Subgroups — C₄×#17985814 #31912713

Order 18176.

434 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₈₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985814	C₂	1	2	yes	maximal
3	C₄×#14556068	C₂	1	1	yes	maximal, characteristic
4	#22586450	C₂	1	2	yes	maximal
5	C₄×#108×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₈₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₄₂	—	2	1	no	—
8	C₂×#14556068	C₂²	1	2	yes	—
9	C₄×Dic₂₈₄	—	2	1	no	—
10	C₄×Dic₂₈₄	—	2	1	no	—
11	C₄×Dic₂₈₄	C₄	1	2	yes	—
12	C₄×Dic₂₈₄	C₄	1	2	yes	—
13	C₂×#14556062	C₂²	1	1	yes	characteristic
14	C₂×#14556053	—	2	1	no	—
15	#17985814	C₄	1	8	yes	direct product factor
16	#17985775	—	2	1	no	—
17	C₄×C₇₁⋊C₁₆	—	2	1	no	—
18	C₂×#108×C₇₁	C₂²	1	2	yes	—
19	C₄×M₁₆×C₇₁	—	2	1	no	—
20	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	#341×C₇₁	—	2	1	no	—
22	C₂²×Dic₁₄₂	—	2	1	no	—
23	C₂×Dic₂₈₄	—	2	1	no	—
24	C₂×Dic₂₈₄	—	2	2	no	—
25	C₂×Dic₂₈₄	C₂×C₄	1	4	yes	—
26	C₂×Dic₂₈₄	C₂×C₄	1	2	yes	—
27	C₄×Dic₁₄₂	—	2	2	no	—
28	C₄×Dic₁₄₂	—	2	2	no	—
29	C₄×Dic₁₄₂	—	4	1	no	—
30	C₄×Dic₁₄₂	—	4	1	no	—
31	C₂×C₄⋊Dic₇₁	—	2	1	no	—
32	C₂×Dic₇₁⋊C₄	—	4	1	no	—
33	C₂×C₄×Dic₇₁	—	4	1	no	—
34	#14556068	—	2	2	no	—
35	#14556068	C₂×C₄	1	4	yes	—
36	C₂×C₇₁⋊C₁₆	—	2	2	no	—
37	#14556062	—	2	1	no	—
38	#14556062	C₂×C₄	1	2	yes	—
39	#14556053	—	2	2	no	—
40	#14556053	—	2	2	no	—
41	C₂×M₁₆×C₇₁	—	2	2	no	—
42	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
45	#108×C₇₁	—	2	2	no	—
46	#108×C₇₁	C₂×C₄	1	4	yes	—
47	#97×C₇₁	—	4	1	no	—
48	C₄×C₈×C₇₁	—	4	1	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	C₂×Dic₁₄₂	—	2	2	no	—
54	C₂×Dic₁₄₂	—	2	4	no	—
55	C₂×Dic₁₄₂	—	4	1	no	—
56	C₂×Dic₁₄₂	—	4	2	no	—
57	C₂²×Dic₇₁	—	4	1	no	—
58	C₂²×Dic₇₁	—	4	1	no	—
59	Dic₂₈₄	—	2	2	no	—
60	Dic₂₈₄	—	2	4	no	—
61	Dic₂₈₄	C₄²	1	4	yes	—
62	C₄⋊Dic₇₁	—	2	2	no	—
63	C₄⋊Dic₇₁	—	4	1	no	—
64	Dic₇₁⋊C₄	—	4	2	no	—
65	Dic₇₁⋊C₄	—	4	2	no	—
66	C₄×Dic₇₁	—	4	2	no	—
67	C₄×Dic₇₁	—	4	2	no	—
68	C₇₁⋊C₁₆	—	2	4	no	—
69	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
72	M₁₆×C₇₁	—	2	4	no	—
73	M₁₆×C₇₁	—	4	2	no	—
74	C₂×C₈×C₇₁	—	2	2	no	—
75	C₂×C₈×C₇₁	—	4	2	no	—
76	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₇₁	#41	1	2	yes	—
82	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₇₁	D₈	1	1	yes	characteristic
85	C₄²×C₇₁	D₈	1	1	yes	characteristic
86	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇₁	—	2	2	no	—
88	C₂×C₄×C₇₁	—	4	1	no	—
89	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
90	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
91	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
92	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
93	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₇₁	#135	1	2	yes	—
97	C₂×Dic₇₁	—	4	2	no	—
98	C₂×Dic₇₁	—	4	4	no	—
99	C₂×Dic₇₁	—	4	2	no	—
100	C₂×Dic₇₁	—	8	2	no	—
101	Dic₁₄₂	—	2	4	no	—
102	Dic₁₄₂	—	4	2	no	—
103	Dic₁₄₂	—	4	4	no	—
104	C₈×C₇₁	—	2	1	no	—
105	C₈×C₇₁	—	2	1	no	—
106	C₈×C₇₁	—	4	4	no	—
107	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
108	C₈×C₇₁	#117	1	1	yes	characteristic
109	C₈×C₇₁	C₄×D₄	1	2	yes	—
110	C₂²×C₇₁	—	2	1	no	—
111	C₂²×C₇₁	—	2	1	no	—
112	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
113	C₂²×C₇₁	C₄×SD₁₆	1	2	yes	—
114	C₄×C₇₁	—	2	1	no	—
115	C₄×C₇₁	—	2	1	no	—
116	C₄×C₇₁	—	2	1	no	—
117	C₄×C₇₁	#362	1	2	yes	complemented
118	C₄×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₇₁	#416	1	1	yes	characteristic
120	C₄×C₇₁	C₄×D₈	1	2	yes	—
121	Dic₇₁	—	4	4	no	—
122	Dic₇₁	—	8	2	no	—
123	C₄×#362	—	71	1	no	maximal, Sylow
124	C₂×C₇₁	—	2	2	no	—
125	C₂×C₇₁	C₄×#102	1	1	yes	characteristic
126	C₂×C₇₁	C₂×#362	1	1	yes	characteristic
127	C₂×C₇₁	#2148	1	1	yes	characteristic
128	C₂×C₄×Q₁₆	—	71	1	no	—
129	C₂×#362	—	71	2	no	—
130	C₄×M₃₂	—	71	1	no	—
131	C₄×#108	D₇₁	1	1	yes	characteristic
132	#1387	—	71	2	no	—
133	C₇₁	C₄×#362	1	1	yes	characteristic, complemented, Sylow
134	C₂²×Q₁₆	—	71	1	no	—
135	C₂×C₄×Q₈	—	142	1	no	—
136	C₂×M₃₂	—	71	2	no	—
137	C₄×Q₁₆	—	71	2	no	—
138	C₄×Q₁₆	—	71	2	no	—
139	C₄×Q₁₆	—	142	1	no	—
140	C₄×Q₁₆	—	142	1	no	—
141	C₂×#108	C₂×D₇₁	1	2	yes	—
142	C₂×#107	—	71	1	no	—
143	C₂×#103	—	142	1	no	—
144	C₄×M₁₆	—	2	1	no	—
145	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
146	#362	—	71	8	no	—
147	#346	—	142	1	no	—
148	C₄×C₁₆	—	142	1	no	—
149	#341	—	2	1	no	—
150	C₂²×Q₈	—	142	1	no	—
151	C₂×Q₁₆	—	71	4	no	—
152	C₂×Q₁₆	—	71	2	no	—
153	C₂×Q₁₆	—	142	1	no	—
154	C₂×Q₁₆	—	142	2	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
158	C₄×Q₈	—	284	1	no	—
159	C₄×Q₈	—	284	1	no	—
160	C₄×Q₈	—	142	2	no	—
161	C₄×Q₈	—	142	2	no	—
162	C₂×C₄⋊C₄	—	284	1	no	—
163	C₂×C₄⋊C₄	—	142	1	no	—
164	C₂×C₄²	—	284	1	no	—
165	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
166	M₃₂	—	71	4	no	—
167	M₃₂	—	142	2	no	—
168	C₂×C₁₆	—	142	2	no	—
169	#108	—	2	2	no	—
170	#108	C₄×D₇₁	1	4	yes	—
171	#107	—	71	2	no	—
172	#107	—	142	1	no	—
173	#103	—	142	2	no	—
174	#103	—	142	2	no	—
175	#97	—	4	1	no	—
176	C₄×C₈	—	4	1	no	—
177	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
178	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
179	C₄×C₈	D₂₈₄	1	1	yes	characteristic
180	C₄×C₈	D₂₈₄	1	1	yes	characteristic
181	C₂×Q₈	—	284	1	no	—
182	C₂×Q₈	—	284	2	no	—
183	C₂×Q₈	—	142	2	no	—
184	C₂×Q₈	—	142	4	no	—
185	C₂²×C₄	—	284	1	no	—
186	C₂²×C₄	—	284	1	no	—
187	C₂²×C₄	#11765852	1	1	yes	characteristic
188	C₂²×C₄	#11765852	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
190	Q₁₆	—	71	4	no	—
191	Q₁₆	—	142	2	no	—
192	Q₁₆	—	142	4	no	—
193	M₁₆	—	2	4	no	—
194	M₁₆	—	4	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
198	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
200	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
201	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
202	C₂×C₈	#11765865	1	2	yes	—
203	C₄⋊C₄	—	284	2	no	—
204	C₄⋊C₄	—	284	2	no	—
205	C₄⋊C₄	—	284	1	no	—
206	C₄⋊C₄	—	142	2	no	—
207	C₄²	—	284	2	no	—
208	C₄²	—	284	2	no	—
209	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
210	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
211	C₄²	#11765850	1	1	yes	characteristic
212	C₄²	#11765850	1	1	yes	characteristic
213	C₁₆	—	142	4	no	—
214	C₂³	C₂×#11765852	1	1	yes	characteristic
215	Q₈	—	284	4	no	—
216	Q₈	—	284	2	no	—
217	Q₈	—	142	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	284	4	no	—
221	C₂×C₄	—	284	2	no	—
222	C₂×C₄	—	284	2	no	—
223	C₂×C₄	—	568	2	no	—
224	C₂×C₄	#14556048	1	1	yes	characteristic
225	C₂×C₄	#14556048	1	1	yes	characteristic
226	C₂×C₄	#14556048	1	1	yes	characteristic
227	C₂×C₄	#14556048	1	1	yes	characteristic
228	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
229	C₂×C₄	C₄×C₂²⋊D₇₁	1	2	yes	—
230	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
231	C₂×C₄	#14556148	1	2	yes	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	4	4	no	—
235	C₈	C₄²×D₇₁	1	1	yes	characteristic
236	C₈	#14556085	1	1	yes	characteristic
237	C₈	C₄×D₂₈₄	1	2	yes	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	C₂×#14556048	1	1	yes	characteristic
241	C₂²	C₄×#11765852	1	2	yes	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	284	4	no	—
246	C₄	—	568	2	no	—
247	C₄	#17985814	1	2	yes	direct product factor
248	C₄	C₄×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
249	C₄	#17985993	1	1	yes	characteristic
250	C₄	C₄×#11765850	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	C₄×#14556048	1	1	yes	characteristic
253	C₂	C₂×#17985814	1	1	yes	characteristic
254	C₂	#22590087	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556048
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₄×#108×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂³×C₇₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₄×#108	D₇₁
hypocenter	C₇₁	C₄×#362
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
15. #17985814	C₄	direct (N × H)
117. C₄×C₇₁	#362	semidirect
133. C₇₁	C₄×#362	semidirect
247. C₄	#17985814	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×C₄×Dic₂₈₄	2
C₂×#17985814	2
C₄×#14556068	2
#22586450	2
C₄×#108×C₇₁	2
C₄×#362	71

Subgroups — C4×#17985814 #31912713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17985814 #31912713