₁₆×#1181	C₂	semidirect
16. #11745348	C₂²	semidirect
17. #11745348	C₂²	semidirect
18. #11745348	C₂²	semidirect
20. #11745346	C₂²	semidirect
28. C₂₉×#137	C₄	semidirect
29. #11745311	C₄	semidirect
30. #11745304	C₄	semidirect
32. #11745280	C₄	semidirect
34. #11745276	C₄	semidirect
35. #11745270	C₄	semidirect
37. #11745226	C₂²	semidirect
38. #11745226	C₂²	semidirect
39. #11745226	C₂²	semidirect
40. #11745226	C₂²	semidirect
41. #11745224	C₂²	semidirect
42. #11745224	C₂²	semidirect
43. #11745212	C₂²	semidirect
44. #11745211	C₂²	semidirect
45. #11745208	C₂²	semidirect
46. C₈×#1181	C₂²	semidirect
89. Q₁₆×C₂₉	C₂×C₄	semidirect
90. SD₁₆×C₂₉	C₂×C₄	semidirect
92. #90785	C₂×C₄	semidirect
93. #90784	C₂×C₄	semidirect
94. #90784	C₂×C₄	semidirect
95. #90782	C₂×C₄	semidirect
97. D₂₃₂	C₂×C₄	semidirect
98. #90773	C₂×C₄	semidirect
185. #137	#1181	semidirect
216. C₂₉	#2956	semidirect
238. Q₁₆	C₂×#1181	semidirect
242. SD₁₆	C₂×#1181	semidirect

Maximal subgroups

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

subgroup	index
#14272243	2
#14272242	2
#14272209	2
#14272208	2
#14272207	2
#14272206	2
#14272178	2
C₂×#11745226	2
D₂₉⋊#104	2
M₁₆×#1181	2
#14271962	2
#2956	29