Subgroups — #334×#64408 #34003976

Order 18688.

470 conjugacy classes of subgroups.

The subgroup lattice has 240 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	#22625473	C₂	1	1	yes	maximal, characteristic
4	#22624760	C₂	1	2	yes	maximal
5	#334×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	#17989449	C₄	1	2	yes	complemented
14	#17989448	C₄	1	2	yes	complemented
15	C₄⋊C₈×D₇₃	C₂²	1	2	yes	—
16	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
17	#17988338	C₄	1	1	yes	characteristic, complemented
18	#17988337	C₄	1	2	yes	complemented
19	#334×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	2	no	—
54	C₄×#64408	C₂×C₈	1	4	yes	—
55	C₄×#64408	M₁₆	1	4	yes	—
56	C₄×#64408	SD₁₆	1	4	yes	—
57	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×#10596831	—	2	2	no	—
61	C₂×#10596831	C₄²	1	1	yes	characteristic
62	C₂×#10596831	C₄²	1	1	yes	characteristic
63	C₈×D₇₃	—	2	2	no	—
64	C₈×D₇₃	—	4	2	no	—
65	C₈×D₇₃	—	4	2	no	—
66	C₈×D₇₃	C₂×C₈	1	4	yes	—
67	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₄×Dic₇₃	#41	1	1	yes	characteristic
69	C₄×Dic₇₃	#41	1	1	yes	characteristic
70	C₂×C₈×C₇₃	—	2	2	no	—
71	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
72	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
73	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×#64408	—	2	2	no	—
76	C₂×#64408	—	2	2	no	—
77	C₂×#64408	C₄⋊C₈	1	4	yes	—
78	C₂×#64408	C₄⋊C₈	1	4	yes	—
79	C₂×#64408	#106	1	4	yes	—
80	C₂×#64408	#108	1	4	yes	—
81	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
84	C₂×Dic₇₃	#117	1	1	yes	characteristic
85	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
86	C₂×Dic₇₃	C₄×Q₈	1	1	yes	characteristic
87	C₄×D₇₃	—	2	2	no	—
88	C₄×D₇₃	C₂²×C₈	1	2	yes	—
89	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
90	C₄×D₇₃	C₂×SD₁₆	1	2	yes	—
91	C₄×D₇₃	#135	1	2	yes	—
92	#10596833	—	2	4	no	—
93	#10596833	—	4	4	no	—
94	C₈×C₇₃	—	2	1	no	—
95	C₈×C₇₃	—	4	2	no	—
96	C₈×C₇₃	C₄×C₈	1	2	yes	complemented
97	#10596831	—	2	1	no	—
98	#10596831	—	4	2	no	—
99	#10596831	C₄×C₈	1	2	yes	complemented
100	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
101	C₂×D₇₃	C₂×C₄⋊C₈	1	2	yes	—
102	C₂×D₇₃	C₂×#106	1	1	yes	characteristic
103	C₂×D₇₃	#427	1	1	yes	characteristic
104	C₂×D₇₃	C₂×#108	1	2	yes	—
105	#64408	—	2	4	no	—
106	#64408	#334	1	8	yes	direct product factor
107	C₄×C₇₃	—	2	1	no	—
108	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
109	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
110	C₄×C₇₃	C₄×SD₁₆	1	2	yes	—
111	Dic₇₃	—	2	1	no	—
112	Dic₇₃	#431	1	2	yes	—
113	Dic₇₃	C₄×SD₁₆	1	2	yes	—
114	C₄×#334	—	73	1	no	maximal, Sylow
115	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
116	C₂×C₇₃	C₄×#106	1	1	yes	characteristic
117	C₂×C₇₃	C₄×#108	1	1	yes	characteristic
118	D₇₃	C₂×#334	1	2	yes	—
119	D₇₃	#1564	1	2	yes	—
120	C₄×C₄⋊C₈	—	73	2	no	—
121	C₄²×C₈	—	73	1	no	—
122	C₂×#334	—	73	1	no	—
123	C₂×#334	—	73	1	no	—
124	#1277	—	73	2	no	—
125	C₇₃	C₄×#334	1	1	yes	characteristic, complemented, Sylow
126	C₂×C₄⋊C₈	—	73	2	no	—
127	C₂×C₄⋊C₈	—	73	2	no	—
128	C₂×C₄×C₈	—	73	2	no	—
129	C₂×C₄×C₈	—	73	1	no	—
130	C₂×C₄×C₈	—	73	2	no	—
131	C₂×C₄×C₈	—	73	1	no	—
132	C₂×C₄×C₈	—	146	2	no	—
133	C₄³	—	73	1	no	—
134	#336	—	146	2	no	—
135	#334	—	73	2	no	—
136	#334	—	73	2	no	—
137	#334	—	73	2	no	—
138	#334	—	73	1	no	—
139	#334	#64408	1	1	yes	characteristic, direct product factor
140	C₂²×C₈	—	73	1	no	—
141	C₂²×C₈	—	73	2	no	—
142	C₂²×C₈	—	73	1	no	—
143	C₂²×C₈	—	146	2	no	—
144	C₂²×C₈	—	146	2	no	—
145	C₂×C₄²	—	73	2	no	—
146	C₂×C₄²	—	73	1	no	—
147	C₂×C₄²	—	73	1	no	—
148	C₂×C₄²	—	73	1	no	—
149	C₂×C₄²	—	73	2	no	—
150	C₄⋊C₈	—	73	4	no	—
151	C₄⋊C₈	—	73	2	no	—
152	C₄⋊C₈	—	146	2	no	—
153	C₄⋊C₈	—	146	2	no	—
154	C₄⋊C₈	C₂×#64408	1	2	yes	—
155	C₄×C₈	—	292	4	no	—
156	C₄×C₈	—	73	8	no	—
157	C₄×C₈	—	73	1	no	—
158	C₄×C₈	—	73	1	no	—
159	C₄×C₈	—	73	1	no	—
160	C₄×C₈	—	146	4	no	—
161	C₄×C₈	—	146	1	no	—
162	C₄×C₈	—	146	1	no	—
163	C₄×C₈	C₂×#64408	1	1	yes	characteristic
164	C₂²×C₄	—	73	2	no	—
165	C₂²×C₄	—	73	1	no	—
166	C₂²×C₄	—	73	1	no	—
167	C₂²×C₄	—	73	2	no	—
168	C₂²×C₄	—	73	1	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	292	2	no	—
171	C₂×C₈	—	292	2	no	—
172	C₂×C₈	—	292	2	no	—
173	C₂×C₈	—	292	2	no	—
174	C₂×C₈	—	73	1	no	—
175	C₂×C₈	—	73	4	no	—
176	C₂×C₈	—	73	4	no	—
177	C₂×C₈	—	73	1	no	—
178	C₂×C₈	—	146	2	no	—
179	C₂×C₈	—	146	4	no	—
180	C₂×C₈	—	146	2	no	—
181	C₂×C₈	—	146	2	no	—
182	C₂×C₈	—	146	2	no	—
183	C₂×C₈	C₄×#64408	1	1	yes	characteristic
184	C₂×C₈	C₄×#64408	1	1	yes	characteristic
185	C₄²	—	73	4	no	—
186	C₄²	—	73	4	no	—
187	C₄²	—	73	4	no	—
188	C₄²	—	73	1	no	—
189	C₄²	—	73	1	no	—
190	C₄²	—	73	1	no	—
191	C₄²	—	146	4	no	—
192	C₄²	—	146	2	no	—
193	C₄²	C₂²×#64408	1	1	yes	characteristic
194	C₂³	—	73	1	no	—
195	C₂×C₄	—	73	4	no	—
196	C₂×C₄	—	73	4	no	—
197	C₂×C₄	—	73	4	no	—
198	C₂×C₄	—	73	1	no	—
199	C₂×C₄	—	73	1	no	—
200	C₂×C₄	—	73	2	no	—
201	C₂×C₄	—	73	2	no	—
202	C₂×C₄	—	73	2	no	—
203	C₂×C₄	—	73	2	no	—
204	C₂×C₄	—	73	4	no	—
205	C₂×C₄	—	73	1	no	—
206	C₂×C₄	—	146	2	no	—
207	C₂×C₄	—	146	2	no	—
208	C₂×C₄	—	146	2	no	—
209	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
210	C₂×C₄	D₄×#64408	1	1	yes	characteristic
211	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	4	2	no	—
214	C₈	—	292	4	no	—
215	C₈	—	292	2	no	—
216	C₈	—	73	2	no	—
217	C₈	—	146	4	no	—
218	C₈	—	146	1	no	—
219	C₈	C₈×#64408	1	2	yes	complemented
220	C₂²	—	73	2	no	—
221	C₂²	—	73	1	no	—
222	C₂²	—	73	1	no	—
223	C₂²	—	73	2	no	—
224	C₂²	C₄⋊C₄×#64408	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	73	8	no	—
227	C₄	—	73	2	no	—
228	C₄	—	73	2	no	—
229	C₄	—	146	4	no	—
230	C₄	—	146	1	no	—
231	C₄	C₂×C₈×#64408	1	1	yes	characteristic
232	C₄	M₁₆×#64408	1	1	yes	characteristic
233	C₄	SD₁₆×#64408	1	2	yes	—
234	C₂	—	73	2	no	—
235	C₂	—	73	2	no	—
236	C₂	C₄⋊C₈×#64408	1	1	yes	characteristic
237	C₂	#106×#64408	1	1	yes	characteristic
238	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	#334×C₇₃	C₄
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂²×D₇₃	C₂×C₄⋊C₄
cosocle	C₄²×D₇₃	C₂³
hypercenter	#334	#64408
hypocenter	C₇₃	C₄×#334
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

Complements

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

N	H ≅ G/N	type
13. #17989449	C₄	semidirect
14. #17989448	C₄	semidirect
17. #17988338	C₄	semidirect
18. #17988337	C₄	semidirect
19. #334×C₇₃	C₄	semidirect
32. C₈×#64408	C₈	semidirect
96. C₈×C₇₃	C₄×C₈	semidirect
99. #10596831	C₄×C₈	semidirect
106. #64408	#334	direct (N × H)
125. C₇₃	C₄×#334	semidirect
139. #334	#64408	direct (N × H)
219. 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
#22625473	2
#22624760	2
#334×D₇₃	2
C₄×#334	73