Subgroups — C₄×#17989676 #34033289

Order 18688.

470 conjugacy classes of subgroups.

The subgroup lattice has 276 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×#17989676	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	#17989676	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	SD₁₆	1	4	yes	—
60	C₄×#64408	C₂²×C₄	1	2	yes	—
61	C₄×#64408	#41	1	2	yes	—
62	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
66	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
67	C₈×D₇₃	—	2	1	no	—
68	C₈×D₇₃	—	2	1	no	—
69	C₈×D₇₃	C₄²	1	2	yes	—
70	C₈×D₇₃	C₂×C₈	1	2	yes	—
71	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
72	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
73	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
74	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
75	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
76	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
77	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×#64408	—	2	1	no	—
79	C₂×#64408	—	2	1	no	—
80	C₂×#64408	—	2	1	no	—
81	C₂×#64408	—	2	1	no	—
82	C₂×#64408	C₄⋊C₈	1	2	yes	—
83	C₂×#64408	C₄⋊C₈	1	2	yes	—
84	C₂×#64408	#106	1	2	yes	—
85	C₂×#64408	#108	1	2	yes	—
86	C₂×#64408	C₄×D₄	1	2	yes	—
87	C₂×#64408	C₄×Q₈	1	2	yes	—
88	C₂×#64408	C₂×SD₁₆	1	2	yes	—
89	C₂×#64408	#135	1	2	yes	—
90	#10596842	—	4	8	no	—
91	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
92	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
93	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
94	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
95	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
96	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
97	C₄×D₇₃	—	2	1	no	—
98	C₄×D₇₃	—	2	1	no	—
99	C₄×D₇₃	#106	1	2	yes	—
100	C₄×D₇₃	#108	1	2	yes	—
101	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
102	C₄×D₇₃	#117	1	1	yes	characteristic
103	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
104	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
105	#10596833	—	2	2	no	—
106	#10596833	—	2	2	no	—
107	C₈×C₇₃	—	2	1	no	—
108	C₈×C₇₃	C₄×C₈	1	2	yes	complemented
109	#10596831	—	2	1	no	—
110	#10596831	C₄×C₈	1	2	yes	complemented
111	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂×D₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
113	C₂×D₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₂×D₇₃	C₂×#106	1	1	yes	characteristic
115	C₂×D₇₃	#427	1	1	yes	characteristic
116	C₂×D₇₃	C₂×#108	1	1	yes	characteristic
117	C₂×D₇₃	C₂×#108	1	1	yes	characteristic
118	#64408	—	2	2	no	—
119	#64408	—	2	2	no	—
120	#64408	#334	1	4	yes	complemented
121	#64408	C₄×SD₁₆	1	4	yes	—
122	C₄×C₇₃	—	2	1	no	—
123	C₄×C₇₃	#334	1	2	yes	complemented
124	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
125	C₄×C₇₃	#431	1	1	yes	characteristic
126	Dic₇₃	—	2	1	no	—
127	Dic₇₃	#334	1	2	yes	complemented
128	Dic₇₃	C₄×M₁₆	1	1	yes	characteristic
129	Dic₇₃	#431	1	1	yes	characteristic
130	C₄×#334	—	73	1	no	maximal, Sylow
131	C₂×C₇₃	C₂×#334	1	1	yes	characteristic
132	C₂×C₇₃	#1564	1	1	yes	characteristic
133	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
134	D₇₃	C₂×#334	1	1	yes	characteristic
135	D₇₃	#1564	1	1	yes	characteristic
136	D₇₃	C₄×#106	1	1	yes	characteristic
137	D₇₃	C₄×#108	1	1	yes	characteristic
138	C₄×C₄⋊C₈	—	73	2	no	—
139	C₄²×C₈	—	73	1	no	—
140	C₂×#334	—	73	2	no	—
141	#1277	—	73	2	no	—
142	C₇₃	C₄×#334	1	1	yes	characteristic, complemented, Sylow
143	C₂×C₄⋊C₈	—	73	4	no	—
144	C₂×C₄×C₈	—	73	1	no	—
145	C₂×C₄×C₈	—	73	1	no	—
146	C₂×C₄×C₈	—	73	1	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	—	146	2	no	—
151	C₄³	—	73	1	no	—
152	#336	—	146	2	no	—
153	#334	—	73	8	no	—
154	C₂²×C₈	—	73	1	no	—
155	C₂²×C₈	—	73	1	no	—
156	C₂²×C₈	—	73	1	no	—
157	C₂²×C₈	—	73	1	no	—
158	C₂²×C₈	—	146	4	no	—
159	C₂×C₄²	—	73	1	no	—
160	C₂×C₄²	—	73	1	no	—
161	C₂×C₄²	—	73	1	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	C₄⋊C₈	—	73	8	no	—
167	C₄⋊C₈	—	146	4	no	—
168	C₄×C₈	—	292	4	no	—
169	C₄×C₈	—	73	4	no	—
170	C₄×C₈	—	73	4	no	—
171	C₄×C₈	—	73	1	no	—
172	C₄×C₈	—	73	1	no	—
173	C₄×C₈	—	73	1	no	—
174	C₄×C₈	—	146	2	no	—
175	C₄×C₈	—	146	2	no	—
176	C₄×C₈	—	146	1	no	—
177	C₄×C₈	—	146	1	no	—
178	C₄×C₈	#10596842	1	1	yes	characteristic, complemented
179	C₂²×C₄	—	73	1	no	—
180	C₂²×C₄	—	73	1	no	—
181	C₂²×C₄	—	73	1	no	—
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	C₂×C₈	—	292	4	no	—
187	C₂×C₈	—	292	4	no	—
188	C₂×C₈	—	73	1	no	—
189	C₂×C₈	—	73	2	no	—
190	C₂×C₈	—	73	2	no	—
191	C₂×C₈	—	73	2	no	—
192	C₂×C₈	—	73	2	no	—
193	C₂×C₈	—	73	1	no	—
194	C₂×C₈	—	146	1	no	—
195	C₂×C₈	—	146	1	no	—
196	C₂×C₈	—	146	8	no	—
197	C₂×C₈	—	146	1	no	—
198	C₂×C₈	—	146	1	no	—
199	C₂×C₈	—	146	1	no	—
200	C₂×C₈	—	146	1	no	—
201	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
202	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
203	C₄²	—	73	2	no	—
204	C₄²	—	73	2	no	—
205	C₄²	—	73	4	no	—
206	C₄²	—	73	2	no	—
207	C₄²	—	73	2	no	—
208	C₄²	—	73	1	no	—
209	C₄²	—	73	1	no	—
210	C₄²	—	73	1	no	—
211	C₄²	—	146	2	no	—
212	C₄²	—	146	2	no	—
213	C₄²	—	146	2	no	—
214	C₄²	C₂×#10596842	1	1	yes	characteristic
215	C₂³	—	73	1	no	—
216	C₂×C₄	—	73	2	no	—
217	C₂×C₄	—	73	2	no	—
218	C₂×C₄	—	73	2	no	—
219	C₂×C₄	—	73	2	no	—
220	C₂×C₄	—	73	2	no	—
221	C₂×C₄	—	73	2	no	—
222	C₂×C₄	—	73	1	no	—
223	C₂×C₄	—	73	1	no	—
224	C₂×C₄	—	73	2	no	—
225	C₂×C₄	—	73	2	no	—
226	C₂×C₄	—	73	1	no	—
227	C₂×C₄	—	73	1	no	—
228	C₂×C₄	—	73	1	no	—
229	C₂×C₄	—	73	1	no	—
230	C₂×C₄	—	73	2	no	—
231	C₂×C₄	—	73	2	no	—
232	C₂×C₄	—	73	1	no	—
233	C₂×C₄	—	146	1	no	—
234	C₂×C₄	—	146	1	no	—
235	C₂×C₄	—	146	1	no	—
236	C₂×C₄	—	146	1	no	—
237	C₂×C₄	—	146	1	no	—
238	C₂×C₄	—	146	1	no	—
239	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
240	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
241	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	292	8	no	—
244	C₈	—	73	2	no	—
245	C₈	—	146	2	no	—
246	C₈	—	146	2	no	—
247	C₈	—	146	1	no	—
248	C₈	C₄×#10596842	1	2	yes	complemented
249	C₂²	—	73	1	no	—
250	C₂²	—	73	1	no	—
251	C₂²	—	73	1	no	—
252	C₂²	—	73	1	no	—
253	C₂²	—	73	1	no	—
254	C₂²	—	73	1	no	—
255	C₂²	C₂×C₄⋊#10596842	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	73	4	no	—
258	C₄	—	73	4	no	—
259	C₄	—	73	2	no	—
260	C₄	—	73	1	no	—
261	C₄	—	73	1	no	—
262	C₄	—	146	2	no	—
263	C₄	—	146	2	no	—
264	C₄	—	146	1	no	—
265	C₄	#17989676	1	2	yes	direct product factor
266	C₄	C₂×C₄×#10596842	1	1	yes	characteristic
267	C₄	#17989700	1	1	yes	characteristic
268	C₂	—	73	1	no	—
269	C₂	—	73	1	no	—
270	C₂	—	73	1	no	—
271	C₂	—	73	1	no	—
272	C₂	C₂×#17989676	1	1	yes	characteristic
273	C₂	#22627055	1	1	yes	characteristic
274	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×#334
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. #17989676	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
108. C₈×C₇₃	C₄×C₈	semidirect
110. #10596831	C₄×C₈	semidirect
120. #64408	#334	semidirect
123. C₄×C₇₃	#334	semidirect
127. Dic₇₃	#334	semidirect
142. C₇₃	C₄×#334	semidirect
178. C₄×C₈	#10596842	semidirect
248. C₈	C₄×#10596842	semidirect
265. C₄	#17989676	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×#17989676	2
#22626937	2
C₄×C₈×#64408	2
C₄×#334	73

Subgroups — C4×#17989676 #34033289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17989676 #34033289