Subgroups — C₂×2₋¹⁺⁴×#64408 #34032701

Order 18688.

11210 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×#64408	C₂	1	32	yes	maximal, direct product factor
2	C₂×#17989656	C₂	1	10	yes	maximal, complemented
3	C₂×#17989655	C₂	1	5	yes	maximal, complemented
4	C₂×#41×#64408	C₂	1	10	yes	maximal, complemented
5	C₂²×Q₈×#64408	C₂	1	5	yes	maximal, complemented
6	C₂×2₋¹⁺⁴×D₇₃	C₂	1	1	yes	maximal, characteristic
7	#17989656	C₂²	1	160	yes	complemented
8	#17989655	C₂²	1	80	yes	complemented
9	#41×#64408	C₂²	1	160	yes	complemented
10	C₂×Q₈×#64408	C₂²	1	10	yes	—
11	C₂×Q₈×#64408	C₂²	1	20	yes	—
12	C₂×Q₈×#64408	C₂²	1	10	yes	—
13	C₂×Q₈×#64408	C₂²	1	80	yes	complemented
14	C₂×D₄×#64408	C₂²	1	10	yes	—
15	C₂×D₄×#64408	C₂²	1	30	yes	—
16	C₂×#14556670	C₂²	1	30	yes	—
17	C₂²×C₄⋊#64408	C₂²	1	15	yes	—
18	C₂²×C₄×#64408	C₂²	1	15	yes	—
19	C₂×#14556639	C₄	1	5	yes	complemented
20	C₂×#14556638	C₄	1	10	yes	complemented
21	2₋¹⁺⁴×D₇₃	C₄	1	16	yes	complemented
22	2₋¹⁺⁴×D₇₃	C₂²	1	16	yes	—
23	C₂×#41×D₇₃	C₂²	1	10	yes	—
24	C₂²×Q₈×D₇₃	C₂²	1	5	yes	—
25	C₂×2₋¹⁺⁴×C₇₃	C₄	1	1	yes	characteristic, complemented
26	C₂³×#64408	C₂³	1	5	yes	—
27	C₂³×#64408	C₂³	1	15	yes	—
28	Q₈×#64408	C₂³	1	80	yes	—
29	Q₈×#64408	C₂³	1	160	yes	—
30	Q₈×#64408	C₂³	1	80	yes	—
31	D₄×#64408	C₂³	1	80	yes	—
32	D₄×#64408	C₂³	1	240	yes	—
33	#14556670	C₂³	1	240	yes	—
34	C₂×C₄⋊#64408	C₂³	1	30	yes	—
35	C₂×C₄⋊#64408	C₂³	1	120	yes	—
36	C₂×C₄⋊#64408	C₂³	1	10	yes	—
37	C₂×C₂²⋊#64408	C₂³	1	20	yes	—
38	C₂×C₄×#64408	C₂³	1	10	yes	—
39	C₂×C₄×#64408	C₂³	1	30	yes	—
40	C₂×C₄×#64408	C₂³	1	120	yes	—
41	#14556639	C₂×C₄	1	80	yes	complemented
42	#14556638	C₂×C₄	1	160	yes	complemented
43	C₂×#11924029	C₂×C₄	1	10	yes	complemented
44	C₂×#11924028	C₂×C₄	1	30	yes	complemented
45	C₂×#11924027	C₂×C₄	1	30	yes	complemented
46	#41×D₇₃	C₂×C₄	1	80	yes	complemented
47	#41×D₇₃	C₂³	1	80	yes	—
48	C₂²×Dic₁₄₆	C₂×C₄	1	15	yes	complemented
49	C₂×Q₈×D₇₃	C₂×C₄	1	20	yes	complemented
50	C₂×Q₈×D₇₃	C₂×C₄	1	40	yes	complemented
51	C₂×Q₈×D₇₃	C₂³	1	10	yes	—
52	C₂×Q₈×D₇₃	C₂³	1	40	yes	—
53	C₂×D₄×D₇₃	C₂³	1	10	yes	—
54	C₂²×C₄×D₇₃	C₂³	1	15	yes	—
55	2₋¹⁺⁴×C₇₃	C₂×C₄	1	16	yes	complemented
56	C₂×#41×C₇₃	C₂×C₄	1	10	yes	complemented
57	C₂²×Q₈×C₇₃	C₂×C₄	1	5	yes	complemented
58	C₂²×#64408	—	2	20	no	—
59	C₂²×#64408	—	4	30	no	—
60	C₂²×#64408	C₂⁴	1	10	yes	—
61	C₂²×#64408	C₂⁴	1	5	yes	—
62	C₂²×#64408	C₂⁴	1	20	yes	—
63	C₂²×#64408	C₂⁴	1	60	yes	—
64	C₂²×#64408	C₂⁴	1	1	yes	characteristic
65	C₄⋊#64408	C₂⁴	1	120	yes	—
66	C₄⋊#64408	C₂⁴	1	40	yes	—
67	C₂²⋊#64408	C₂⁴	1	80	yes	—
68	C₄×#64408	C₂⁴	1	40	yes	—
69	C₄×#64408	C₂⁴	1	120	yes	—
70	C₂³×D₇₃	C₂⁴	1	5	yes	—
71	#11924029	C₂²×C₄	1	80	yes	complemented
72	#11924028	C₂²×C₄	1	240	yes	complemented
73	#11924027	C₂²×C₄	1	240	yes	complemented
74	C₂×Dic₁₄₆	C₂²×C₄	1	30	yes	—
75	C₂×Dic₁₄₆	C₂²×C₄	1	120	yes	complemented
76	Q₈×D₇₃	C₂²×C₄	1	40	yes	—
77	Q₈×D₇₃	C₂²×C₄	1	160	yes	complemented
78	Q₈×D₇₃	C₂⁴	1	40	yes	—
79	C₂×D₂₉₂	C₂²×C₄	1	10	yes	—
80	C₂×C₂²⋊D₇₃	C₂²×C₄	1	20	yes	—
81	D₄×D₇₃	C₂²×C₄	1	40	yes	—
82	D₄×D₇₃	C₂⁴	1	40	yes	—
83	C₂²×Dic₇₃	C₂²×C₄	1	15	yes	—
84	C₂×C₄×D₇₃	C₂²×C₄	1	60	yes	—
85	C₂×C₄×D₇₃	C₂²×C₄	1	30	yes	—
86	C₂×C₄×D₇₃	C₂⁴	1	10	yes	—
87	C₂×C₄×D₇₃	C₂⁴	1	60	yes	—
88	#41×C₇₃	C₂²×C₄	1	80	yes	complemented
89	C₂×Q₈×C₇₃	C₂²×C₄	1	10	yes	—
90	C₂×Q₈×C₇₃	C₂²×C₄	1	40	yes	complemented
91	C₂×D₄×C₇₃	C₂²×C₄	1	10	yes	—
92	C₂²×C₄×C₇₃	C₂²×C₄	1	15	yes	—
93	C₂³×C₇₃	C₂³×C₄	1	5	yes	—
94	C₂²×D₇₃	—	2	10	no	—
95	C₂²×D₇₃	—	2	10	no	—
96	C₂²×D₇₃	C₂³×C₄	1	10	yes	—
97	C₂²×D₇₃	C₂³×C₄	1	5	yes	—
98	C₂²×D₇₃	C₂⁵	1	10	yes	—
99	C₂²×D₇₃	C₂⁵	1	1	yes	characteristic
100	C₂×#64408	—	2	10	no	—
101	C₂×#64408	—	2	20	no	—
102	C₂×#64408	—	2	40	no	—
103	C₂×#64408	—	4	60	no	—
104	C₂×#64408	2₋¹⁺⁴	1	4	yes	direct product factor
105	C₂×#64408	C₂⁵	1	10	yes	—
106	C₂×#64408	C₂⁵	1	20	yes	—
107	C₂×#64408	C₂⁵	1	2	yes	—
108	Q₈×C₇₃	C₂³×C₄	1	40	yes	—
109	D₄×C₇₃	C₂³×C₄	1	40	yes	—
110	C₂×C₄×C₇₃	C₂³×C₄	1	10	yes	—
111	C₂×C₄×C₇₃	C₂³×C₄	1	60	yes	—
112	C₂²⋊D₇₃	C₂³×C₄	1	80	yes	—
113	C₂×Dic₇₃	C₂³×C₄	1	10	yes	—
114	C₂×Dic₇₃	C₂³×C₄	1	60	yes	—
115	D₂₉₂	C₂³×C₄	1	40	yes	—
116	C₄×D₇₃	C₂³×C₄	1	20	yes	—
117	C₄×D₇₃	C₂³×C₄	1	120	yes	—
118	C₄×D₇₃	C₂⁵	1	20	yes	—
119	Dic₁₄₆	C₂³×C₄	1	120	yes	—
120	C₂²×C₇₃	—	2	10	no	—
121	C₂²×C₇₃	C₂⁴×C₄	1	10	yes	—
122	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
123	C₂×D₇₃	—	2	10	no	—
124	C₂×D₇₃	—	2	10	no	—
125	C₂×D₇₃	—	2	10	no	—
126	C₂×D₇₃	—	2	20	no	—
127	C₂×D₇₃	C₂⁴×C₄	1	10	yes	—
128	C₂×D₇₃	C₂⁴×C₄	1	1	yes	characteristic
129	C₂×D₇₃	C₂×2₋¹⁺⁴	1	2	yes	—
130	C₂×D₇₃	#585	1	2	yes	—
131	C₂×D₇₃	C₂⁶	1	1	yes	characteristic
132	#64408	—	2	10	no	—
133	#64408	—	2	20	no	—
134	#64408	C₂×2₋¹⁺⁴	1	4	yes	direct product factor
135	C₄×C₇₃	C₂⁴×C₄	1	20	yes	—
136	Dic₇₃	C₂⁴×C₄	1	20	yes	—
137	C₂×C₄×2₋¹⁺⁴	—	73	1	no	maximal, Sylow
138	C₂×C₇₃	—	2	10	no	—
139	C₂×C₇₃	C₄×2₋¹⁺⁴	1	2	yes	complemented
140	C₂×C₇₃	C₂⁵×C₄	1	1	yes	characteristic
141	D₇₃	—	2	10	no	—
142	D₇₃	C₄×2₋¹⁺⁴	1	2	yes	complemented
143	D₇₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
144	D₇₃	C₂×#585	1	1	yes	characteristic
145	C₂²×2₋¹⁺⁴	—	73	1	no	—
146	C₄×2₋¹⁺⁴	—	73	32	no	—
147	C₂×#520	—	73	10	no	—
148	C₂×#519	—	73	5	no	—
149	C₂×C₄×#41	—	73	10	no	—
150	C₂²×C₄×Q₈	—	73	5	no	—
151	C₇₃	C₂×C₄×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
152	C₂×2₋¹⁺⁴	—	73	16	no	—
153	C₂×2₋¹⁺⁴	—	73	16	no	—
154	C₂×2₋¹⁺⁴	—	73	10	no	—
155	C₂×2₋¹⁺⁴	—	73	5	no	—
156	C₂×2₋¹⁺⁴	#64408	1	1	yes	characteristic, direct product factor
157	C₂²×#41	—	73	10	no	—
158	C₂³×Q₈	—	73	5	no	—
159	#520	—	73	160	no	—
160	#519	—	73	80	no	—
161	C₄×#41	—	73	160	no	—
162	C₂×C₄×Q₈	—	73	10	no	—
163	C₂×C₄×Q₈	—	73	20	no	—
164	C₂×C₄×Q₈	—	73	10	no	—
165	C₂×C₄×Q₈	—	73	80	no	—
166	C₂×C₄×D₄	—	73	10	no	—
167	C₂×C₄×D₄	—	73	30	no	—
168	C₂×#117	—	73	30	no	—
169	C₂²×C₄⋊C₄	—	73	15	no	—
170	C₂²×C₄²	—	73	15	no	—
171	2₋¹⁺⁴	—	73	160	no	—
172	2₋¹⁺⁴	—	73	80	no	—
173	2₋¹⁺⁴	C₂×#64408	1	16	yes	direct product factor
174	C₂×#41	—	73	80	no	—
175	C₂×#41	—	73	80	no	—
176	C₂×#41	—	73	10	no	—
177	C₂×#41	—	73	30	no	—
178	C₂×#41	—	73	30	no	—
179	C₂×#41	C₂×#64408	1	10	yes	complemented
180	C₂²×Q₈	—	73	10	no	—
181	C₂²×Q₈	—	73	20	no	—
182	C₂²×Q₈	—	73	40	no	—
183	C₂²×Q₈	—	73	40	no	—
184	C₂²×Q₈	—	73	15	no	—
185	C₂²×Q₈	C₂×#64408	1	5	yes	complemented
186	C₂²×D₄	—	73	10	no	—
187	C₂³×C₄	—	73	15	no	—
188	C₂³×C₄	—	73	15	no	—
189	C₂³×C₄	—	73	5	no	—
190	C₄×Q₈	—	73	80	no	—
191	C₄×Q₈	—	73	160	no	—
192	C₄×Q₈	—	73	80	no	—
193	C₄×D₄	—	73	80	no	—
194	C₄×D₄	—	73	240	no	—
195	#117	—	73	240	no	—
196	C₂×C₄⋊C₄	—	73	30	no	—
197	C₂×C₄⋊C₄	—	73	120	no	—
198	C₂×C₄⋊C₄	—	73	10	no	—
199	C₂×C₂²⋊C₄	—	73	20	no	—
200	C₂×C₄²	—	73	30	no	—
201	C₂×C₄²	—	73	10	no	—
202	C₂×C₄²	—	73	120	no	—
203	C₂⁴	—	73	5	no	—
204	#41	—	73	80	no	—
205	#41	—	73	240	no	—
206	#41	—	73	240	no	—
207	#41	C₂²×#64408	1	80	yes	complemented
208	C₂×Q₈	—	73	40	no	—
209	C₂×Q₈	—	73	160	no	—
210	C₂×Q₈	—	73	40	no	—
211	C₂×Q₈	—	73	30	no	—
212	C₂×Q₈	—	73	120	no	—
213	C₂×Q₈	C₂²×#64408	1	10	yes	—
214	C₂×Q₈	C₂²×#64408	1	40	yes	complemented
215	C₂×D₄	—	73	20	no	—
216	C₂×D₄	—	73	40	no	—
217	C₂×D₄	—	73	40	no	—
218	C₂×D₄	—	73	10	no	—
219	C₂×D₄	C₂²×#64408	1	10	yes	—
220	C₂²×C₄	—	292	30	no	—
221	C₂²×C₄	—	73	60	no	—
222	C₂²×C₄	—	73	1	no	—
223	C₂²×C₄	—	73	15	no	—
224	C₂²×C₄	—	73	60	no	—
225	C₂²×C₄	—	73	30	no	—
226	C₂²×C₄	—	73	10	no	—
227	C₂²×C₄	—	73	60	no	—
228	C₂²×C₄	—	73	5	no	—
229	C₂²×C₄	—	73	20	no	—
230	C₂²×C₄	—	73	10	no	—
231	C₂²×C₄	—	146	20	no	—
232	C₂²×C₄	C₂²×#64408	1	15	yes	—
233	C₄⋊C₄	—	73	120	no	—
234	C₄⋊C₄	—	73	40	no	—
235	C₂²⋊C₄	—	73	80	no	—
236	C₄²	—	73	40	no	—
237	C₄²	—	73	120	no	—
238	C₂³	—	73	1	no	—
239	C₂³	—	73	10	no	—
240	C₂³	—	73	10	no	—
241	C₂³	—	73	5	no	—
242	C₂³	—	146	10	no	—
243	C₂³	—	146	10	no	—
244	C₂³	C₂³×#64408	1	5	yes	—
245	Q₈	—	73	120	no	—
246	Q₈	C₂³×#64408	1	40	yes	—
247	D₄	—	73	80	no	—
248	D₄	—	73	40	no	—
249	D₄	C₂³×#64408	1	40	yes	—
250	C₂×C₄	—	292	60	no	—
251	C₂×C₄	—	73	2	no	—
252	C₂×C₄	—	73	4	no	—
253	C₂×C₄	—	73	60	no	—
254	C₂×C₄	—	73	10	no	—
255	C₂×C₄	—	73	20	no	—
256	C₂×C₄	—	73	20	no	—
257	C₂×C₄	—	73	120	no	—
258	C₂×C₄	—	73	10	no	—
259	C₂×C₄	—	73	20	no	—
260	C₂×C₄	—	146	40	no	—
261	C₂×C₄	—	146	20	no	—
262	C₂×C₄	—	146	10	no	—
263	C₂×C₄	C₂³×#64408	1	60	yes	—
264	C₂×C₄	C₂³×#64408	1	10	yes	—
265	C₂²	—	2	10	no	—
266	C₂²	—	73	2	no	—
267	C₂²	—	73	2	no	—
268	C₂²	—	73	1	no	—
269	C₂²	—	73	1	no	—
270	C₂²	—	73	10	no	—
271	C₂²	—	146	10	no	—
272	C₂²	—	146	10	no	—
273	C₂²	—	146	20	no	—
274	C₂²	—	146	10	no	—
275	C₂²	C₂⁴×#64408	1	10	yes	—
276	C₂²	C₂⁴×#64408	1	1	yes	characteristic
277	C₄	—	73	4	no	—
278	C₄	—	73	20	no	—
279	C₄	—	146	20	no	—
280	C₄	—	146	10	no	—
281	C₄	C₂⁴×#64408	1	20	yes	—
282	C₂	—	2	10	no	—
283	C₂	—	73	1	no	—
284	C₂	—	73	1	no	—
285	C₂	—	73	2	no	—
286	C₂	—	146	10	no	—
287	C₂	2₋¹⁺⁴×#64408	1	2	yes	direct product factor
288	C₂	C₂⁵×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂⁴×#64408
commutator	C₂×C₇₃	C₂⁵×C₄
Frattini	C₂	C₂⁵×#64408
Fitting	C₂×2₋¹⁺⁴×C₇₃	C₄
socle	C₂²×C₇₃	C₂⁴×C₄
prime radical	C₂×2₋¹⁺⁴×D₇₃	C₂
cosocle	C₂×D₇₃	C₂⁶
hypercenter	C₂×2₋¹⁺⁴	#64408
hypocenter	C₇₃	C₂×C₄×2₋¹⁺⁴
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×#64408	C₂	direct (N × H)
2. C₂×#17989656	C₂	semidirect
3. C₂×#17989655	C₂	semidirect
4. C₂×#41×#64408	C₂	semidirect
5. C₂²×Q₈×#64408	C₂	semidirect
7. #17989656	C₂²	semidirect
8. #17989655	C₂²	semidirect
9. #41×#64408	C₂²	semidirect
13. C₂×Q₈×#64408	C₂²	semidirect
19. C₂×#14556639	C₄	semidirect
20. C₂×#14556638	C₄	semidirect
21. 2₋¹⁺⁴×D₇₃	C₄	semidirect
25. C₂×2₋¹⁺⁴×C₇₃	C₄	semidirect
41. #14556639	C₂×C₄	semidirect
42. #14556638	C₂×C₄	semidirect
43. C₂×#11924029	C₂×C₄	semidirect
44. C₂×#11924028	C₂×C₄	semidirect
45. C₂×#11924027	C₂×C₄	semidirect
46. #41×D₇₃	C₂×C₄	semidirect
48. C₂²×Dic₁₄₆	C₂×C₄	semidirect
49. C₂×Q₈×D₇₃	C₂×C₄	semidirect
50. C₂×Q₈×D₇₃	C₂×C₄	semidirect
55. 2₋¹⁺⁴×C₇₃	C₂×C₄	semidirect
56. C₂×#41×C₇₃	C₂×C₄	semidirect
57. C₂²×Q₈×C₇₃	C₂×C₄	semidirect
71. #11924029	C₂²×C₄	semidirect
72. #11924028	C₂²×C₄	semidirect
73. #11924027	C₂²×C₄	semidirect
75. C₂×Dic₁₄₆	C₂²×C₄	semidirect
77. Q₈×D₇₃	C₂²×C₄	semidirect
88. #41×C₇₃	C₂²×C₄	semidirect
90. C₂×Q₈×C₇₃	C₂²×C₄	semidirect
104. C₂×#64408	2₋¹⁺⁴	direct (N × H)
134. #64408	C₂×2₋¹⁺⁴	direct (N × H)
139. C₂×C₇₃	C₄×2₋¹⁺⁴	semidirect
142. D₇₃	C₄×2₋¹⁺⁴	semidirect
151. C₇₃	C₂×C₄×2₋¹⁺⁴	semidirect
156. C₂×2₋¹⁺⁴	#64408	direct (N × H)
173. 2₋¹⁺⁴	C₂×#64408	direct (N × H)
179. C₂×#41	C₂×#64408	semidirect
185. C₂²×Q₈	C₂×#64408	semidirect
207. #41	C₂²×#64408	semidirect
214. C₂×Q₈	C₂²×#64408	semidirect
287. C₂	2₋¹⁺⁴×#64408	direct (N × H)

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴×#64408	2
C₂×#17989656	2
C₂×#17989655	2
C₂×#41×#64408	2
C₂²×Q₈×#64408	2
C₂×2₋¹⁺⁴×D₇₃	2
C₂×C₄×2₋¹⁺⁴	73

Subgroups — C2×2−1+4×#64408 #34032701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×2₋¹⁺⁴×#64408 #34032701