Subgroups — C₄×#17989677 #34033290

Order 18688.

470 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#10596842	C₂	1	2	yes	maximal
2	C₂×#17989677	C₂	1	2	yes	maximal
3	#22626937	C₂	1	2	yes	maximal
4	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10596842	C₂²	1	4	yes	—
6	C₂×C₄×#10596842	—	2	2	no	—
7	(C₂×#10596842)⋊C₄	—	2	2	no	—
8	#17989677	C₄	1	8	yes	direct product factor
9	C₄×#11924036	C₄	1	1	yes	characteristic
10	C₂×C₈×#64408	C₄	1	1	yes	characteristic
11	C₂×C₈×#64408	C₄	1	1	yes	characteristic
12	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
13	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
14	C₄²×#64408	C₂²	1	1	yes	characteristic
15	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
16	C₂²×#10596842	—	2	4	no	—
17	C₄⋊#10596842	—	2	4	no	—
18	C₄⋊#10596842	C₂×C₄	1	8	yes	—
19	C₄×#10596842	—	4	4	no	—
20	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
21	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×#64408	D₄	1	1	yes	characteristic
24	C₂×C₄×#64408	D₄	1	1	yes	characteristic
25	C₂×C₄×#64408	Q₈	1	1	yes	characteristic
26	C₂×C₄×#64408	Q₈	1	1	yes	characteristic
27	C₂×C₄×#64408	C₂³	1	1	yes	characteristic
28	C₄×#10596833	—	2	1	no	—
29	C₄×#10596833	—	2	1	no	—
30	C₈×#64408	—	2	2	no	—
31	C₈×#64408	—	2	2	no	—
32	C₈×#64408	C₈	1	4	yes	complemented
33	C₈×#64408	C₂×C₄	1	4	yes	—
34	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
37	C₄×#10596831	C₈	1	1	yes	characteristic, complemented
38	C₈×Dic₇₃	C₈	1	1	yes	characteristic, complemented
39	C₈×Dic₇₃	C₈	1	1	yes	characteristic, complemented
40	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic, complemented
41	C₂×#10596842	—	2	8	no	—
42	C₂×#10596842	—	4	4	no	—
43	C₂×#10596842	—	4	4	no	—
44	C₂²×#64408	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×#64408	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×#64408	C₂×D₄	1	1	yes	characteristic
47	C₂²×#64408	C₂×Q₈	1	1	yes	characteristic
48	#11924036	—	2	1	no	—
49	#11924036	—	2	1	no	—
50	#11924036	C₄²	1	2	yes	—
51	#11924036	C₂×C₈	1	2	yes	—
52	C₂×#10596833	—	2	1	no	—
53	C₂×#10596833	—	2	1	no	—
54	C₄×#64408	—	2	2	no	—
55	C₄×#64408	—	2	2	no	—
56	C₄×#64408	—	2	2	no	—
57	C₄×#64408	C₂×C₈	1	2	yes	—
58	C₄×#64408	M₁₆	1	2	yes	—
59	C₄×#64408	D₈	1	2	yes	—
60	C₄×#64408	Q₁₆	1	2	yes	—
61	C₄×#64408	C₂²×C₄	1	2	yes	—
62	C₄×#64408	#41	1	2	yes	—
63	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
67	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
68	C₈×D₇₃	—	2	1	no	—
69	C₈×D₇₃	—	2	1	no	—
70	C₈×D₇₃	C₄²	1	2	yes	—
71	C₈×D₇₃	C₂×C₈	1	2	yes	—
72	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
73	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
74	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
75	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
76	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
77	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
78	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×#64408	—	2	1	no	—
80	C₂×#64408	—	2	1	no	—
81	C₂×#64408	—	2	1	no	—
82	C₂×#64408	—	2	1	no	—
83	C₂×#64408	C₄⋊C₈	1	2	yes	—
84	C₂×#64408	C₄⋊C₈	1	2	yes	—
85	C₂×#64408	#107	1	2	yes	—
86	C₂×#64408	#108	1	2	yes	—
87	C₂×#64408	C₄×D₄	1	2	yes	—
88	C₂×#64408	C₄×Q₈	1	2	yes	—
89	C₂×#64408	C₂×D₈	1	1	yes	characteristic
90	C₂×#64408	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×#64408	#135	1	2	yes	—
92	#10596842	—	4	8	no	—
93	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
97	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
98	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
99	C₄×D₇₃	—	2	1	no	—
100	C₄×D₇₃	—	2	1	no	—
101	C₄×D₇₃	#107	1	2	yes	—
102	C₄×D₇₃	#108	1	2	yes	—
103	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
104	C₄×D₇₃	#117	1	1	yes	characteristic
105	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
106	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
107	#10596833	—	2	2	no	—
108	#10596833	—	2	2	no	—
109	C₈×C₇₃	—	2	1	no	—
110	C₈×C₇₃	C₄×C₈	1	2	yes	complemented
111	#10596831	—	2	1	no	—
112	#10596831	C₄×C₈	1	2	yes	complemented
113	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₂×D₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
115	C₂×D₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₂×D₇₃	C₂×#107	1	1	yes	characteristic
117	C₂×D₇₃	#427	1	1	yes	characteristic
118	C₂×D₇₃	C₂×#108	1	1	yes	characteristic
119	C₂×D₇₃	C₂×#108	1	1	yes	characteristic
120	#64408	—	2	2	no	—
121	#64408	—	2	2	no	—
122	#64408	#335	1	4	yes	complemented
123	#64408	C₄×D₈	1	2	yes	—
124	#64408	C₄×Q₁₆	1	2	yes	—
125	C₄×C₇₃	—	2	1	no	—
126	C₄×C₇₃	#335	1	2	yes	complemented
127	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
128	C₄×C₇₃	#431	1	1	yes	characteristic
129	Dic₇₃	—	2	1	no	—
130	Dic₇₃	#335	1	2	yes	complemented
131	Dic₇₃	C₄×M₁₆	1	1	yes	characteristic
132	Dic₇₃	#431	1	1	yes	characteristic
133	C₄×#335	—	73	1	no	maximal, Sylow
134	C₂×C₇₃	C₂×#335	1	1	yes	characteristic
135	C₂×C₇₃	#1564	1	1	yes	characteristic
136	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
137	D₇₃	C₂×#335	1	1	yes	characteristic
138	D₇₃	#1564	1	1	yes	characteristic
139	D₇₃	C₄×#107	1	1	yes	characteristic
140	D₇₃	C₄×#108	1	1	yes	characteristic
141	C₄×C₄⋊C₈	—	73	2	no	—
142	C₄²×C₈	—	73	1	no	—
143	C₂×#335	—	73	2	no	—
144	#1277	—	73	2	no	—
145	C₇₃	C₄×#335	1	1	yes	characteristic, complemented, Sylow
146	C₂×C₄⋊C₈	—	73	4	no	—
147	C₂×C₄×C₈	—	73	1	no	—
148	C₂×C₄×C₈	—	73	1	no	—
149	C₂×C₄×C₈	—	73	1	no	—
150	C₂×C₄×C₈	—	73	1	no	—
151	C₂×C₄×C₈	—	73	1	no	—
152	C₂×C₄×C₈	—	73	1	no	—
153	C₂×C₄×C₈	—	146	2	no	—
154	C₄³	—	73	1	no	—
155	#336	—	146	2	no	—
156	#335	—	73	8	no	—
157	C₂²×C₈	—	73	1	no	—
158	C₂²×C₈	—	73	1	no	—
159	C₂²×C₈	—	73	1	no	—
160	C₂²×C₈	—	73	1	no	—
161	C₂²×C₈	—	146	4	no	—
162	C₂×C₄²	—	73	1	no	—
163	C₂×C₄²	—	73	1	no	—
164	C₂×C₄²	—	73	1	no	—
165	C₂×C₄²	—	73	1	no	—
166	C₂×C₄²	—	73	1	no	—
167	C₂×C₄²	—	73	1	no	—
168	C₂×C₄²	—	73	1	no	—
169	C₄⋊C₈	—	73	8	no	—
170	C₄⋊C₈	—	146	4	no	—
171	C₄×C₈	—	292	4	no	—
172	C₄×C₈	—	73	4	no	—
173	C₄×C₈	—	73	4	no	—
174	C₄×C₈	—	73	1	no	—
175	C₄×C₈	—	73	1	no	—
176	C₄×C₈	—	73	1	no	—
177	C₄×C₈	—	146	2	no	—
178	C₄×C₈	—	146	2	no	—
179	C₄×C₈	—	146	1	no	—
180	C₄×C₈	—	146	1	no	—
181	C₄×C₈	#10596842	1	1	yes	characteristic, complemented
182	C₂²×C₄	—	73	1	no	—
183	C₂²×C₄	—	73	1	no	—
184	C₂²×C₄	—	73	1	no	—
185	C₂²×C₄	—	73	1	no	—
186	C₂²×C₄	—	73	1	no	—
187	C₂²×C₄	—	73	1	no	—
188	C₂²×C₄	—	73	1	no	—
189	C₂×C₈	—	292	4	no	—
190	C₂×C₈	—	292	4	no	—
191	C₂×C₈	—	73	1	no	—
192	C₂×C₈	—	73	2	no	—
193	C₂×C₈	—	73	2	no	—
194	C₂×C₈	—	73	2	no	—
195	C₂×C₈	—	73	2	no	—
196	C₂×C₈	—	73	1	no	—
197	C₂×C₈	—	146	1	no	—
198	C₂×C₈	—	146	1	no	—
199	C₂×C₈	—	146	8	no	—
200	C₂×C₈	—	146	1	no	—
201	C₂×C₈	—	146	1	no	—
202	C₂×C₈	—	146	1	no	—
203	C₂×C₈	—	146	1	no	—
204	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
205	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
206	C₄²	—	73	2	no	—
207	C₄²	—	73	2	no	—
208	C₄²	—	73	2	no	—
209	C₄²	—	73	2	no	—
210	C₄²	—	73	2	no	—
211	C₄²	—	73	2	no	—
212	C₄²	—	73	1	no	—
213	C₄²	—	73	1	no	—
214	C₄²	—	73	1	no	—
215	C₄²	—	146	2	no	—
216	C₄²	—	146	2	no	—
217	C₄²	—	146	2	no	—
218	C₄²	C₂×#10596842	1	1	yes	characteristic
219	C₂³	—	73	1	no	—
220	C₂×C₄	—	73	1	no	—
221	C₂×C₄	—	73	2	no	—
222	C₂×C₄	—	73	2	no	—
223	C₂×C₄	—	73	2	no	—
224	C₂×C₄	—	73	1	no	—
225	C₂×C₄	—	73	2	no	—
226	C₂×C₄	—	73	2	no	—
227	C₂×C₄	—	73	1	no	—
228	C₂×C₄	—	73	1	no	—
229	C₂×C₄	—	73	2	no	—
230	C₂×C₄	—	73	2	no	—
231	C₂×C₄	—	73	1	no	—
232	C₂×C₄	—	73	1	no	—
233	C₂×C₄	—	73	1	no	—
234	C₂×C₄	—	73	1	no	—
235	C₂×C₄	—	73	2	no	—
236	C₂×C₄	—	73	2	no	—
237	C₂×C₄	—	73	1	no	—
238	C₂×C₄	—	146	1	no	—
239	C₂×C₄	—	146	1	no	—
240	C₂×C₄	—	146	1	no	—
241	C₂×C₄	—	146	1	no	—
242	C₂×C₄	—	146	1	no	—
243	C₂×C₄	—	146	1	no	—
244	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
245	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
246	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
247	C₈	—	2	1	no	—
248	C₈	—	292	8	no	—
249	C₈	—	73	2	no	—
250	C₈	—	146	2	no	—
251	C₈	—	146	2	no	—
252	C₈	—	146	1	no	—
253	C₈	C₄×#10596842	1	2	yes	complemented
254	C₂²	—	73	1	no	—
255	C₂²	—	73	1	no	—
256	C₂²	—	73	1	no	—
257	C₂²	—	73	1	no	—
258	C₂²	—	73	1	no	—
259	C₂²	—	73	1	no	—
260	C₂²	C₂×C₄⋊#10596842	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	73	2	no	—
263	C₄	—	73	4	no	—
264	C₄	—	73	2	no	—
265	C₄	—	73	2	no	—
266	C₄	—	73	1	no	—
267	C₄	—	73	1	no	—
268	C₄	—	146	2	no	—
269	C₄	—	146	2	no	—
270	C₄	—	146	1	no	—
271	C₄	#17989677	1	2	yes	direct product factor
272	C₄	C₂×C₄×#10596842	1	1	yes	characteristic
273	C₄	#17989700	1	1	yes	characteristic
274	C₂	—	73	1	no	—
275	C₂	—	73	1	no	—
276	C₂	—	73	1	no	—
277	C₂	—	73	1	no	—
278	C₂	C₂×#17989677	1	1	yes	characteristic
279	C₂	#22627055	1	1	yes	characteristic
280	C₂	C₄×C₄⋊#10596842	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#10596842
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#10596842
Fitting	C₄×C₈×C₇₃	C₈
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂²×D₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×#64408	C₂³
hypercenter	C₄×C₈	#10596842
hypocenter	C₇₃	C₄×#335
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
8. #17989677	C₄	direct (N × H)
32. C₈×#64408	C₈	semidirect
37. C₄×#10596831	C₈	semidirect
38. C₈×Dic₇₃	C₈	semidirect
39. C₈×Dic₇₃	C₈	semidirect
40. C₄×C₈×C₇₃	C₈	semidirect
110. C₈×C₇₃	C₄×C₈	semidirect
112. #10596831	C₄×C₈	semidirect
122. #64408	#335	semidirect
126. C₄×C₇₃	#335	semidirect
130. Dic₇₃	#335	semidirect
145. C₇₃	C₄×#335	semidirect
181. C₄×C₈	#10596842	semidirect
253. C₈	C₄×#10596842	semidirect
271. C₄	#17989677	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#10596842	2
C₂×#17989677	2
#22626937	2
C₄×C₈×#64408	2
C₄×#335	73

Subgroups — C4×#17989677 #34033290

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17989677 #34033290