Subgroups — #335×#64408 #34003979

Order 18688.

470 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₈×#64408	C₂	1	2	yes	maximal
2	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
3	#22625476	C₂	1	1	yes	maximal, characteristic
4	#22624760	C₂	1	2	yes	maximal
5	#335×D₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989588	C₂²	1	2	yes	—
7	C₄×#11924036	C₂²	1	1	yes	characteristic
8	C₂×C₈×#64408	—	2	2	no	—
9	C₂×C₈×#64408	C₄	1	2	yes	—
10	C₂×C₈×#64408	C₄	1	2	yes	—
11	C₄²×#64408	C₂²	1	1	yes	characteristic
12	#17989452	—	2	2	no	—
13	#17989451	C₄	1	2	yes	complemented
14	#17989450	C₄	1	2	yes	complemented
15	C₄⋊C₈×D₇₃	C₂²	1	2	yes	—
16	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
17	#17988340	C₄	1	1	yes	characteristic, complemented
18	#17988339	C₄	1	2	yes	complemented
19	#335×C₇₃	C₄	1	1	yes	characteristic, complemented
20	C₂×#11924036	—	2	2	no	—
21	C₂×#11924036	C₂×C₄	1	2	yes	—
22	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
23	C₂×C₄×#64408	D₄	1	1	yes	characteristic
24	C₂×C₄×#64408	D₄	1	1	yes	characteristic
25	C₂×C₄×#64408	Q₈	1	2	yes	—
26	#14556658	—	2	2	no	—
27	C₄⋊#10596833	C₂×C₄	1	4	yes	—
28	C₄×#10596833	—	2	1	no	—
29	C₄×#10596833	—	2	1	no	—
30	C₈×#64408	—	2	4	no	—
31	C₈×#64408	—	4	4	no	—
32	C₈×#64408	C₈	1	8	yes	complemented
33	C₂×C₈×D₇₃	—	2	2	no	—
34	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×D₇₃	C₂³	1	1	yes	characteristic
37	C₄⋊#10596831	C₂×C₄	1	2	yes	—
38	#14556520	—	2	2	no	—
39	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
41	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×#64408	C₄⋊C₄	1	2	yes	—
45	C₂²×#64408	C₄⋊C₄	1	2	yes	—
46	#11924036	—	2	2	no	—
47	#11924036	—	4	2	no	—
48	#11924036	—	4	2	no	—
49	#11924036	C₂×C₈	1	4	yes	—
50	C₂×#10596833	—	2	2	no	—
51	C₂×#10596833	—	2	4	no	—
52	C₄×#64408	—	2	4	no	—
53	C₄×#64408	—	2	1	no	—
54	C₄×#64408	—	2	1	no	—
55	C₄×#64408	C₂×C₈	1	4	yes	—
56	C₄×#64408	M₁₆	1	4	yes	—
57	C₄×#64408	D₈	1	2	yes	—
58	C₄×#64408	Q₁₆	1	2	yes	—
59	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×#10596831	—	2	2	no	—
63	C₂×#10596831	C₄²	1	1	yes	characteristic
64	C₂×#10596831	C₄²	1	1	yes	characteristic
65	C₈×D₇₃	—	2	2	no	—
66	C₈×D₇₃	—	4	2	no	—
67	C₈×D₇₃	—	4	2	no	—
68	C₈×D₇₃	C₂×C₈	1	4	yes	—
69	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
70	C₄×Dic₇₃	#41	1	1	yes	characteristic
71	C₄×Dic₇₃	#41	1	1	yes	characteristic
72	C₂×C₈×C₇₃	—	2	2	no	—
73	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
74	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
75	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×#64408	—	2	2	no	—
78	C₂×#64408	—	2	2	no	—
79	C₂×#64408	C₄⋊C₈	1	4	yes	—
80	C₂×#64408	C₄⋊C₈	1	4	yes	—
81	C₂×#64408	#107	1	4	yes	—
82	C₂×#64408	#108	1	4	yes	—
83	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
86	C₂×Dic₇₃	#117	1	1	yes	characteristic
87	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
88	C₂×Dic₇₃	C₄×Q₈	1	1	yes	characteristic
89	C₄×D₇₃	—	2	2	no	—
90	C₄×D₇₃	C₂²×C₈	1	2	yes	—
91	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
92	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
93	C₄×D₇₃	C₂×Q₁₆	1	1	yes	characteristic
94	C₄×D₇₃	#135	1	1	yes	characteristic
95	C₄×D₇₃	#135	1	1	yes	characteristic
96	#10596833	—	2	4	no	—
97	#10596833	—	4	4	no	—
98	C₈×C₇₃	—	2	1	no	—
99	C₈×C₇₃	—	4	2	no	—
100	C₈×C₇₃	C₄×C₈	1	2	yes	complemented
101	#10596831	—	2	1	no	—
102	#10596831	—	4	2	no	—
103	#10596831	C₄×C₈	1	2	yes	complemented
104	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
105	C₂×D₇₃	C₂×C₄⋊C₈	1	2	yes	—
106	C₂×D₇₃	C₂×#107	1	1	yes	characteristic
107	C₂×D₇₃	#427	1	1	yes	characteristic
108	C₂×D₇₃	C₂×#108	1	2	yes	—
109	#64408	—	2	4	no	—
110	#64408	#335	1	8	yes	direct product factor
111	C₄×C₇₃	—	2	1	no	—
112	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
113	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
114	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
115	C₄×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
116	Dic₇₃	—	2	1	no	—
117	Dic₇₃	#431	1	2	yes	—
118	Dic₇₃	C₄×D₈	1	1	yes	characteristic
119	Dic₇₃	C₄×Q₁₆	1	1	yes	characteristic
120	C₄×#335	—	73	1	no	maximal, Sylow
121	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
122	C₂×C₇₃	C₄×#107	1	1	yes	characteristic
123	C₂×C₇₃	C₄×#108	1	1	yes	characteristic
124	D₇₃	C₂×#335	1	2	yes	—
125	D₇₃	#1564	1	2	yes	—
126	C₄×C₄⋊C₈	—	73	2	no	—
127	C₄²×C₈	—	73	1	no	—
128	C₂×#335	—	73	1	no	—
129	C₂×#335	—	73	1	no	—
130	#1277	—	73	2	no	—
131	C₇₃	C₄×#335	1	1	yes	characteristic, complemented, Sylow
132	C₂×C₄⋊C₈	—	73	2	no	—
133	C₂×C₄⋊C₈	—	73	2	no	—
134	C₂×C₄×C₈	—	73	2	no	—
135	C₂×C₄×C₈	—	73	1	no	—
136	C₂×C₄×C₈	—	73	2	no	—
137	C₂×C₄×C₈	—	73	1	no	—
138	C₂×C₄×C₈	—	146	2	no	—
139	C₄³	—	73	1	no	—
140	#336	—	146	2	no	—
141	#335	—	73	2	no	—
142	#335	—	73	2	no	—
143	#335	—	73	2	no	—
144	#335	—	73	1	no	—
145	#335	#64408	1	1	yes	characteristic, direct product factor
146	C₂²×C₈	—	73	1	no	—
147	C₂²×C₈	—	73	2	no	—
148	C₂²×C₈	—	73	1	no	—
149	C₂²×C₈	—	146	2	no	—
150	C₂²×C₈	—	146	2	no	—
151	C₂×C₄²	—	73	2	no	—
152	C₂×C₄²	—	73	1	no	—
153	C₂×C₄²	—	73	1	no	—
154	C₂×C₄²	—	73	1	no	—
155	C₂×C₄²	—	73	2	no	—
156	C₄⋊C₈	—	73	4	no	—
157	C₄⋊C₈	—	73	2	no	—
158	C₄⋊C₈	—	146	2	no	—
159	C₄⋊C₈	—	146	2	no	—
160	C₄⋊C₈	C₂×#64408	1	2	yes	—
161	C₄×C₈	—	292	4	no	—
162	C₄×C₈	—	73	8	no	—
163	C₄×C₈	—	73	1	no	—
164	C₄×C₈	—	73	1	no	—
165	C₄×C₈	—	73	1	no	—
166	C₄×C₈	—	146	4	no	—
167	C₄×C₈	—	146	1	no	—
168	C₄×C₈	—	146	1	no	—
169	C₄×C₈	C₂×#64408	1	1	yes	characteristic
170	C₂²×C₄	—	73	2	no	—
171	C₂²×C₄	—	73	1	no	—
172	C₂²×C₄	—	73	1	no	—
173	C₂²×C₄	—	73	2	no	—
174	C₂²×C₄	—	73	1	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	292	2	no	—
177	C₂×C₈	—	292	2	no	—
178	C₂×C₈	—	292	2	no	—
179	C₂×C₈	—	292	2	no	—
180	C₂×C₈	—	73	1	no	—
181	C₂×C₈	—	73	4	no	—
182	C₂×C₈	—	73	4	no	—
183	C₂×C₈	—	73	1	no	—
184	C₂×C₈	—	146	2	no	—
185	C₂×C₈	—	146	4	no	—
186	C₂×C₈	—	146	2	no	—
187	C₂×C₈	—	146	2	no	—
188	C₂×C₈	—	146	2	no	—
189	C₂×C₈	C₄×#64408	1	1	yes	characteristic
190	C₂×C₈	C₄×#64408	1	1	yes	characteristic
191	C₄²	—	73	4	no	—
192	C₄²	—	73	4	no	—
193	C₄²	—	73	2	no	—
194	C₄²	—	73	2	no	—
195	C₄²	—	73	1	no	—
196	C₄²	—	73	1	no	—
197	C₄²	—	73	1	no	—
198	C₄²	—	146	4	no	—
199	C₄²	—	146	1	no	—
200	C₄²	—	146	1	no	—
201	C₄²	C₂²×#64408	1	1	yes	characteristic
202	C₂³	—	73	1	no	—
203	C₂×C₄	—	73	4	no	—
204	C₂×C₄	—	73	4	no	—
205	C₂×C₄	—	73	4	no	—
206	C₂×C₄	—	73	1	no	—
207	C₂×C₄	—	73	1	no	—
208	C₂×C₄	—	73	2	no	—
209	C₂×C₄	—	73	2	no	—
210	C₂×C₄	—	73	1	no	—
211	C₂×C₄	—	73	1	no	—
212	C₂×C₄	—	73	1	no	—
213	C₂×C₄	—	73	1	no	—
214	C₂×C₄	—	73	4	no	—
215	C₂×C₄	—	73	1	no	—
216	C₂×C₄	—	146	2	no	—
217	C₂×C₄	—	146	2	no	—
218	C₂×C₄	—	146	2	no	—
219	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
220	C₂×C₄	D₄×#64408	1	1	yes	characteristic
221	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
222	C₈	—	2	1	no	—
223	C₈	—	4	2	no	—
224	C₈	—	292	4	no	—
225	C₈	—	292	2	no	—
226	C₈	—	73	2	no	—
227	C₈	—	146	4	no	—
228	C₈	—	146	1	no	—
229	C₈	C₈×#64408	1	2	yes	complemented
230	C₂²	—	73	2	no	—
231	C₂²	—	73	1	no	—
232	C₂²	—	73	1	no	—
233	C₂²	—	73	2	no	—
234	C₂²	C₄⋊C₄×#64408	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	—	73	8	no	—
237	C₄	—	73	2	no	—
238	C₄	—	73	1	no	—
239	C₄	—	73	1	no	—
240	C₄	—	146	4	no	—
241	C₄	—	146	1	no	—
242	C₄	C₂×C₈×#64408	1	1	yes	characteristic
243	C₄	M₁₆×#64408	1	1	yes	characteristic
244	C₄	D₈×#64408	1	1	yes	characteristic
245	C₄	Q₁₆×#64408	1	1	yes	characteristic
246	C₂	—	73	2	no	—
247	C₂	—	73	2	no	—
248	C₂	C₄⋊C₈×#64408	1	1	yes	characteristic
249	C₂	#107×#64408	1	1	yes	characteristic
250	C₂	#108×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×#64408
Fitting	#335×C₇₃	C₄
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂²×D₇₃	C₂×C₄⋊C₄
cosocle	C₄²×D₇₃	C₂³
hypercenter	#335	#64408
hypocenter	C₇₃	C₄×#335
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

Complements

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

N	H ≅ G/N	type
13. #17989451	C₄	semidirect
14. #17989450	C₄	semidirect
17. #17988340	C₄	semidirect
18. #17988339	C₄	semidirect
19. #335×C₇₃	C₄	semidirect
32. C₈×#64408	C₈	semidirect
100. C₈×C₇₃	C₄×C₈	semidirect
103. #10596831	C₄×C₈	semidirect
110. #64408	#335	direct (N × H)
131. C₇₃	C₄×#335	semidirect
145. #335	#64408	direct (N × H)
229. C₈	C₈×#64408	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₈×#64408	2
C₄×C₈×#64408	2
#22625476	2
#22624760	2
#335×D₇₃	2
C₄×#335	73