Subgroups — C₁₆×C₂²⋊#64408 #34012411

Order 18688.

510 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	#22626156	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×#64408	C₂	1	2	yes	maximal, complemented
3	C₈×C₂²⋊#64408	C₂	1	1	yes	maximal, characteristic
4	#22624968	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556664	C₂²	1	2	yes	—
7	#17989555	C₄	1	1	yes	characteristic
8	C₂×C₈×#64408	C₂²	1	2	yes	—
9	C₄×C₂²⋊#64408	C₄	1	1	yes	characteristic
10	#17989482	C₄	1	2	yes	complemented
11	C₂²⋊#11924031	C₄	1	2	yes	complemented
12	C₁₆×#64408	—	2	4	no	—
13	#17989452	C₄	1	2	yes	—
14	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
15	C₂²×#11924003	C₄	1	1	yes	characteristic, complemented
16	C₂×C₁₆×D₇₃	—	2	4	no	—
17	C₂×C₁₆×D₇₃	C₄	1	1	yes	characteristic, complemented
18	C₂×C₁₆×D₇₃	C₄	1	1	yes	characteristic, complemented
19	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic, complemented
22	C₂×#11924036	C₂×C₄	1	2	yes	—
23	C₂×C₂²⋊#64408	C₈	1	2	yes	—
24	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
25	#14556664	—	2	2	no	—
26	#14556664	—	2	2	no	—
27	C₂×#11924031	C₂×C₄	1	4	yes	—
28	#14556648	C₈	1	2	yes	—
29	C₂²⋊#10596833	C₈	1	2	yes	—
30	C₈×#64408	—	2	2	no	—
31	C₈×#64408	—	2	2	no	—
32	(C₂×#64408)⋊C₄	C₈	1	2	yes	—
33	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₇₃	—	2	2	no	—
36	C₂×C₈×D₇₃	—	2	2	no	—
37	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
41	C₂×#11924003	—	2	2	no	—
42	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
43	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
44	C₁₆×D₇₃	—	2	2	no	—
45	C₁₆×D₇₃	—	2	2	no	—
46	C₁₆×D₇₃	—	2	2	no	—
47	C₁₆×D₇₃	D₄	1	4	yes	—
48	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₆×C₇₃	—	2	2	no	—
50	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×#64408	C₂×C₈	1	4	yes	—
53	#11924036	—	2	2	no	—
54	#11924036	—	2	2	no	—
55	C₂×#10596833	C₂×C₈	1	4	yes	—
56	C₂²⋊#64408	C₁₆	1	8	yes	direct product factor
57	C₄×#64408	—	2	2	no	—
58	C₄×#64408	—	2	2	no	—
59	#11924031	—	2	4	no	—
60	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
62	C₂×C₄×D₇₃	—	2	2	no	—
63	C₂×C₄×D₇₃	—	2	2	no	—
64	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
65	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
66	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
67	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×#10596831	—	2	2	no	—
69	C₂×#10596831	C₄²	1	1	yes	characteristic
70	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
71	C₈×D₇₃	—	2	2	no	—
72	C₈×D₇₃	—	2	1	no	—
73	C₈×D₇₃	—	2	1	no	—
74	C₈×D₇₃	—	2	2	no	—
75	C₈×D₇₃	C₂×D₄	1	2	yes	—
76	C₈×D₇₃	#41	1	2	yes	—
77	#11924003	—	2	1	no	—
78	#11924003	C₂²⋊C₄	1	2	yes	complemented
79	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
80	C₂×C₈×C₇₃	—	2	1	no	—
81	C₂×C₈×C₇₃	—	2	1	no	—
82	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
83	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
84	C₁₆×C₇₃	—	2	1	no	—
85	C₁₆×C₇₃	C₂²⋊C₄	1	2	yes	complemented
86	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
87	C₂²×D₇₃	—	2	2	no	—
88	C₂²×D₇₃	—	2	2	no	—
89	C₂²×D₇₃	C₄×C₈	1	1	yes	characteristic
90	C₂²×D₇₃	C₂×C₁₆	1	4	yes	—
91	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×#64408	—	2	4	no	—
93	C₂×#64408	—	2	4	no	—
94	C₂×#64408	C₂×C₁₆	1	8	yes	complemented
95	C₂×C₄×C₇₃	—	2	1	no	—
96	C₂×C₄×C₇₃	—	2	1	no	—
97	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
98	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
99	C₂×Dic₇₃	—	2	2	no	—
100	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
101	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
102	C₄×D₇₃	—	2	2	no	—
103	C₄×D₇₃	—	2	1	no	—
104	C₄×D₇₃	—	2	1	no	—
105	C₄×D₇₃	—	2	2	no	—
106	C₄×D₇₃	C₄×D₄	1	2	yes	—
107	C₄×D₇₃	#131	1	2	yes	—
108	#10596833	—	2	4	no	—
109	C₈×C₇₃	—	2	1	no	—
110	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₈×C₇₃	#117	1	1	yes	characteristic
112	#10596831	—	2	1	no	—
113	#10596831	C₄×D₄	1	2	yes	—
114	C₂²×C₇₃	—	2	1	no	—
115	C₂²×C₇₃	—	2	1	no	—
116	C₂²×C₇₃	C₄×C₁₆	1	2	yes	complemented
117	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
118	C₂×D₇₃	—	2	1	no	—
119	C₂×D₇₃	—	2	1	no	—
120	C₂×D₇₃	—	2	8	no	—
121	C₂×D₇₃	C₄×C₁₆	1	2	yes	complemented
122	C₂×D₇₃	C₈×D₄	1	2	yes	—
123	C₂×D₇₃	C₂²×C₁₆	1	2	yes	—
124	C₂×D₇₃	#504	1	2	yes	—
125	#64408	—	2	8	no	—
126	C₄×C₇₃	—	2	1	no	—
127	C₄×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
128	C₄×C₇₃	#405	1	1	yes	characteristic
129	Dic₇₃	—	2	1	no	—
130	Dic₇₃	C₈×D₄	1	2	yes	—
131	C₁₆×C₂²⋊C₄	—	73	1	no	maximal, Sylow
132	C₂×C₇₃	—	2	2	no	—
133	C₂×C₇₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₇₃	C₂×C₄×C₁₆	1	1	yes	characteristic
135	C₂×C₇₃	#2108	1	1	yes	characteristic
136	D₇₃	—	2	2	no	—
137	D₇₃	D₄×C₁₆	1	4	yes	—
138	C₂³×C₁₆	—	73	1	no	—
139	C₂×C₂²⋊C₁₆	—	73	1	no	—
140	C₂×C₄×C₁₆	—	73	2	no	—
141	C₈×C₂²⋊C₄	—	73	1	no	—
142	#1374	—	73	2	no	—
143	C₇₃	C₁₆×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
144	C₂³×C₈	—	73	1	no	—
145	C₂²×C₁₆	—	73	1	no	—
146	C₂²×C₁₆	—	73	2	no	—
147	C₂²×C₁₆	—	73	1	no	—
148	C₂²×C₁₆	—	73	1	no	—
149	C₂²×C₁₆	—	73	1	no	—
150	C₂²×C₁₆	—	73	1	no	—
151	C₂²×C₁₆	—	146	4	no	—
152	C₂²×C₁₆	#64408	1	1	yes	characteristic, complemented
153	C₂×C₂²⋊C₈	—	73	1	no	—
154	C₂×C₄×C₈	—	73	2	no	—
155	C₄×C₂²⋊C₄	—	73	1	no	—
156	C₂²⋊C₁₆	—	73	2	no	—
157	C₂²⋊C₁₆	—	73	2	no	—
158	C₄×C₁₆	—	146	4	no	—
159	#336	—	73	2	no	—
160	C₂³×C₄	—	73	1	no	—
161	C₂²×C₈	—	73	1	no	—
162	C₂²×C₈	—	73	2	no	—
163	C₂²×C₈	—	73	1	no	—
164	C₂²×C₈	—	73	1	no	—
165	C₂²×C₈	—	73	1	no	—
166	C₂²×C₈	—	73	1	no	—
167	C₂²×C₈	—	146	2	no	—
168	C₂²×C₈	—	146	2	no	—
169	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	—	73	2	no	—
171	C₂×C₄²	—	73	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	73	1	no	—
174	C₂×C₁₆	—	73	4	no	—
175	C₂×C₁₆	—	73	4	no	—
176	C₂×C₁₆	—	73	1	no	—
177	C₂×C₁₆	—	146	2	no	—
178	C₂×C₁₆	—	146	2	no	—
179	C₂×C₁₆	—	146	2	no	—
180	C₂×C₁₆	—	146	2	no	—
181	C₂×C₁₆	—	146	2	no	—
182	C₂×C₁₆	—	146	2	no	—
183	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
184	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
185	C₂²⋊C₈	—	73	2	no	—
186	C₂²⋊C₈	—	73	2	no	—
187	C₄×C₈	—	146	2	no	—
188	C₄×C₈	—	146	2	no	—
189	#95	—	73	2	no	—
190	C₂⁴	—	73	1	no	—
191	C₂²×C₄	—	73	1	no	—
192	C₂²×C₄	—	73	4	no	—
193	C₂²×C₄	—	73	1	no	—
194	C₂²×C₄	—	73	1	no	—
195	C₂²×C₄	—	73	1	no	—
196	C₂²×C₄	—	73	1	no	—
197	C₂²×C₄	—	146	2	no	—
198	C₂²×C₄	—	146	2	no	—
199	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	73	1	no	—
203	C₂×C₈	—	73	2	no	—
204	C₂×C₈	—	73	2	no	—
205	C₂×C₈	—	73	4	no	—
206	C₂×C₈	—	73	1	no	—
207	C₂×C₈	—	146	2	no	—
208	C₂×C₈	—	146	2	no	—
209	C₂×C₈	—	146	2	no	—
210	C₂×C₈	—	146	2	no	—
211	C₂×C₈	—	146	2	no	—
212	C₂×C₈	—	146	1	no	—
213	C₂×C₈	—	146	1	no	—
214	C₂×C₈	C₄×#64408	1	1	yes	characteristic
215	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
216	C₂²⋊C₄	—	73	8	no	—
217	C₄²	—	146	2	no	—
218	C₄²	—	146	2	no	—
219	C₁₆	—	2	1	no	—
220	C₁₆	—	73	2	no	—
221	C₁₆	—	146	4	no	—
222	C₁₆	—	146	1	no	—
223	C₁₆	C₂²⋊#64408	1	2	yes	direct product factor
224	C₂³	—	73	1	no	—
225	C₂³	—	73	4	no	—
226	C₂³	—	73	1	no	—
227	C₂³	—	146	2	no	—
228	C₂³	—	146	2	no	—
229	C₂³	C₈×#64408	1	1	yes	characteristic
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	73	1	no	—
233	C₂×C₄	—	73	2	no	—
234	C₂×C₄	—	73	2	no	—
235	C₂×C₄	—	73	8	no	—
236	C₂×C₄	—	73	1	no	—
237	C₂×C₄	—	146	2	no	—
238	C₂×C₄	—	146	2	no	—
239	C₂×C₄	—	146	2	no	—
240	C₂×C₄	—	146	4	no	—
241	C₂×C₄	—	146	4	no	—
242	C₂×C₄	—	146	1	no	—
243	C₂×C₄	—	146	1	no	—
244	C₂×C₄	C₈×#64408	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
246	C₈	—	2	1	no	—
247	C₈	—	73	2	no	—
248	C₈	—	146	4	no	—
249	C₈	—	146	1	no	—
250	C₈	C₂×C₂²⋊#64408	1	1	yes	characteristic
251	C₈	#14556670	1	1	yes	characteristic
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	73	2	no	—
255	C₂²	—	73	2	no	—
256	C₂²	—	73	2	no	—
257	C₂²	—	73	2	no	—
258	C₂²	—	146	8	no	—
259	C₂²	—	146	1	no	—
260	C₂²	—	146	1	no	—
261	C₂²	C₁₆×#64408	1	2	yes	complemented
262	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	73	2	no	—
265	C₄	—	146	8	no	—
266	C₄	—	146	1	no	—
267	C₄	C₄×C₂²⋊#64408	1	1	yes	characteristic
268	C₄	#17989550	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	73	4	no	—
271	C₂	—	146	2	no	—
272	C₂	C₈×C₂²⋊#64408	1	1	yes	characteristic
273	C₂	C₂×C₁₆×#64408	1	1	yes	characteristic
274	C₂	#22626139	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×#64408
commutator	C₂×C₇₃	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₂²×C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	C₂²×C₁₆	#64408
hypocenter	C₇₃	C₁₆×C₂²⋊C₄
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₆×#64408	C₂	semidirect
4. #22624968	C₂	semidirect
10. #17989482	C₄	semidirect
11. C₂²⋊#11924031	C₄	semidirect
15. C₂²×#11924003	C₄	semidirect
17. C₂×C₁₆×D₇₃	C₄	semidirect
18. C₂×C₁₆×D₇₃	C₄	semidirect
21. C₂²×C₁₆×C₇₃	C₄	semidirect
56. C₂²⋊#64408	C₁₆	direct (N × H)
78. #11924003	C₂²⋊C₄	semidirect
85. C₁₆×C₇₃	C₂²⋊C₄	semidirect
94. C₂×#64408	C₂×C₁₆	semidirect
116. C₂²×C₇₃	C₄×C₁₆	semidirect
121. C₂×D₇₃	C₄×C₁₆	semidirect
143. C₇₃	C₁₆×C₂²⋊C₄	semidirect
152. C₂²×C₁₆	#64408	semidirect
223. C₁₆	C₂²⋊#64408	direct (N × H)
261. C₂²	C₁₆×#64408	semidirect

Maximal subgroups

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

subgroup	index
#22626156	2
C₂×C₁₆×#64408	2
C₈×C₂²⋊#64408	2
#22624968	2
C₂²×C₁₆×D₇₃	2
C₁₆×C₂²⋊C₄	73

Subgroups — C16×C22⋊#64408 #34012411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×C₂²⋊#64408 #34012411