Subgroups — C₄×#17985812 #31912705

Order 18176.

498 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985812	C₂	1	2	yes	maximal
2	C₄×#14556068	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14556064	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586448	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×D₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	2	yes	—
7	C₂×#14556064	C₂²	1	2	yes	—
8	C₄×#11765849	—	2	1	no	—
9	#17985812	C₄	1	8	yes	direct product factor
10	#17985775	—	2	1	no	—
11	C₄×C₇₁⋊C₁₆	—	2	1	no	—
12	#17985764	—	2	1	no	—
13	C₂²×D₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₇₁	—	2	1	no	—
15	C₄×D₈×C₇₁	—	2	1	no	—
16	C₄×D₈×C₇₁	—	2	1	no	—
17	C₄×D₈×C₇₁	C₄	1	2	yes	—
18	C₄×D₈×C₇₁	C₄	1	2	yes	—
19	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×#102×C₇₁	—	2	1	no	—
21	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×#11765849	—	2	2	no	—
23	#14556068	—	2	2	no	—
24	#14556068	C₂×C₄	1	4	yes	complemented
25	C₂×C₇₁⋊C₁₆	—	2	2	no	—
26	#14556064	—	2	2	no	—
27	#14556064	C₂×C₄	1	4	yes	complemented
28	#14556039	—	4	1	no	—
29	C₄×C₇₁⋊C₈	—	4	1	no	—
30	C₂²×D₄×C₇₁	—	2	1	no	—
31	C₂³×C₄×C₇₁	—	4	1	no	—
32	C₂×D₈×C₇₁	—	2	1	no	—
33	C₂×D₈×C₇₁	—	2	2	no	—
34	C₂×D₈×C₇₁	C₂×C₄	1	4	yes	—
35	C₂×D₈×C₇₁	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
38	C₄×D₄×C₇₁	—	2	2	no	—
39	C₄×D₄×C₇₁	—	2	2	no	—
40	C₄×D₄×C₇₁	—	4	1	no	—
41	C₄×D₄×C₇₁	—	4	1	no	—
42	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
43	C₂×C₂²⋊C₄×C₇₁	—	4	1	no	—
44	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
45	#107×C₇₁	—	2	1	no	—
46	#107×C₇₁	C₂×C₄	1	2	yes	—
47	#102×C₇₁	—	2	2	no	—
48	#102×C₇₁	—	2	2	no	—
49	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
52	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
53	#11765849	—	2	4	no	—
54	#11765849	—	4	2	no	—
55	C₂×C₇₁⋊C₈	—	4	2	no	—
56	C₇₁⋊C₁₆	—	2	4	no	—
57	C₂⁴×C₇₁	—	4	1	no	—
58	C₂×D₄×C₇₁	—	2	2	no	—
59	C₂×D₄×C₇₁	—	2	4	no	—
60	C₂×D₄×C₇₁	—	4	1	no	—
61	C₂×D₄×C₇₁	—	4	2	no	—
62	C₂²×C₄×C₇₁	—	4	1	no	—
63	C₂²×C₄×C₇₁	—	4	2	no	—
64	C₂²×C₄×C₇₁	—	4	2	no	—
65	C₂²×C₄×C₇₁	—	8	4	no	—
66	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
69	D₈×C₇₁	—	2	2	no	—
70	D₈×C₇₁	—	2	4	no	—
71	D₈×C₇₁	C₄²	1	4	yes	—
72	C₂×C₈×C₇₁	—	2	2	no	—
73	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₇₁	#41	1	2	yes	—
79	C₄⋊C₄×C₇₁	—	2	2	no	—
80	C₄⋊C₄×C₇₁	—	4	1	no	—
81	C₂²⋊C₄×C₇₁	—	4	2	no	—
82	C₂²⋊C₄×C₇₁	—	4	2	no	—
83	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
85	C₄²×C₇₁	D₈	1	1	yes	characteristic
86	C₄²×C₇₁	D₈	1	1	yes	characteristic
87	C₂³×C₇₁	—	4	2	no	—
88	C₂³×C₇₁	—	4	4	no	—
89	C₂³×C₇₁	—	8	2	no	—
90	C₂³×C₇₁	—	8	2	no	—
91	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
92	D₄×C₇₁	—	2	4	no	—
93	D₄×C₇₁	—	4	2	no	—
94	D₄×C₇₁	—	4	4	no	—
95	C₂×C₄×C₇₁	—	2	2	no	—
96	C₂×C₄×C₇₁	—	4	1	no	—
97	C₂×C₄×C₇₁	—	4	2	no	—
98	C₂×C₄×C₇₁	—	8	4	no	—
99	C₂×C₄×C₇₁	—	8	2	no	—
100	C₂×C₄×C₇₁	—	8	4	no	—
101	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
102	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
103	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
104	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₇₁	#135	1	2	yes	—
109	C₈×C₇₁	—	2	1	no	—
110	C₈×C₇₁	—	2	1	no	—
111	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
112	C₈×C₇₁	#117	1	1	yes	characteristic
113	C₈×C₇₁	C₄×D₄	1	2	yes	—
114	C₇₁⋊C₈	—	4	4	no	—
115	C₂²×C₇₁	—	2	1	no	—
116	C₂²×C₇₁	—	2	1	no	—
117	C₂²×C₇₁	—	4	4	no	—
118	C₂²×C₇₁	—	8	8	no	—
119	C₂²×C₇₁	—	8	2	no	—
120	C₂²×C₇₁	—	8	2	no	—
121	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
122	C₂²×C₇₁	C₄×SD₁₆	1	2	yes	—
123	C₄×C₇₁	—	2	1	no	—
124	C₄×C₇₁	—	2	1	no	—
125	C₄×C₇₁	—	2	1	no	—
126	C₄×C₇₁	—	8	2	no	—
127	C₄×C₇₁	#361	1	2	yes	complemented
128	C₄×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
129	C₄×C₇₁	#416	1	1	yes	characteristic
130	C₄×C₇₁	C₄×D₈	1	2	yes	—
131	C₄×#361	—	71	1	no	maximal, Sylow
132	C₂×C₇₁	—	2	2	no	—
133	C₂×C₇₁	—	8	4	no	—
134	C₂×C₇₁	C₄×#102	1	1	yes	characteristic
135	C₂×C₇₁	C₂×#361	1	1	yes	characteristic
136	C₂×C₇₁	#2148	1	1	yes	characteristic
137	C₂×C₄×D₈	D₇₁	1	1	yes	characteristic
138	C₂×#361	—	71	2	no	—
139	C₄×M₃₂	—	71	1	no	—
140	C₄×#108	—	71	1	no	—
141	#1387	—	71	2	no	—
142	C₇₁	C₄×#361	1	1	yes	characteristic, complemented, Sylow
143	C₂²×D₈	C₂×D₇₁	1	1	yes	characteristic
144	C₂×C₄×D₄	—	2	1	no	—
145	C₂×M₃₂	—	71	2	no	—
146	C₄×D₈	—	2	1	no	—
147	C₄×D₈	—	2	1	no	—
148	C₄×D₈	Dic₇₁	1	2	yes	—
149	C₄×D₈	Dic₇₁	1	2	yes	—
150	C₂×#108	—	71	2	no	—
151	C₂×#107	C₂×D₇₁	1	1	yes	characteristic
152	C₂×#102	—	2	1	no	—
153	C₄×M₁₆	—	142	1	no	—
154	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
155	#361	—	71	8	no	—
156	#346	—	142	1	no	—
157	C₄×C₁₆	—	142	1	no	—
158	#341	—	142	1	no	—
159	C₂²×D₄	—	2	1	no	—
160	C₂³×C₄	—	4	1	no	—
161	C₂×D₈	—	2	1	no	—
162	C₂×D₈	—	2	2	no	—
163	C₂×D₈	C₄×D₇₁	1	4	yes	—
164	C₂×D₈	C₂×Dic₇₁	1	2	yes	—
165	C₂×M₁₆	—	142	2	no	—
166	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
168	C₄×D₄	—	2	2	no	—
169	C₄×D₄	—	2	2	no	—
170	C₄×D₄	—	4	1	no	—
171	C₄×D₄	—	4	1	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₂²⋊C₄	—	4	1	no	—
174	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
175	M₃₂	—	71	4	no	—
176	M₃₂	—	142	2	no	—
177	C₂×C₁₆	—	142	2	no	—
178	#108	—	71	4	no	—
179	#108	—	142	2	no	—
180	#107	—	2	1	no	—
181	#107	C₂×Dic₇₁	1	2	yes	—
182	#102	—	2	2	no	—
183	#102	—	2	2	no	—
184	#97	—	284	1	no	—
185	C₄×C₈	—	284	1	no	—
186	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
187	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
188	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
189	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
190	C₂⁴	—	4	1	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	8	4	no	—
199	C₂²×C₄	#11765853	1	1	yes	characteristic
200	C₂²×C₄	#11765853	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
202	D₈	—	2	2	no	—
203	D₈	—	2	4	no	—
204	D₈	C₄×Dic₇₁	1	4	yes	—
205	M₁₆	—	284	2	no	—
206	M₁₆	—	142	4	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	284	2	no	—
209	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
212	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
213	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
214	C₂×C₈	#11765865	1	2	yes	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
220	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
221	C₄²	#11765850	1	1	yes	characteristic
222	C₄²	#11765850	1	1	yes	characteristic
223	C₁₆	—	142	4	no	—
224	C₂³	—	4	4	no	—
225	C₂³	—	4	2	no	—
226	C₂³	—	8	2	no	—
227	C₂³	—	8	2	no	—
228	C₂³	C₂×#11765853	1	1	yes	characteristic
229	D₄	—	2	4	no	—
230	D₄	—	4	4	no	—
231	D₄	—	4	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	#14556049	1	1	yes	characteristic
239	C₂×C₄	#14556049	1	1	yes	characteristic
240	C₂×C₄	#14556049	1	1	yes	characteristic
241	C₂×C₄	#14556049	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
243	C₂×C₄	C₄×C₂²⋊D₇₁	1	2	yes	—
244	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
245	C₂×C₄	#14556148	1	2	yes	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	284	4	no	—
249	C₈	C₂×C₄×Dic₇₁	1	1	yes	characteristic
250	C₈	#14556087	1	1	yes	characteristic
251	C₈	C₄×C₂²⋊D₇₁	1	2	yes	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	8	8	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	C₂×#14556049	1	1	yes	characteristic
259	C₂²	C₄×#11765853	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	8	2	no	—
264	C₄	#17985812	1	2	yes	direct product factor
265	C₄	C₄×C₂²⋊Dic₇₁	1	1	yes	characteristic
266	C₄	#17985995	1	1	yes	characteristic
267	C₄	C₄×#11765850	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	8	4	no	—
270	C₂	C₄×#14556049	1	1	yes	characteristic
271	C₂	C₂×#17985812	1	1	yes	characteristic
272	C₂	#22590089	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556049
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×C₄×D₈×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂²×D₈×C₇₁	C₂²
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×C₄×D₈	D₇₁
hypocenter	C₇₁	C₄×#361
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₄×#14556068	C₂	semidirect
3. C₄×#14556064	C₂	semidirect
4. #22586448	C₂	semidirect
9. #17985812	C₄	direct (N × H)
24. #14556068	C₂×C₄	semidirect
27. #14556064	C₂×C₄	semidirect
127. C₄×C₇₁	#361	semidirect
142. C₇₁	C₄×#361	semidirect
264. C₄	#17985812	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×#17985812	2
C₄×#14556068	2
C₄×#14556064	2
#22586448	2
C₂×C₄×D₈×C₇₁	2
C₄×#361	71

Subgroups — C4×#17985812 #31912705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17985812 #31912705