Subgroups — C₂×#22624999 #34012190

Order 18688.

586 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989597	C₂	1	2	yes	maximal
2	#22624999	C₂	1	4	yes	maximal, direct product factor
3	C₂²×#14556530	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556670	—	2	2	no	—
5	#17989597	—	2	2	no	—
6	#17989597	—	2	2	no	—
7	#17989597	C₄	1	4	yes	complemented
8	#17989597	C₂²	1	4	yes	—
9	C₂×#14556660	—	2	2	no	—
10	C₂×#14556659	C₄	1	4	yes	complemented
11	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	C₂×#14556530	—	2	2	no	—
13	C₂×#14556530	—	2	2	no	—
14	C₂×#14556530	C₄	1	2	yes	complemented
15	C₂×#14556530	C₂²	1	2	yes	—
16	C₂²×#11924003	C₄	1	1	yes	characteristic, complemented
17	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic, complemented
18	#14556670	—	2	4	no	—
19	#14556670	—	2	4	no	—
20	#14556670	—	4	2	no	—
21	#14556670	—	4	2	no	—
22	C₂×C₄⋊#64408	—	2	4	no	—
23	C₂×C₂²⋊#64408	—	4	2	no	—
24	C₂×C₄×#64408	—	4	2	no	—
25	#14556660	—	2	4	no	—
26	#14556660	—	2	4	no	—
27	#14556659	—	2	4	no	—
28	#14556659	C₂×C₄	1	8	yes	complemented
29	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
30	C₂²×#10596831	Q₈	1	1	yes	characteristic
31	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₇₃	D₄	1	2	yes	—
35	C₂×C₈×D₇₃	D₄	1	2	yes	—
36	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
37	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
38	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
39	#14556530	—	2	4	no	—
40	#14556530	—	2	4	no	—
41	C₂×#11924003	—	2	1	no	—
42	C₂×#11924003	—	2	1	no	—
43	C₂×#11924003	C₂×C₄	1	2	yes	complemented
44	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₆×C₇₃	—	2	1	no	—
46	C₂×C₁₆×C₇₃	—	2	1	no	—
47	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	complemented
48	C₂²×#64408	—	4	4	no	—
49	C₄⋊#64408	—	2	8	no	—
50	C₄⋊#64408	—	4	4	no	—
51	C₂²⋊#64408	—	4	4	no	—
52	C₂²⋊#64408	—	4	4	no	—
53	C₄×#64408	—	4	4	no	—
54	C₄×#64408	—	4	4	no	—
55	C₂³×D₇₃	D₈	1	1	yes	characteristic
56	C₂²×Dic₇₃	Q₁₆	1	1	yes	characteristic
57	C₂×C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
61	C₂×C₄×D₇₃	SD₁₆	1	2	yes	—
62	C₂×C₄×D₇₃	SD₁₆	1	2	yes	—
63	C₂×C₄×D₇₃	Q₁₆	1	1	yes	characteristic
64	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
65	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
66	C₂×#10596831	C₄⋊C₄	1	2	yes	—
67	C₂×#10596831	C₄⋊C₄	1	2	yes	—
68	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
69	C₈×D₇₃	C₄²	1	2	yes	—
70	C₈×D₇₃	C₂²⋊C₄	1	2	yes	—
71	C₈×D₇₃	C₂²⋊C₄	1	2	yes	—
72	C₈×D₇₃	C₂²⋊C₄	1	2	yes	—
73	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
74	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
75	C₈×D₇₃	C₂²×C₄	1	2	yes	—
76	C₈×D₇₃	C₂×D₄	1	2	yes	—
77	#11924003	—	2	2	no	—
78	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
80	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
81	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
85	C₁₆×C₇₃	—	2	2	no	—
86	C₂³×C₇₃	#107	1	1	yes	characteristic
87	C₂²×D₇₃	—	4	2	no	—
88	C₂²×D₇₃	#102	1	2	yes	—
89	C₂²×D₇₃	#102	1	2	yes	—
90	C₂²×D₇₃	#107	1	1	yes	characteristic
91	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
92	C₂×#64408	—	4	8	no	—
93	C₂×#64408	—	8	8	no	—
94	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
95	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
96	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
97	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
98	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
99	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×Dic₇₃	#103	1	2	yes	—
101	C₂×Dic₇₃	#103	1	2	yes	—
102	C₂×Dic₇₃	#107	1	1	yes	characteristic
103	C₂×Dic₇₃	C₂×Q₁₆	1	1	yes	characteristic
104	C₄×D₇₃	#95	1	2	yes	—
105	C₄×D₇₃	#102	1	2	yes	—
106	C₄×D₇₃	#102	1	2	yes	—
107	C₄×D₇₃	#103	1	2	yes	—
108	C₄×D₇₃	#103	1	2	yes	—
109	C₄×D₇₃	#106	1	2	yes	—
110	C₄×D₇₃	C₂×C₂²⋊C₄	1	2	yes	—
111	C₄×D₇₃	C₂×SD₁₆	1	2	yes	—
112	C₈×C₇₃	#95	1	1	yes	characteristic
113	C₈×C₇₃	#95	1	1	yes	characteristic
114	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
115	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	#10596831	#95	1	2	yes	—
117	#10596831	C₂×C₄⋊C₄	1	2	yes	—
118	C₂²×C₇₃	—	2	2	no	—
119	C₂²×C₇₃	#340	1	1	yes	characteristic
120	C₂²×C₇₃	#340	1	1	yes	characteristic
121	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
122	C₂×D₇₃	—	2	4	no	—
123	C₂×D₇₃	—	4	2	no	—
124	C₂×D₇₃	—	4	1	no	—
125	C₂×D₇₃	—	4	1	no	—
126	C₂×D₇₃	#340	1	2	yes	—
127	C₂×D₇₃	C₂×#102	1	2	yes	—
128	#64408	—	8	8	no	—
129	C₄×C₇₃	#340	1	1	yes	characteristic
130	C₄×C₇₃	#340	1	1	yes	characteristic
131	C₄×C₇₃	C₂×#95	1	1	yes	characteristic
132	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
133	Dic₇₃	#340	1	2	yes	—
134	Dic₇₃	C₂×#103	1	2	yes	—
135	C₂×#1384	—	73	1	no	maximal, Sylow
136	C₂×C₇₃	—	2	1	no	—
137	C₂×C₇₃	—	2	1	no	—
138	C₂×C₇₃	#1384	1	2	yes	complemented
139	C₂×C₇₃	C₂×#340	1	1	yes	characteristic
140	D₇₃	—	2	2	no	—
141	D₇₃	—	2	2	no	—
142	C₂²×M₃₂	—	73	1	no	—
143	C₂×#427	—	73	2	no	—
144	#1384	—	73	4	no	—
145	C₇₃	C₂×#1384	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₈	—	73	1	no	—
147	C₂×#117	—	146	2	no	—
148	C₂×M₃₂	—	73	2	no	—
149	C₂×M₃₂	—	73	2	no	—
150	C₂×M₃₂	—	146	2	no	—
151	C₂×M₃₂	—	146	2	no	—
152	C₂²×C₁₆	—	73	1	no	—
153	C₂²×C₁₆	#64408	1	1	yes	characteristic, complemented
154	#427	—	73	4	no	—
155	#427	—	73	4	no	—
156	#427	—	146	2	no	—
157	#427	—	146	2	no	—
158	C₂×#107	—	146	2	no	—
159	C₂×#106	—	73	4	no	—
160	C₂³×C₄	—	73	1	no	—
161	C₂²×C₈	—	73	1	no	—
162	C₂²×C₈	—	73	2	no	—
163	C₂²×C₈	—	73	2	no	—
164	C₂²×C₈	—	73	2	no	—
165	C₂²×C₈	—	73	2	no	—
166	C₂²×C₈	—	73	1	no	—
167	C₂²×C₈	—	73	1	no	—
168	C₂²×C₈	—	73	1	no	—
169	C₂²×C₈	—	73	1	no	—
170	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
171	#117	—	292	2	no	—
172	#117	—	292	2	no	—
173	#117	—	146	4	no	—
174	#117	—	146	4	no	—
175	C₂×C₄⋊C₄	—	146	4	no	—
176	C₂×C₂²⋊C₄	—	292	2	no	—
177	C₂×C₄²	—	292	2	no	—
178	M₃₂	—	146	4	no	—
179	M₃₂	—	146	4	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	73	2	no	—
183	C₂×C₁₆	—	146	1	no	—
184	C₂×C₁₆	—	146	1	no	—
185	C₂×C₁₆	C₂×#64408	1	2	yes	complemented
186	#107	—	146	4	no	—
187	#107	—	146	4	no	—
188	#106	—	73	8	no	—
189	#106	—	146	4	no	—
190	C₂⁴	—	73	1	no	—
191	C₂²×C₄	—	292	4	no	—
192	C₂²×C₄	—	73	1	no	—
193	C₂²×C₄	—	73	2	no	—
194	C₂²×C₄	—	73	2	no	—
195	C₂²×C₄	—	73	2	no	—
196	C₂²×C₄	—	73	2	no	—
197	C₂²×C₄	—	73	1	no	—
198	C₂²×C₄	—	73	1	no	—
199	C₂²×C₄	—	73	1	no	—
200	C₂²×C₄	—	73	1	no	—
201	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
202	C₂×C₈	—	73	2	no	—
203	C₂×C₈	—	73	2	no	—
204	C₂×C₈	—	73	1	no	—
205	C₂×C₈	—	73	1	no	—
206	C₂×C₈	—	73	2	no	—
207	C₂×C₈	—	73	2	no	—
208	C₂×C₈	—	73	2	no	—
209	C₂×C₈	—	73	2	no	—
210	C₂×C₈	—	73	2	no	—
211	C₂×C₈	—	73	2	no	—
212	C₂×C₈	—	73	2	no	—
213	C₂×C₈	—	73	2	no	—
214	C₂×C₈	C₄×#64408	1	1	yes	characteristic
215	C₂×C₈	C₄×#64408	1	1	yes	characteristic
216	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
217	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
218	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
219	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
220	C₄⋊C₄	—	292	4	no	—
221	C₄⋊C₄	—	146	8	no	—
222	C₂²⋊C₄	—	292	4	no	—
223	C₂²⋊C₄	—	292	4	no	—
224	C₄²	—	292	4	no	—
225	C₄²	—	292	4	no	—
226	C₁₆	—	2	2	no	—
227	C₁₆	—	146	2	no	—
228	C₂³	—	292	2	no	—
229	C₂³	—	73	2	no	—
230	C₂³	—	73	1	no	—
231	C₂³	—	73	1	no	—
232	C₂³	—	73	2	no	—
233	C₂³	#14556660	1	1	yes	characteristic
234	C₂×C₄	—	292	8	no	—
235	C₂×C₄	—	584	8	no	—
236	C₂×C₄	—	73	2	no	—
237	C₂×C₄	—	73	2	no	—
238	C₂×C₄	—	73	1	no	—
239	C₂×C₄	—	73	1	no	—
240	C₂×C₄	—	73	2	no	—
241	C₂×C₄	—	73	2	no	—
242	C₂×C₄	—	73	2	no	—
243	C₂×C₄	—	73	2	no	—
244	C₂×C₄	—	73	2	no	—
245	C₂×C₄	—	73	2	no	—
246	C₂×C₄	—	73	2	no	—
247	C₂×C₄	—	73	2	no	—
248	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
249	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
250	C₂×C₄	#14556659	1	1	yes	characteristic
251	C₂×C₄	#14556659	1	1	yes	characteristic
252	C₂×C₄	#14556660	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
254	C₈	—	73	2	no	—
255	C₈	—	73	2	no	—
256	C₈	(C₂×#64408)⋊C₄	1	1	yes	characteristic
257	C₈	(C₂×#64408)⋊C₄	1	1	yes	characteristic
258	C₈	C₂×C₄×#64408	1	1	yes	characteristic
259	C₈	C₂×C₂²⋊#64408	1	1	yes	characteristic
260	C₂²	—	2	2	no	—
261	C₂²	—	292	2	no	—
262	C₂²	—	292	1	no	—
263	C₂²	—	292	1	no	—
264	C₂²	—	73	2	no	—
265	C₂²	—	73	2	no	—
266	C₂²	—	146	4	no	—
267	C₂²	#17989461	1	1	yes	characteristic
268	C₂²	#17989461	1	1	yes	characteristic
269	C₂²	C₂×#14556660	1	1	yes	characteristic
270	C₄	—	584	8	no	—
271	C₄	—	73	2	no	—
272	C₄	—	73	2	no	—
273	C₄	#17989461	1	1	yes	characteristic
274	C₄	#17989461	1	1	yes	characteristic
275	C₄	C₂×(C₂×#64408)⋊C₄	1	1	yes	characteristic
276	C₄	C₂×#14556659	1	1	yes	characteristic
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	146	2	no	—
280	C₂	—	146	2	no	—
281	C₂	#22624999	1	2	yes	direct product factor
282	C₂	C₂×#17989461	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556659
commutator	C₈×C₇₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₂²×C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂³×D₇₃	D₈
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	C₂²×C₁₆	#64408
hypocenter	C₇₃	C₂×#1384
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22624999	C₂	direct (N × H)
7. #17989597	C₄	semidirect
10. C₂×#14556659	C₄	semidirect
14. C₂×#14556530	C₄	semidirect
16. C₂²×#11924003	C₄	semidirect
17. C₂²×C₁₆×C₇₃	C₄	semidirect
28. #14556659	C₂×C₄	semidirect
43. C₂×#11924003	C₂×C₄	semidirect
47. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
138. C₂×C₇₃	#1384	semidirect
145. C₇₃	C₂×#1384	semidirect
153. C₂²×C₁₆	#64408	semidirect
185. C₂×C₁₆	C₂×#64408	semidirect
281. C₂	#22624999	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17989597	2
#22624999	2
C₂²×#14556530	2
C₂×#1384	73

Subgroups — C2×#22624999 #34012190

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22624999 #34012190