Subgroups — C₄×#11745271 #16828315

Order 3712.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11745271	C₂	1	1	yes	maximal, characteristic
2	C₄×#90805	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₅₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271271	C₂	1	2	yes	maximal, complemented
5	#14271265	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₁₁₆	C₂	1	2	yes	maximal, complemented
7	C₄×#90773	C₂	1	2	yes	maximal, complemented
8	#14271205	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271173	C₂	1	1	yes	maximal, characteristic, complemented
10	#14271165	C₂	1	1	yes	maximal, characteristic
11	C₄×M₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#90805	C₂²	1	1	yes	characteristic
13	C₂²×Dic₅₈	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
17	C₄×C₂²⋊D₂₉	—	2	1	no	—
18	C₄×C₂²⋊D₂₉	—	2	1	no	—
19	#11745288	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
21	C₂×#90779	—	2	1	no	—
22	C₂×C₄×Dic₂₉	—	2	1	no	—
23	#11745271	C₄	1	8	yes	direct product factor
24	C₂×Dic₁₁₆	C₂²	1	2	yes	—
25	C₂×#90773	C₂²	1	2	yes	—
26	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
27	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
28	#11745234	—	2	1	no	—
29	C₄²×D₂₉	—	2	1	no	—
30	C₄×Dic₅₈	—	2	1	no	—
31	C₄×Dic₅₈	—	2	1	no	—
32	C₄×Dic₅₈	C₂²	1	2	yes	—
33	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
34	C₄×Dic₅₈	C₂²	1	2	yes	—
35	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
36	C₂₉×#97	C₂²	1	2	yes	—
37	C₄×C₈×C₂₉	C₂²	1	2	yes	—
38	#11745169	C₂²	1	2	yes	—
39	#11745166	C₂²	1	2	yes	—
40	#11745165	C₂²	1	2	yes	—
41	#11745164	C₂²	1	2	yes	—
42	#11745164	C₂²	1	2	yes	—
43	#11745164	C₂²	1	2	yes	—
44	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₂₉	—	2	1	no	—
48	C₂²×Dic₂₉	—	2	1	no	—
49	C₂²×Dic₂₉	—	2	1	no	—
50	#90805	—	2	2	no	—
51	#90805	C₂×C₄	1	4	yes	—
52	C₂×D₁₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₂₉	—	2	1	no	—
54	C₂×C₄×D₂₉	—	2	1	no	—
55	C₂×C₄×D₂₉	—	2	1	no	—
56	C₂×Dic₅₈	—	2	1	no	—
57	C₂×Dic₅₈	—	2	2	no	—
58	C₂×Dic₅₈	C₂×C₄	1	4	yes	complemented
59	C₂×Dic₅₈	C₂³	1	2	yes	—
60	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
61	M₁₆×C₂₉	—	2	2	no	—
62	M₁₆×C₂₉	C₂×C₄	1	4	yes	complemented
63	C₂×C₈×C₂₉	C₂³	1	2	yes	—
64	C₂×C₈×C₂₉	C₂³	1	2	yes	—
65	C₄²×C₂₉	D₄	1	1	yes	characteristic
66	C₄²×C₂₉	D₄	1	1	yes	characteristic
67	C₄²×C₂₉	C₂³	1	1	yes	characteristic
68	C₄²×C₂₉	C₂³	1	1	yes	characteristic
69	C₂²⋊Dic₂₉	—	2	1	no	—
70	#90781	—	2	1	no	—
71	#90781	—	2	1	no	—
72	C₄⋊Dic₂₉	—	2	1	no	—
73	C₄⋊Dic₂₉	C₂³	1	2	yes	—
74	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
76	#90779	—	2	2	no	—
77	#90779	—	2	2	no	—
78	#90779	—	2	1	no	—
79	#90779	—	2	1	no	—
80	C₄×Dic₂₉	—	2	2	no	—
81	C₄×Dic₂₉	—	2	2	no	—
82	C₄×Dic₂₉	—	2	1	no	—
83	C₄×Dic₂₉	—	2	1	no	—
84	C₄×Dic₂₉	—	2	1	no	—
85	Dic₁₁₆	C₂×C₄	1	8	yes	complemented
86	#90773	C₂×C₄	1	8	yes	complemented
87	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×D₂₉	—	2	1	no	—
89	C₂×C₄×C₂₉	—	2	2	no	—
90	C₂×C₄×C₂₉	—	4	1	no	—
91	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₂₉	#41	1	2	yes	—
98	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
99	C₂²⋊D₂₉	—	2	4	no	—
100	C₂×Dic₂₉	—	2	4	no	—
101	C₂×Dic₂₉	—	2	2	no	—
102	C₂×Dic₂₉	—	2	1	no	—
103	C₂×Dic₂₉	—	2	2	no	—
104	C₂×Dic₂₉	—	2	1	no	—
105	C₂×Dic₂₉	—	2	1	no	—
106	C₂×Dic₂₉	—	4	2	no	—
107	D₁₁₆	—	2	1	no	—
108	D₁₁₆	C₂²×C₄	1	2	yes	—
109	C₄×D₂₉	—	2	4	no	—
110	C₄×D₂₉	—	4	2	no	—
111	C₄×D₂₉	—	4	2	no	—
112	Dic₅₈	—	2	4	no	—
113	Dic₅₈	—	2	2	no	—
114	Dic₅₈	—	2	1	no	—
115	Dic₅₈	C₂²×C₄	1	4	yes	—
116	Dic₅₈	C₂²×C₄	1	2	yes	—
117	C₈×C₂₉	—	2	2	no	—
118	C₈×C₂₉	C₂²×C₄	1	4	yes	—
119	C₄×#137	—	29	1	no	maximal, Sylow
120	C₂²×C₂₉	—	2	1	no	—
121	C₂²×C₂₉	—	2	1	no	—
122	C₂²×C₂₉	C₄×D₄	1	2	yes	—
123	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×D₂₉	—	2	2	no	—
125	C₂×D₂₉	—	4	1	no	—
126	C₂×D₂₉	—	4	1	no	—
127	C₄×C₂₉	—	2	1	no	—
128	C₄×C₂₉	—	2	1	no	—
129	C₄×C₂₉	—	2	1	no	—
130	C₄×C₂₉	C₄×D₄	1	2	yes	—
131	C₄×C₂₉	#137	1	2	yes	complemented
132	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
133	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
134	Dic₂₉	—	2	4	no	—
135	Dic₂₉	—	2	2	no	—
136	Dic₂₉	—	4	1	no	—
137	Dic₂₉	—	4	1	no	—
138	Dic₂₉	—	4	2	no	—
139	C₂×#137	—	29	1	no	—
140	C₄×#41	—	29	1	no	—
141	C₂×C₄×Q₈	—	29	1	no	—
142	#441	—	29	2	no	—
143	#440	—	29	2	no	—
144	C₄×Q₁₆	—	29	2	no	—
145	C₄×SD₁₆	—	29	2	no	—
146	#428	—	29	1	no	—
147	#419	—	29	1	no	—
148	#417	—	29	1	no	—
149	C₄×M₁₆	D₂₉	1	1	yes	characteristic, complemented
150	C₂×C₂₉	—	2	2	no	—
151	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂×C₂₉	C₂×#137	1	1	yes	characteristic
153	C₂×C₂₉	#575	1	1	yes	characteristic
154	D₂₉	—	4	2	no	—
155	C₂×#41	—	29	1	no	—
156	C₂²×Q₈	—	29	1	no	—
157	#137	—	29	8	no	—
158	C₂×Q₁₆	—	29	2	no	—
159	C₂×SD₁₆	—	29	2	no	—
160	C₂×M₁₆	C₂×D₂₉	1	1	yes	characteristic
161	C₂×M₁₆	C₂×D₂₉	1	1	yes	characteristic
162	C₄×Q₈	—	29	2	no	—
163	C₄×Q₈	—	29	2	no	—
164	C₄×Q₈	—	29	1	no	—
165	C₄×Q₈	—	29	1	no	—
166	C₄×Q₈	—	58	1	no	—
167	C₄×Q₈	—	58	1	no	—
168	C₄×D₄	—	29	1	no	—
169	C₄×D₄	—	29	1	no	—
170	C₄×D₄	—	58	1	no	—
171	C₄×D₄	—	58	1	no	—
172	#117	—	29	1	no	—
173	#117	—	58	1	no	—
174	C₂×C₄⋊C₄	—	29	1	no	—
175	C₂×C₄⋊C₄	—	58	1	no	—
176	C₂×C₄²	—	58	1	no	—
177	C₂×C₄²	—	58	1	no	—
178	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
179	#107	—	29	2	no	—
180	#106	—	29	2	no	—
181	#103	—	29	2	no	—
182	#103	—	29	2	no	—
183	#103	—	29	2	no	—
184	#102	—	29	2	no	—
185	#97	C₂×D₂₉	1	2	yes	—
186	C₄×C₈	C₂×D₂₉	1	2	yes	—
187	C₂₉	C₄×#137	1	1	yes	characteristic, complemented, Sylow
188	#41	—	29	4	no	—
189	#41	—	58	2	no	—
190	C₂×Q₈	—	29	4	no	—
191	C₂×Q₈	—	29	2	no	—
192	C₂×Q₈	—	29	1	no	—
193	C₂×Q₈	—	58	1	no	—
194	C₂×Q₈	—	58	2	no	—
195	C₂×D₄	—	29	1	no	—
196	C₂×D₄	—	58	1	no	—
197	C₂²×C₄	—	58	1	no	—
198	C₂²×C₄	—	58	1	no	—
199	C₂²×C₄	—	58	1	no	—
200	C₂²×C₄	—	58	1	no	—
201	C₂²×C₄	—	58	1	no	—
202	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
203	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
205	Q₁₆	—	29	8	no	—
206	SD₁₆	—	29	8	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	C₄×D₂₉	1	4	yes	complemented
209	C₂×C₈	C₂²×D₂₉	1	2	yes	—
210	C₂×C₈	C₂²×D₂₉	1	2	yes	—
211	C₄⋊C₄	—	29	2	no	—
212	C₄⋊C₄	—	29	1	no	—
213	C₄⋊C₄	—	29	1	no	—
214	C₄⋊C₄	—	58	1	no	—
215	C₄⋊C₄	—	58	2	no	—
216	C₄⋊C₄	—	58	2	no	—
217	C₄⋊C₄	—	58	1	no	—
218	C₄⋊C₄	—	58	1	no	—
219	C₂²⋊C₄	—	58	1	no	—
220	C₂²⋊C₄	—	58	1	no	—
221	C₂²⋊C₄	—	58	1	no	—
222	C₄²	—	58	1	no	—
223	C₄²	—	58	1	no	—
224	C₄²	—	58	2	no	—
225	C₄²	—	58	1	no	—
226	C₄²	—	58	2	no	—
227	C₄²	D₁₁₆	1	1	yes	characteristic
228	C₄²	D₁₁₆	1	1	yes	characteristic
229	C₄²	C₂²×D₂₉	1	1	yes	characteristic
230	C₄²	C₂²×D₂₉	1	1	yes	characteristic
231	C₂³	—	58	1	no	—
232	C₂³	C₂×D₁₁₆	1	1	yes	characteristic
233	Q₈	—	29	4	no	—
234	Q₈	—	29	2	no	—
235	Q₈	—	58	2	no	—
236	Q₈	—	58	1	no	—
237	Q₈	—	58	4	no	—
238	D₄	—	29	2	no	—
239	D₄	—	58	4	no	—
240	D₄	—	58	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	58	1	no	—
244	C₂×C₄	—	58	2	no	—
245	C₂×C₄	—	58	4	no	—
246	C₂×C₄	—	58	4	no	—
247	C₂×C₄	—	58	1	no	—
248	C₂×C₄	—	58	1	no	—
249	C₂×C₄	—	58	2	no	—
250	C₂×C₄	—	116	2	no	—
251	C₂×C₄	—	116	2	no	—
252	C₂×C₄	—	116	2	no	—
253	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
254	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
255	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
256	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
257	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
258	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
259	C₂×C₄	#90805	1	2	yes	—
260	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	C₂×C₄×D₂₉	1	4	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	58	2	no	—
266	C₂²	—	116	1	no	—
267	C₂²	—	116	1	no	—
268	C₂²	C₄×D₁₁₆	1	2	yes	—
269	C₂²	C₂²×D₁₁₆	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	58	2	no	—
274	C₄	—	58	4	no	—
275	C₄	—	116	1	no	—
276	C₄	—	116	1	no	—
277	C₄	—	116	2	no	—
278	C₄	C₄×D₁₁₆	1	2	yes	—
279	C₄	#11745271	1	2	yes	direct product factor
280	C₄	C₂²×C₄×D₂₉	1	1	yes	characteristic
281	C₄	C₂×#90805	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	116	2	no	—
284	C₂	C₂×C₄×D₁₁₆	1	1	yes	characteristic
285	C₂	C₂×#11745271	1	1	yes	characteristic
286	C₂	#14271963	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₁₆
commutator	C₄×C₂₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₄×M₁₆×C₂₉	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂×#90805	C₂²
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₄×M₁₆	D₂₉
hypocenter	C₂₉	C₄×#137
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Dic₅₈	C₂	semidirect
4. #14271271	C₂	semidirect
5. #14271265	C₂	semidirect
6. C₄×Dic₁₁₆	C₂	semidirect
7. C₄×#90773	C₂	semidirect
8. #14271205	C₂	semidirect
9. #14271173	C₂	semidirect
11. C₄×M₁₆×C₂₉	C₂	semidirect
23. #11745271	C₄	direct (N × H)
58. C₂×Dic₅₈	C₂×C₄	semidirect
62. M₁₆×C₂₉	C₂×C₄	semidirect
85. Dic₁₁₆	C₂×C₄	semidirect
86. #90773	C₂×C₄	semidirect
131. C₄×C₂₉	#137	semidirect
149. C₄×M₁₆	D₂₉	semidirect
187. C₂₉	C₄×#137	semidirect
208. M₁₆	C₄×D₂₉	semidirect
279. C₄	#11745271	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×#11745271	2
C₄×#90805	2
C₂×C₄×Dic₅₈	2
#14271271	2
#14271265	2
C₄×Dic₁₁₆	2
C₄×#90773	2
#14271205	2
#14271173	2
#14271165	2
C₄×M₁₆×C₂₉	2
C₄×#137	29

Subgroups — C4×#11745271 #16828315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11745271 #16828315