Subgroups — #41×D₁₁₆ #16832983

Order 3712.

1648 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₂₉	C₂	1	2	yes	maximal, complemented
2	#14271835	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×D₁₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271825	C₂	1	2	yes	maximal, complemented
5	#14271817	C₂	1	3	yes	maximal, complemented
6	#14271808	C₂	1	6	yes	maximal, complemented
7	D₄×D₁₁₆	C₂	1	3	yes	maximal, complemented
8	#14271713	C₂	1	3	yes	maximal, complemented
9	#14271675	C₂	1	6	yes	maximal, complemented
10	C₂×C₄×D₁₁₆	C₂	1	3	yes	maximal, complemented
11	C₄×#41×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
13	#41×D₂₉	—	2	16	no	—
14	C₂×#90809	C₂²	1	2	yes	complemented
15	C₂×Q₈×D₂₉	C₂²	1	2	yes	complemented
16	C₂×#90807	C₂²	1	6	yes	complemented
17	C₂×D₄×D₂₉	C₂²	1	6	yes	complemented
18	C₂×#90805	C₂²	1	6	yes	complemented
19	C₂²×D₁₁₆	C₂²	1	3	yes	—
20	C₂²×C₄×D₂₉	C₂²	1	6	yes	complemented
21	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic, complemented
22	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic, complemented
23	C₄×D₄×C₂₉	C₂²	1	3	yes	complemented
24	C₄×D₄×C₂₉	C₂²	1	3	yes	complemented
25	C₂₉×#117	C₂²	1	3	yes	complemented
26	C₂×C₄²×C₂₉	C₂²	1	3	yes	complemented
27	#11745292	C₂²	1	6	yes	complemented
28	#11745292	C₂²	1	6	yes	complemented
29	C₂×#90781	C₂²	1	6	yes	complemented
30	C₂×C₄⋊Dic₂₉	C₂²	1	3	yes	complemented
31	#11745259	C₂²	1	6	yes	complemented
32	#11745259	C₂²	1	6	yes	complemented
33	#11745257	C₂²	1	6	yes	complemented
34	#11745257	C₂²	1	6	yes	complemented
35	#11745248	C₂²	1	12	yes	complemented
36	#11745244	C₂²	1	12	yes	complemented
37	#11745237	C₂²	1	6	yes	complemented
38	#11745236	C₂²	1	3	yes	complemented
39	C₄×D₁₁₆	C₂²	1	3	yes	—
40	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
41	C₄×D₁₁₆	C₂²	1	6	yes	complemented
42	C₄×D₁₁₆	C₂²	1	3	yes	complemented
43	C₄×D₁₁₆	C₂²	1	6	yes	complemented
44	C₄×D₁₁₆	C₂²	1	3	yes	complemented
45	C₄×D₁₁₆	C₂²	1	6	yes	complemented
46	#11745231	C₂²	1	3	yes	complemented
47	C₂³×D₂₉	C₂³	1	6	yes	—
48	#41×C₂₉	D₄	1	8	yes	complemented
49	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₂₉	C₂³	1	3	yes	—
51	C₂²×C₄×C₂₉	C₂³	1	3	yes	—
52	C₂²×C₄×C₂₉	C₂³	1	3	yes	—
53	C₂²×C₄×C₂₉	C₂³	1	3	yes	—
54	C₂×C₂²⋊D₂₉	C₂³	1	12	yes	—
55	C₂²×Dic₂₉	C₂³	1	6	yes	—
56	#90809	—	2	8	no	—
57	Q₈×D₂₉	—	2	8	no	—
58	#90807	—	2	24	no	—
59	D₄×D₂₉	—	2	24	no	—
60	#90805	—	2	24	no	—
61	C₂×D₁₁₆	—	2	12	no	—
62	C₂×D₁₁₆	C₂³	1	3	yes	—
63	C₂×D₁₁₆	C₂³	1	6	yes	—
64	C₂×D₁₁₆	C₂³	1	6	yes	—
65	C₂×D₁₁₆	C₂³	1	6	yes	—
66	C₂×D₁₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₂₉	—	2	24	no	—
68	C₂×C₄×D₂₉	C₂³	1	2	yes	—
69	C₂×C₄×D₂₉	C₂³	1	6	yes	—
70	C₂×C₄×D₂₉	C₂³	1	6	yes	—
71	C₂×C₄×D₂₉	C₂³	1	6	yes	—
72	C₂×Dic₅₈	C₂³	1	6	yes	—
73	C₄⋊C₄×C₂₉	C₂³	1	3	yes	—
74	C₄⋊C₄×C₂₉	C₂³	1	3	yes	—
75	C₂²⋊C₄×C₂₉	C₂³	1	6	yes	—
76	C₄²×C₂₉	C₂³	1	1	yes	characteristic
77	C₄²×C₂₉	C₂³	1	3	yes	—
78	C₄²×C₂₉	C₂³	1	3	yes	—
79	C₄²×C₂₉	C₂³	1	3	yes	—
80	#90781	C₂³	1	2	yes	—
81	#90781	C₂³	1	6	yes	—
82	#90781	C₂³	1	6	yes	—
83	#90781	C₂³	1	6	yes	—
84	#90781	C₂³	1	12	yes	—
85	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₂₉	C₂³	1	3	yes	—
87	C₄⋊Dic₂₉	C₂³	1	6	yes	—
88	C₄⋊Dic₂₉	C₂³	1	6	yes	—
89	C₂³×C₂₉	C₂⁴	1	3	yes	—
90	C₂²×D₂₉	—	2	12	no	—
91	C₂²×D₂₉	—	2	12	no	—
92	C₂²×D₂₉	—	4	6	no	—
93	C₂²×D₂₉	C₂⁴	1	6	yes	—
94	C₂²×D₂₉	C₂⁴	1	2	yes	—
95	Q₈×C₂₉	C₂×D₄	1	4	yes	complemented
96	D₄×C₂₉	C₂×D₄	1	12	yes	complemented
97	C₂×C₄×C₂₉	—	2	6	no	—
98	C₂×C₄×C₂₉	—	4	3	no	—
99	C₂×C₄×C₂₉	C₂×D₄	1	12	yes	complemented
100	C₂×C₄×C₂₉	C₂⁴	1	3	yes	—
101	C₂×C₄×C₂₉	C₂⁴	1	3	yes	—
102	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
103	C₂×C₄×C₂₉	C₂⁴	1	3	yes	—
104	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
105	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₂₉	—	2	24	no	—
107	C₂×Dic₂₉	—	2	12	no	—
108	C₂×Dic₂₉	C₂⁴	1	6	yes	—
109	C₂×Dic₂₉	C₂⁴	1	2	yes	—
110	D₁₁₆	—	2	12	no	—
111	D₁₁₆	—	2	12	no	—
112	D₁₁₆	—	2	6	no	—
113	D₁₁₆	—	4	6	no	—
114	D₁₁₆	#41	1	4	yes	direct product factor
115	C₄×D₂₉	—	2	4	no	—
116	C₄×D₂₉	—	2	12	no	—
117	C₄×D₂₉	—	2	12	no	—
118	C₄×D₂₉	—	2	12	no	—
119	Dic₅₈	—	2	12	no	—
120	D₄×#41	—	29	1	no	maximal, Sylow
121	C₂²×C₂₉	—	2	3	no	—
122	C₂²×C₂₉	—	2	3	no	—
123	C₂²×C₂₉	C₂²×D₄	1	6	yes	—
124	C₂²×C₂₉	C₂⁵	1	1	yes	characteristic
125	C₂×D₂₉	—	2	2	no	—
126	C₂×D₂₉	—	2	2	no	—
127	C₂×D₂₉	—	2	6	no	—
128	C₂×D₂₉	—	2	6	no	—
129	C₂×D₂₉	—	2	6	no	—
130	C₂×D₂₉	—	4	24	no	—
131	C₂×D₂₉	C₂×#41	1	4	yes	complemented
132	C₄×C₂₉	—	2	3	no	—
133	C₄×C₂₉	—	2	3	no	—
134	C₄×C₂₉	—	2	1	no	—
135	C₄×C₂₉	C₂²×D₄	1	6	yes	—
136	C₄×C₂₉	C₂²×D₄	1	2	yes	—
137	C₄×C₂₉	C₂×#41	1	2	yes	complemented
138	Dic₂₉	—	2	6	no	—
139	Dic₂₉	—	2	2	no	—
140	C₂²×#41	—	29	2	no	—
141	#550	—	29	1	no	—
142	D₄×Q₈	—	29	1	no	—
143	#548	—	29	2	no	—
144	#547	—	29	3	no	—
145	#546	—	29	6	no	—
146	D₄²	—	29	3	no	—
147	#532	—	29	3	no	—
148	#525	—	29	6	no	—
149	C₄×#41	D₂₉	1	1	yes	characteristic, complemented
150	C₂×C₄×D₄	—	29	3	no	—
151	C₂×C₂₉	—	2	6	no	—
152	C₂×C₂₉	C₂³×D₄	1	1	yes	characteristic
153	C₂×C₂₉	C₂²×#41	1	1	yes	characteristic
154	C₂×C₂₉	#585	1	1	yes	characteristic
155	D₂₉	—	2	4	no	—
156	D₂₉	—	4	6	no	—
157	C₂×#41	—	29	2	no	—
158	C₂×#41	—	29	6	no	—
159	C₂×#41	—	29	6	no	—
160	C₂×#41	—	58	16	no	—
161	C₂×#41	C₂×D₂₉	1	1	yes	characteristic
162	C₂²×Q₈	—	29	2	no	—
163	C₂²×D₄	—	29	6	no	—
164	C₂²×D₄	—	29	3	no	—
165	C₂³×C₄	—	29	6	no	—
166	C₄⋊Q₈	—	29	3	no	—
167	#127	—	29	3	no	—
168	#124	—	29	6	no	—
169	#123	—	29	12	no	—
170	C₂²⋊Q₈	—	29	6	no	—
171	C₂²⋊Q₈	—	29	6	no	—
172	#121	—	29	6	no	—
173	#121	—	29	6	no	—
174	#121	—	29	6	no	—
175	#121	—	29	6	no	—
176	#120	—	29	12	no	—
177	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic, complemented
178	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic, complemented
179	C₄×D₄	—	29	1	no	—
180	C₄×D₄	—	29	6	no	—
181	C₄×D₄	—	29	3	no	—
182	C₄×D₄	—	29	3	no	—
183	C₄×D₄	—	29	6	no	—
184	C₄×D₄	—	29	3	no	—
185	C₄×D₄	—	29	6	no	—
186	C₄×D₄	C₂×D₂₉	1	3	yes	complemented
187	C₄×D₄	C₂×D₂₉	1	3	yes	complemented
188	#117	C₂×D₂₉	1	3	yes	complemented
189	C₂×C₄⋊C₄	—	29	3	no	—
190	C₂×C₂²⋊C₄	—	29	6	no	—
191	C₂×C₄²	C₂×D₂₉	1	3	yes	complemented
192	C₂₉	D₄×#41	1	1	yes	characteristic, complemented, Sylow
193	C₂⁴	—	29	6	no	—
194	#41	—	58	8	no	—
195	#41	—	58	24	no	—
196	#41	—	58	24	no	—
197	#41	D₁₁₆	1	8	yes	direct product factor
198	C₂×Q₈	—	29	6	no	—
199	C₂×Q₈	—	58	8	no	—
200	C₂×Q₈	C₂²×D₂₉	1	1	yes	characteristic
201	C₂×D₄	—	29	12	no	—
202	C₂×D₄	—	29	1	no	—
203	C₂×D₄	—	29	6	no	—
204	C₂×D₄	—	29	3	no	—
205	C₂×D₄	—	29	6	no	—
206	C₂×D₄	—	29	6	no	—
207	C₂×D₄	—	58	12	no	—
208	C₂×D₄	—	58	24	no	—
209	C₂×D₄	C₂²×D₂₉	1	3	yes	—
210	C₂²×C₄	—	29	6	no	—
211	C₂²×C₄	—	29	6	no	—
212	C₂²×C₄	—	29	6	no	—
213	C₂²×C₄	—	29	2	no	—
214	C₂²×C₄	—	29	6	no	—
215	C₂²×C₄	—	58	24	no	—
216	C₂²×C₄	C₂²×D₂₉	1	3	yes	—
217	C₂²×C₄	C₂²×D₂₉	1	3	yes	—
218	C₂²×C₄	C₂²×D₂₉	1	3	yes	—
219	C₄⋊C₄	—	29	1	no	—
220	C₄⋊C₄	—	29	3	no	—
221	C₄⋊C₄	—	29	6	no	—
222	C₄⋊C₄	—	29	6	no	—
223	C₄⋊C₄	C₂²×D₂₉	1	3	yes	—
224	C₄⋊C₄	C₂²×D₂₉	1	3	yes	—
225	C₂²⋊C₄	—	29	2	no	—
226	C₂²⋊C₄	—	29	6	no	—
227	C₂²⋊C₄	—	29	6	no	—
228	C₂²⋊C₄	—	29	6	no	—
229	C₂²⋊C₄	—	29	12	no	—
230	C₂²⋊C₄	C₂²×D₂₉	1	6	yes	—
231	C₄²	C₂²×D₂₉	1	3	yes	—
232	C₄²	C₂²×D₂₉	1	3	yes	—
233	C₄²	C₂²×D₂₉	1	1	yes	characteristic
234	C₄²	C₂²×D₂₉	1	3	yes	—
235	C₂³	—	29	2	no	—
236	C₂³	—	29	6	no	—
237	C₂³	—	58	12	no	—
238	C₂³	—	58	12	no	—
239	C₂³	—	116	6	no	—
240	C₂³	C₂³×D₂₉	1	3	yes	—
241	Q₈	—	58	12	no	—
242	Q₈	C₂×D₁₁₆	1	4	yes	complemented
243	D₄	—	29	4	no	—
244	D₄	—	58	12	no	—
245	D₄	—	58	24	no	—
246	D₄	—	58	12	no	—
247	D₄	—	58	6	no	—
248	D₄	—	116	6	no	—
249	D₄	C₂×D₁₁₆	1	12	yes	complemented
250	C₂×C₄	—	2	6	no	—
251	C₂×C₄	—	4	3	no	—
252	C₂×C₄	—	29	2	no	—
253	C₂×C₄	—	29	6	no	—
254	C₂×C₄	—	58	12	no	—
255	C₂×C₄	—	58	4	no	—
256	C₂×C₄	—	58	12	no	—
257	C₂×C₄	—	58	12	no	—
258	C₂×C₄	—	58	12	no	—
259	C₂×C₄	C₂×D₁₁₆	1	12	yes	complemented
260	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
261	C₂×C₄	C₂³×D₂₉	1	3	yes	—
262	C₂×C₄	C₂³×D₂₉	1	3	yes	—
263	C₂×C₄	C₂³×D₂₉	1	3	yes	—
264	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
265	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
266	C₂²	—	2	3	no	—
267	C₂²	—	2	3	no	—
268	C₂²	—	29	4	no	—
269	C₂²	—	58	2	no	—
270	C₂²	—	58	2	no	—
271	C₂²	—	58	6	no	—
272	C₂²	—	58	6	no	—
273	C₂²	—	58	6	no	—
274	C₂²	—	116	24	no	—
275	C₂²	C₂²×D₁₁₆	1	6	yes	—
276	C₂²	C₂⁴×D₂₉	1	1	yes	characteristic
277	C₄	—	2	3	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	3	no	—
280	C₄	—	58	2	no	—
281	C₄	—	58	6	no	—
282	C₄	C₂²×D₁₁₆	1	2	yes	—
283	C₄	C₂²×D₁₁₆	1	6	yes	—
284	C₄	#41×D₂₉	1	2	yes	complemented
285	C₂	—	2	6	no	—
286	C₂	—	58	4	no	—
287	C₂	—	116	6	no	—
288	C₂	C₂³×D₁₁₆	1	1	yes	characteristic
289	C₂	C₂×#41×D₂₉	1	1	yes	characteristic
290	C₂	#14271999	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₂₉
commutator, socle, cosocle	C₂²×C₂₉	C₂⁵
Frattini	C₂²	C₂⁴×D₂₉
Fitting	C₄×#41×C₂₉	C₂
hypercenter	C₄×#41	D₂₉
hypocenter	C₂₉	D₄×#41
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#41×D₂₉	C₂	semidirect
2. #14271835	C₂	semidirect
3. Q₈×D₁₁₆	C₂	semidirect
4. #14271825	C₂	semidirect
5. #14271817	C₂	semidirect
6. #14271808	C₂	semidirect
7. D₄×D₁₁₆	C₂	semidirect
8. #14271713	C₂	semidirect
9. #14271675	C₂	semidirect
10. C₂×C₄×D₁₁₆	C₂	semidirect
11. C₄×#41×C₂₉	C₂	semidirect
14. C₂×#90809	C₂²	semidirect
15. C₂×Q₈×D₂₉	C₂²	semidirect
16. C₂×#90807	C₂²	semidirect
17. C₂×D₄×D₂₉	C₂²	semidirect
18. C₂×#90805	C₂²	semidirect
20. C₂²×C₄×D₂₉	C₂²	semidirect
21. C₄×Q₈×C₂₉	C₂²	semidirect
22. C₄×Q₈×C₂₉	C₂²	semidirect
23. C₄×D₄×C₂₉	C₂²	semidirect
24. C₄×D₄×C₂₉	C₂²	semidirect
25. C₂₉×#117	C₂²	semidirect
26. C₂×C₄²×C₂₉	C₂²	semidirect
27. #11745292	C₂²	semidirect
28. #11745292	C₂²	semidirect
29. C₂×#90781	C₂²	semidirect
30. C₂×C₄⋊Dic₂₉	C₂²	semidirect
31. #11745259	C₂²	semidirect
32. #11745259	C₂²	semidirect
33. #11745257	C₂²	semidirect
34. #11745257	C₂²	semidirect
35. #11745248	C₂²	semidirect
36. #11745244	C₂²	semidirect
37. #11745237	C₂²	semidirect
38. #11745236	C₂²	semidirect
41. C₄×D₁₁₆	C₂²	semidirect
42. C₄×D₁₁₆	C₂²	semidirect
43. C₄×D₁₁₆	C₂²	semidirect
44. C₄×D₁₁₆	C₂²	semidirect
45. C₄×D₁₁₆	C₂²	semidirect
46. #11745231	C₂²	semidirect
48. #41×C₂₉	D₄	semidirect
95. Q₈×C₂₉	C₂×D₄	semidirect
96. D₄×C₂₉	C₂×D₄	semidirect
99. C₂×C₄×C₂₉	C₂×D₄	semidirect
114. D₁₁₆	#41	direct (N × H)
131. C₂×D₂₉	C₂×#41	semidirect
137. C₄×C₂₉	C₂×#41	semidirect
149. C₄×#41	D₂₉	semidirect
177. C₄×Q₈	C₂×D₂₉	semidirect
178. C₄×Q₈	C₂×D₂₉	semidirect
186. C₄×D₄	C₂×D₂₉	semidirect
187. C₄×D₄	C₂×D₂₉	semidirect
188. #117	C₂×D₂₉	semidirect
191. C₂×C₄²	C₂×D₂₉	semidirect
192. C₂₉	D₄×#41	semidirect
197. #41	D₁₁₆	direct (N × H)
242. Q₈	C₂×D₁₁₆	semidirect
249. D₄	C₂×D₁₁₆	semidirect
259. C₂×C₄	C₂×D₁₁₆	semidirect
284. C₄	#41×D₂₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×D₂₉	2
#14271835	2
Q₈×D₁₁₆	2
#14271825	2
#14271817	2
#14271808	2
D₄×D₁₁₆	2
#14271713	2
#14271675	2
C₂×C₄×D₁₁₆	2
C₄×#41×C₂₉	2
D₄×#41	29

Subgroups — #41×D116 #16832983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #41×D₁₁₆ #16832983