Subgroups — C₄×#11745294 #16828300

Order 3712.

548 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11745294	C₂	1	1	yes	maximal, characteristic
2	C₄×#90805	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271273	C₂	1	2	yes	maximal, complemented
4	#14271263	C₂	1	2	yes	maximal, complemented
5	C₄×#90783	C₂	1	2	yes	maximal, complemented
6	C₄×#90782	C₂	1	2	yes	maximal, complemented
7	#14271203	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271168	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271162	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#90777	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₂₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₉	—	2	1	no	—
14	C₂×#90805	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₂₉	—	2	1	no	—
16	C₄×D₄×C₂₉	—	2	1	no	—
17	C₄×D₄×C₂₉	C₂²	1	2	yes	—
18	C₄×D₄×C₂₉	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₉	—	2	1	no	—
21	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
22	C₂×#90783	C₂²	1	2	yes	—
23	#11745294	C₄	1	8	yes	direct product factor
24	C₂×#90782	C₂²	1	2	yes	—
25	C₄×C₂²⋊D₂₉	—	2	1	no	—
26	C₄×C₂²⋊D₂₉	—	2	1	no	—
27	#11745288	C₂²	1	1	yes	characteristic
28	C₂×#90777	C₂²	1	1	yes	characteristic
29	C₂×#90777	C₂²	1	1	yes	characteristic
30	C₄×D₁₁₆	C₂²	1	1	yes	characteristic, complemented
31	C₄×D₁₁₆	C₂²	1	1	yes	characteristic, complemented
32	#11745234	—	2	1	no	—
33	C₄²×D₂₉	—	2	1	no	—
34	C₄×Dic₅₈	C₂²	1	1	yes	characteristic, complemented
35	C₄×Dic₅₈	C₂²	1	1	yes	characteristic, complemented
36	#11745180	C₂²	1	2	yes	—
37	#11745180	C₂²	1	2	yes	—
38	#11745158	C₂²	1	2	yes	complemented
39	#11745157	C₂²	1	2	yes	complemented
40	#11745156	C₂²	1	2	yes	complemented
41	#11745155	C₂²	1	2	yes	complemented
42	#11745151	C₂²	1	2	yes	complemented
43	C₄×#6597	C₂²	1	2	yes	complemented
44	C₂⁴×C₂₉	—	2	1	no	—
45	C₂×D₄×C₂₉	—	2	1	no	—
46	C₂×D₄×C₂₉	—	2	2	no	—
47	C₂×D₄×C₂₉	C₂×C₄	1	4	yes	complemented
48	C₂×D₄×C₂₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₉	—	2	1	no	—
50	C₂²×C₄×C₂₉	—	2	2	no	—
51	C₂²×C₄×C₂₉	—	2	2	no	—
52	C₂²×C₄×C₂₉	—	4	4	no	—
53	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
56	C₂×C₂²⋊D₂₉	—	2	1	no	—
57	#90805	—	2	2	no	—
58	#90805	C₂×C₄	1	4	yes	complemented
59	C₂×D₁₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₄×D₂₉	—	2	1	no	—
61	C₂×C₄×D₂₉	—	2	1	no	—
62	C₂×C₄×D₂₉	—	2	1	no	—
63	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₂₉	—	2	1	no	—
65	C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
66	C₂²⋊C₄×C₂₉	—	2	2	no	—
67	C₂²⋊C₄×C₂₉	—	2	2	no	—
68	C₄²×C₂₉	D₄	1	1	yes	characteristic
69	C₄²×C₂₉	D₄	1	1	yes	characteristic
70	C₄²×C₂₉	C₂³	1	1	yes	characteristic
71	C₄²×C₂₉	C₂³	1	1	yes	characteristic
72	C₂²⋊Dic₂₉	—	2	1	no	—
73	#90783	C₂×C₄	1	8	yes	complemented
74	#90782	C₂×C₄	1	8	yes	complemented
75	#90781	—	2	1	no	—
76	#90781	—	2	1	no	—
77	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
79	#90779	—	2	1	no	—
80	#90779	—	2	1	no	—
81	C₄×Dic₂₉	—	2	1	no	—
82	C₄×Dic₂₉	—	2	1	no	—
83	C₄×Dic₂₉	—	2	1	no	—
84	#90777	—	2	2	no	—
85	#90777	C₂×C₄	1	4	yes	complemented
86	C₂×#6597	C₂³	1	2	yes	—
87	C₂×#6597	C₂³	1	2	yes	—
88	C₂³×C₂₉	—	2	4	no	—
89	C₂³×C₂₉	—	2	2	no	—
90	C₂³×C₂₉	—	4	2	no	—
91	C₂³×C₂₉	—	4	2	no	—
92	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×D₂₉	—	2	1	no	—
94	D₄×C₂₉	—	2	4	no	—
95	D₄×C₂₉	—	2	2	no	—
96	D₄×C₂₉	C₂²×C₄	1	4	yes	—
97	C₂×C₄×C₂₉	—	2	2	no	—
98	C₂×C₄×C₂₉	—	2	2	no	—
99	C₂×C₄×C₂₉	—	4	1	no	—
100	C₂×C₄×C₂₉	—	4	4	no	—
101	C₂×C₄×C₂₉	—	4	2	no	—
102	C₂×C₄×C₂₉	—	4	4	no	—
103	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₂₉	#41	1	2	yes	—
110	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
111	C₂²⋊D₂₉	—	2	4	no	—
112	C₂×Dic₂₉	—	2	1	no	—
113	C₂×Dic₂₉	—	2	1	no	—
114	C₂×Dic₂₉	—	2	1	no	—
115	D₁₁₆	—	2	1	no	—
116	D₁₁₆	C₂²×C₄	1	2	yes	complemented
117	C₄×D₂₉	—	2	4	no	—
118	C₄×D₂₉	—	4	2	no	—
119	C₄×D₂₉	—	4	2	no	—
120	Dic₅₈	—	2	1	no	—
121	Dic₅₈	C₂²×C₄	1	2	yes	complemented
122	#6597	—	2	2	no	—
123	#6597	C₂²×C₄	1	4	yes	complemented
124	C₄×AGL(1,ℤ/8)	—	29	1	no	maximal, Sylow
125	C₂²×C₂₉	—	2	1	no	—
126	C₂²×C₂₉	—	2	1	no	—
127	C₂²×C₂₉	—	2	4	no	—
128	C₂²×C₂₉	—	4	8	no	—
129	C₂²×C₂₉	—	4	2	no	—
130	C₂²×C₂₉	—	4	2	no	—
131	C₂²×C₂₉	C₄×D₄	1	2	yes	—
132	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
133	C₂×D₂₉	—	2	2	no	—
134	C₂×D₂₉	—	4	1	no	—
135	C₂×D₂₉	—	4	1	no	—
136	C₄×C₂₉	—	2	1	no	—
137	C₄×C₂₉	—	2	1	no	—
138	C₄×C₂₉	—	2	1	no	—
139	C₄×C₂₉	—	4	2	no	—
140	C₄×C₂₉	C₄×D₄	1	2	yes	—
141	C₄×C₂₉	AGL(1,ℤ/8)	1	2	yes	complemented
142	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
143	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
144	Dic₂₉	—	2	2	no	—
145	Dic₂₉	—	4	1	no	—
146	Dic₂₉	—	4	1	no	—
147	C₂×AGL(1,ℤ/8)	—	29	1	no	—
148	C₄×#41	—	29	1	no	—
149	C₂×C₄×D₄	D₂₉	1	1	yes	characteristic, complemented
150	#442	—	29	2	no	—
151	#440	—	29	2	no	—
152	C₄×SD₁₆	—	29	2	no	—
153	C₄×D₈	—	29	2	no	—
154	#428	—	29	1	no	—
155	#418	—	29	1	no	—
156	#417	—	29	1	no	—
157	C₄×M₁₆	—	29	1	no	—
158	C₂×C₂₉	—	2	2	no	—
159	C₂×C₂₉	—	4	4	no	—
160	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
161	C₂×C₂₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂×C₂₉	#575	1	1	yes	characteristic
163	D₂₉	—	4	2	no	—
164	C₂×#41	—	29	1	no	—
165	C₂²×D₄	C₂×D₂₉	1	1	yes	characteristic
166	C₂³×C₄	—	2	1	no	—
167	AGL(1,ℤ/8)	—	29	8	no	—
168	C₂×SD₁₆	—	29	2	no	—
169	C₂×D₈	—	29	2	no	—
170	C₂×M₁₆	—	29	1	no	—
171	C₂×M₁₆	—	29	1	no	—
172	C₄×Q₈	—	29	1	no	—
173	C₄×Q₈	—	29	1	no	—
174	C₄×D₄	—	2	1	no	—
175	C₄×D₄	—	2	1	no	—
176	C₄×D₄	—	29	1	no	—
177	C₄×D₄	—	29	1	no	—
178	C₄×D₄	—	58	1	no	—
179	C₄×D₄	—	58	1	no	—
180	C₄×D₄	C₂×D₂₉	1	2	yes	—
181	C₄×D₄	C₂×D₂₉	1	2	yes	—
182	#117	—	29	1	no	—
183	#117	—	58	1	no	—
184	C₂×C₄⋊C₄	C₂×D₂₉	1	1	yes	characteristic
185	C₂×C₂²⋊C₄	—	2	1	no	—
186	C₂×C₄²	—	58	1	no	—
187	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
188	#107	—	29	2	no	—
189	#106	—	29	2	no	—
190	#103	—	29	2	no	—
191	#102	—	29	2	no	—
192	#102	—	29	2	no	—
193	#102	—	29	2	no	—
194	#97	—	29	2	no	—
195	C₄×C₈	—	29	2	no	—
196	C₂₉	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
197	C₂⁴	—	2	1	no	—
198	#41	—	29	4	no	—
199	#41	—	58	2	no	—
200	C₂×Q₈	—	29	1	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	29	1	no	—
204	C₂×D₄	—	58	1	no	—
205	C₂×D₄	C₄×D₂₉	1	4	yes	complemented
206	C₂×D₄	C₂²×D₂₉	1	2	yes	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	58	1	no	—
212	C₂²×C₄	—	58	1	no	—
213	C₂²×C₄	—	58	1	no	—
214	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
215	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
216	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
217	SD₁₆	—	29	8	no	—
218	D₈	—	29	8	no	—
219	M₁₆	—	29	4	no	—
220	M₁₆	—	58	2	no	—
221	C₂×C₈	—	29	2	no	—
222	C₂×C₈	—	29	2	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	29	1	no	—
225	C₄⋊C₄	—	29	1	no	—
226	C₄⋊C₄	—	58	1	no	—
227	C₄⋊C₄	—	58	1	no	—
228	C₄⋊C₄	C₂²×D₂₉	1	2	yes	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	58	1	no	—
232	C₂²⋊C₄	—	58	1	no	—
233	C₂²⋊C₄	—	58	1	no	—
234	C₄²	—	58	1	no	—
235	C₄²	—	58	1	no	—
236	C₄²	—	58	1	no	—
237	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
238	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
239	C₄²	C₂²×D₂₉	1	1	yes	characteristic
240	C₄²	C₂²×D₂₉	1	1	yes	characteristic
241	C₂³	—	2	4	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	58	1	no	—
246	C₂³	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
247	Q₈	—	29	2	no	—
248	Q₈	—	58	1	no	—
249	D₄	—	2	4	no	—
250	D₄	—	2	2	no	—
251	D₄	—	29	2	no	—
252	D₄	—	58	4	no	—
253	D₄	—	58	1	no	—
254	D₄	C₂×C₄×D₂₉	1	4	yes	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	58	1	no	—
262	C₂×C₄	—	58	4	no	—
263	C₂×C₄	—	58	1	no	—
264	C₂×C₄	—	58	1	no	—
265	C₂×C₄	—	116	2	no	—
266	C₂×C₄	—	116	2	no	—
267	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
268	C₂×C₄	#90805	1	2	yes	—
269	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
271	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
272	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
273	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
274	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
275	C₈	—	29	4	no	—
276	C₈	—	58	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	4	8	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	58	2	no	—
284	C₂²	—	116	1	no	—
285	C₂²	—	116	1	no	—
286	C₂²	C₄×C₂²⋊D₂₉	1	2	yes	—
287	C₂²	C₂²×C₂²⋊D₂₉	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	4	2	no	—
292	C₄	—	58	2	no	—
293	C₄	—	116	1	no	—
294	C₄	—	116	1	no	—
295	C₄	C₄×C₂²⋊D₂₉	1	2	yes	—
296	C₄	#11745294	1	2	yes	direct product factor
297	C₄	C₂²×C₄×D₂₉	1	1	yes	characteristic
298	C₄	C₂×#90805	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	4	4	no	—
301	C₂	—	116	2	no	—
302	C₂	C₂×C₄×C₂²⋊D₂₉	1	1	yes	characteristic
303	C₂	C₂×#11745294	1	1	yes	characteristic
304	C₂	#14271961	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₉
commutator	C₄×C₂₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂×C₄×D₄×C₂₉	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂×#11745294	C₂
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₂×C₄×D₄	D₂₉
hypocenter	C₂₉	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₄×#90805	C₂	semidirect
3. #14271273	C₂	semidirect
4. #14271263	C₂	semidirect
5. C₄×#90783	C₂	semidirect
6. C₄×#90782	C₂	semidirect
7. #14271203	C₂	semidirect
8. #14271168	C₂	semidirect
9. #14271162	C₂	semidirect
10. C₄×#90777	C₂	semidirect
11. C₂×C₄×D₄×C₂₉	C₂	semidirect
23. #11745294	C₄	direct (N × H)
30. C₄×D₁₁₆	C₂²	semidirect
31. C₄×D₁₁₆	C₂²	semidirect
34. C₄×Dic₅₈	C₂²	semidirect
35. C₄×Dic₅₈	C₂²	semidirect
38. #11745158	C₂²	semidirect
39. #11745157	C₂²	semidirect
40. #11745156	C₂²	semidirect
41. #11745155	C₂²	semidirect
42. #11745151	C₂²	semidirect
43. C₄×#6597	C₂²	semidirect
47. C₂×D₄×C₂₉	C₂×C₄	semidirect
58. #90805	C₂×C₄	semidirect
73. #90783	C₂×C₄	semidirect
74. #90782	C₂×C₄	semidirect
85. #90777	C₂×C₄	semidirect
116. D₁₁₆	C₂²×C₄	semidirect
121. Dic₅₈	C₂²×C₄	semidirect
123. #6597	C₂²×C₄	semidirect
141. C₄×C₂₉	AGL(1,ℤ/8)	semidirect
149. C₂×C₄×D₄	D₂₉	semidirect
196. C₂₉	C₄×AGL(1,ℤ/8)	semidirect
205. C₂×D₄	C₄×D₂₉	semidirect
296. C₄	#11745294	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11745294	2
C₄×#90805	2
#14271273	2
#14271263	2
C₄×#90783	2
C₄×#90782	2
#14271203	2
#14271168	2
#14271162	2
C₄×#90777	2
C₂×C₄×D₄×C₂₉	2
C₄×AGL(1,ℤ/8)	29

Subgroups — C4×#11745294 #16828300

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11745294 #16828300