Subgroups — C₄×#17986415 #32150094

Order 18176.

834 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986415	C₂	1	2	yes	maximal
2	C₄×#14556134	C₂	1	1	yes	maximal, characteristic
3	#22589936	C₂	1	2	yes	maximal, complemented
4	#22589935	C₂	1	4	yes	maximal, complemented
5	C₄×#14556068	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14556067	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₆×D₇₁	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556134	C₄	1	1	yes	characteristic
10	C₂×#14556134	C₂²	1	1	yes	characteristic
11	C₄×#11765865	C₄	1	1	yes	characteristic
12	#17986415	C₄	1	16	yes	direct product factor
13	C₂×#14556068	C₂²	1	2	yes	—
14	C₂×#14556067	C₂²	1	4	yes	—
15	C₂×C₁₆×D₇₁	C₂²	1	4	yes	—
16	#17985954	C₄	1	1	yes	characteristic
17	#17985954	C₂²	1	1	yes	characteristic
18	#17985953	C₄	1	2	yes	—
19	#17985953	C₂²	1	2	yes	—
20	C₄×#11765849	C₄	1	1	yes	characteristic
21	C₄×#11765848	C₄	1	2	yes	—
22	C₄×C₈×D₇₁	C₂²	1	2	yes	—
23	#17985775	C₂²	1	2	yes	—
24	#17985774	C₂²	1	4	yes	—
25	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
26	C₁₆×Dic₇₁	C₂²	1	4	yes	—
27	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
29	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
30	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
31	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
32	C₂×#11765865	C₂×C₄	1	1	yes	characteristic
33	#14556134	C₈	1	8	yes	—
34	#14556134	C₂×C₄	1	8	yes	—
35	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
36	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
37	C₂×#11765848	C₂×C₄	1	2	yes	—
38	C₂×#11765848	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₇₁	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₇₁	C₂³	1	2	yes	—
41	C₄×Dic₁₄₂	C₈	1	2	yes	—
42	C₄×D₂₈₄	C₈	1	2	yes	—
43	C₄×C₂²⋊D₇₁	C₈	1	4	yes	—
44	#14556087	C₂×C₄	1	1	yes	characteristic
45	#14556085	C₂×C₄	1	2	yes	—
46	C₄²×D₇₁	C₂×C₄	1	2	yes	—
47	#14556068	C₂×C₄	1	8	yes	complemented
48	#14556067	C₂×C₄	1	16	yes	complemented
49	C₂×C₇₁⋊C₁₆	C₂³	1	4	yes	—
50	C₁₆×D₇₁	C₂×C₄	1	16	yes	complemented
51	#14556039	C₂×C₄	1	2	yes	—
52	#14556038	C₂×C₄	1	2	yes	—
53	#14556038	C₂×C₄	1	2	yes	—
54	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
55	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
56	C₈×Dic₇₁	C₂³	1	2	yes	—
57	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₁₆×C₇₁	C₂×C₄	1	8	yes	complemented
61	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
62	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
63	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
67	#11765865	C₄²	1	4	yes	—
68	#11765865	C₂×C₈	1	4	yes	—
69	C₂×Dic₁₄₂	C₂×C₈	1	1	yes	characteristic
70	C₂×D₂₈₄	C₂×C₈	1	1	yes	characteristic
71	C₂×C₂²⋊D₇₁	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₇₁	C₂×C₈	1	4	yes	—
73	C₂×C₄×D₇₁	C₂²×C₄	1	2	yes	—
74	#11765849	C₄²	1	4	yes	—
75	#11765849	C₂×C₈	1	4	yes	—
76	#11765848	C₄²	1	8	yes	—
77	#11765848	C₂×C₈	1	8	yes	—
78	C₂×C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
79	C₂×C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
80	C₈×D₇₁	C₂×C₈	1	8	yes	—
81	C₈×D₇₁	C₂²×C₄	1	8	yes	—
82	C₄⋊Dic₇₁	C₂×C₈	1	2	yes	—
83	Dic₇₁⋊C₄	C₂×C₈	1	4	yes	—
84	C₂²⋊Dic₇₁	C₂×C₈	1	2	yes	—
85	(C₂×D₇₁)⋊C₄	C₂×C₈	1	4	yes	—
86	C₄×Dic₇₁	C₂×C₈	1	4	yes	—
87	C₄×Dic₇₁	C₂²×C₄	1	2	yes	—
88	C₇₁⋊C₁₆	C₂²×C₄	1	8	yes	—
89	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
90	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
92	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₇₁	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
97	C₄²×C₇₁	C₂×C₈	1	2	yes	—
98	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
100	C₁₆×C₇₁	C₂²×C₄	1	4	yes	—
101	C₁₆×C₇₁	C₂²×C₄	1	4	yes	—
102	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₇₁	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₇₁	—	2	4	no	—
105	C₂×C₄×C₇₁	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
111	C₂²⋊D₇₁	C₄×C₈	1	8	yes	—
112	C₂×Dic₇₁	C₂²×C₈	1	4	yes	—
113	C₂×Dic₇₁	C₂²×C₈	1	2	yes	—
114	D₂₈₄	C₄×C₈	1	4	yes	—
115	C₄×D₇₁	—	2	8	no	—
116	C₄×D₇₁	C₂×C₄²	1	4	yes	—
117	C₄×D₇₁	C₂²×C₈	1	4	yes	—
118	Dic₁₄₂	C₄×C₈	1	4	yes	—
119	C₈×C₇₁	C₂×C₄²	1	2	yes	—
120	C₈×C₇₁	C₂²×C₈	1	2	yes	—
121	C₈×C₇₁	C₂²×C₈	1	2	yes	—
122	C₈×C₇₁	C₂³×C₄	1	2	yes	—
123	C₇₁⋊C₈	C₂×C₄²	1	4	yes	—
124	C₇₁⋊C₈	C₂²×C₈	1	4	yes	—
125	C₂²×C₇₁	—	2	2	no	—
126	C₂²×C₇₁	C₂×C₄×C₈	1	2	yes	—
127	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
128	C₂×D₇₁	—	2	4	no	—
129	C₂×D₇₁	C₂×C₄×C₈	1	4	yes	—
130	C₄×C₇₁	—	2	2	no	—
131	C₄×C₇₁	C₂×C₄×C₈	1	2	yes	—
132	C₄×C₇₁	#504	1	4	yes	complemented
133	C₄×C₇₁	C₂²×C₄²	1	1	yes	characteristic
134	C₄×C₇₁	C₂³×C₈	1	1	yes	characteristic
135	Dic₇₁	—	2	4	no	—
136	Dic₇₁	C₂×C₄×C₈	1	4	yes	—
137	C₄×#504	—	71	1	no	maximal, Sylow
138	C₂×C₇₁	—	2	2	no	—
139	C₂×C₇₁	C₂²×C₄×C₈	1	1	yes	characteristic
140	C₂×C₇₁	C₂×#504	1	2	yes	—
141	D₇₁	—	2	4	no	—
142	C₂×#504	—	71	2	no	—
143	C₄×#131	—	71	1	no	—
144	#2108	—	71	2	no	—
145	#2108	—	71	4	no	—
146	C₄×M₃₂	—	71	1	no	—
147	C₄×M₃₂	—	71	2	no	—
148	C₂×C₄×C₁₆	—	71	2	no	—
149	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic, complemented
150	C₇₁	C₄×#504	1	1	yes	characteristic, complemented, Sylow
151	C₂×#131	—	71	1	no	—
152	C₂×#131	—	71	1	no	—
153	C₄×#41	—	71	1	no	—
154	#504	—	71	16	no	—
155	C₂×M₃₂	—	71	2	no	—
156	C₂×M₃₂	—	71	4	no	—
157	C₂²×C₁₆	—	71	4	no	—
158	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
159	#405	—	71	1	no	—
160	#405	—	71	1	no	—
161	#405	—	71	2	no	—
162	#405	—	71	2	no	—
163	C₄×M₁₆	—	71	1	no	—
164	C₄×M₁₆	—	71	2	no	—
165	C₂×C₄×C₈	—	71	2	no	—
166	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
167	#346	—	71	4	no	—
168	#346	—	71	2	no	—
169	C₄×C₁₆	—	71	4	no	—
170	C₄×C₁₆	—	71	2	no	—
171	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
172	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
173	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
174	C₂×#41	—	71	1	no	—
175	#131	—	71	8	no	—
176	#131	—	71	8	no	—
177	C₂×M₁₆	—	71	1	no	—
178	C₂×M₁₆	—	71	1	no	—
179	C₂×M₁₆	—	71	2	no	—
180	C₂×M₁₆	—	71	2	no	—
181	C₂²×C₈	—	71	2	no	—
182	C₂²×C₈	—	71	2	no	—
183	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
185	C₄×Q₈	—	71	2	no	—
186	C₄×D₄	—	71	4	no	—
187	C₄×D₄	—	71	2	no	—
188	#117	—	71	1	no	—
189	#117	—	71	2	no	—
190	C₂×C₄²	—	71	2	no	—
191	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
192	M₃₂	—	71	8	no	—
193	M₃₂	—	71	16	no	—
194	C₂×C₁₆	—	71	4	no	—
195	C₂×C₁₆	—	71	16	no	—
196	C₂×C₁₆	C₄×D₇₁	1	8	yes	complemented
197	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
198	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
199	#97	—	71	2	no	—
200	#97	—	71	2	no	—
201	#97	—	71	2	no	—
202	C₄×C₈	—	71	2	no	—
203	C₄×C₈	—	71	2	no	—
204	C₄×C₈	—	71	2	no	—
205	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
206	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
209	#41	—	71	4	no	—
210	#41	—	71	4	no	—
211	C₂×Q₈	—	71	1	no	—
212	C₂×D₄	—	71	2	no	—
213	C₂×D₄	—	71	1	no	—
214	C₂²×C₄	—	71	2	no	—
215	C₂²×C₄	—	71	4	no	—
216	C₂²×C₄	C₈×D₇₁	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
218	M₁₆	—	71	4	no	—
219	M₁₆	—	71	4	no	—
220	M₁₆	—	71	8	no	—
221	M₁₆	—	71	8	no	—
222	C₂×C₈	—	71	2	no	—
223	C₂×C₈	—	71	2	no	—
224	C₂×C₈	—	71	8	no	—
225	C₂×C₈	—	71	8	no	—
226	C₂×C₈	C₈×D₇₁	1	4	yes	—
227	C₂×C₈	C₂×C₄×D₇₁	1	4	yes	—
228	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
231	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
232	C₄⋊C₄	—	71	4	no	—
233	C₄⋊C₄	—	71	2	no	—
234	C₂²⋊C₄	—	71	2	no	—
235	C₂²⋊C₄	—	71	4	no	—
236	C₄²	—	71	2	no	—
237	C₄²	—	71	4	no	—
238	C₄²	C₈×D₇₁	1	2	yes	—
239	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
240	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
241	C₁₆	—	71	8	no	—
242	C₁₆	C₂×C₄×D₇₁	1	4	yes	—
243	C₁₆	C₂×C₄×D₇₁	1	4	yes	—
244	C₂³	—	71	2	no	—
245	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
246	Q₈	—	71	4	no	—
247	D₄	—	71	8	no	—
248	D₄	—	71	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	71	2	no	—
251	C₂×C₄	—	71	4	no	—
252	C₂×C₄	—	71	4	no	—
253	C₂×C₄	—	71	4	no	—
254	C₂×C₄	—	142	8	no	—
255	C₂×C₄	C₄²×D₇₁	1	2	yes	—
256	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
257	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
258	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
259	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
260	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
261	C₈	—	71	4	no	—
262	C₈	—	71	4	no	—
263	C₈	C₄²×D₇₁	1	2	yes	—
264	C₈	C₂×C₈×D₇₁	1	2	yes	—
265	C₈	C₂×C₈×D₇₁	1	2	yes	—
266	C₈	C₂²×C₄×D₇₁	1	2	yes	—
267	C₂²	—	2	2	no	—
268	C₂²	—	71	4	no	—
269	C₂²	—	142	4	no	—
270	C₂²	C₄×C₈×D₇₁	1	2	yes	—
271	C₂²	C₂²×C₈×D₇₁	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	71	4	no	—
274	C₄	—	142	4	no	—
275	C₄	C₄×C₈×D₇₁	1	2	yes	—
276	C₄	#17986415	1	4	yes	direct product factor
277	C₄	C₂×C₄²×D₇₁	1	1	yes	characteristic
278	C₄	C₂²×C₈×D₇₁	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	142	4	no	—
281	C₂	C₂×C₄×C₈×D₇₁	1	1	yes	characteristic
282	C₂	C₂×#17986415	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆	C₂×D₇₁
commutator	C₂×C₇₁	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂³×C₈
prime radical	C₂×#11765865	C₂×C₄
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	C₄×#504
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #22589936	C₂	semidirect
4. #22589935	C₂	semidirect
5. C₄×#14556068	C₂	semidirect
6. C₄×#14556067	C₂	semidirect
7. C₄×C₁₆×D₇₁	C₂	semidirect
8. C₂×C₄×C₁₆×C₇₁	C₂	semidirect
12. #17986415	C₄	direct (N × H)
47. #14556068	C₂×C₄	semidirect
48. #14556067	C₂×C₄	semidirect
50. C₁₆×D₇₁	C₂×C₄	semidirect
60. C₂×C₁₆×C₇₁	C₂×C₄	semidirect
132. C₄×C₇₁	#504	semidirect
149. C₂×C₄×C₁₆	D₇₁	semidirect
150. C₇₁	C₄×#504	semidirect
196. C₂×C₁₆	C₄×D₇₁	semidirect
276. C₄	#17986415	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×#17986415	2
C₄×#14556134	2
#22589936	2
#22589935	2
C₄×#14556068	2
C₄×#14556067	2
C₄×C₁₆×D₇₁	2
C₂×C₄×C₁₆×C₇₁	2
C₄×#504	71

Subgroups — C4×#17986415 #32150094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17986415 #32150094