Subgroups — C₂²×#17988820 #33953073

Order 18688.

3694 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14556562	C₂	1	2	yes	maximal, complemented
3	C₂×#17988820	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#14556525	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#14556512	C₂	1	2	yes	maximal, complemented
6	C₂³×D₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₂²⋊D₇₃	—	2	2	no	—
8	C₂³×C₄×D₇₃	C₂²	1	1	yes	characteristic
9	C₂³×#10596831	C₂²	1	1	yes	characteristic
10	C₂²×C₈×D₇₃	—	2	8	no	—
11	C₂²×C₈×D₇₃	C₂²	1	12	yes	complemented
12	C₂×#14556562	—	2	12	no	—
13	C₂×#14556562	C₂²	1	24	yes	complemented
14	C₂²×C₄⋊Dic₇₃	C₂²	1	2	yes	—
15	C₂²×C₂²⋊Dic₇₃	—	2	2	no	—
16	#17988820	C₂²	1	64	yes	direct product factor
17	C₂×#14556525	C₂²	1	12	yes	complemented
18	C₂×#14556512	C₂²	1	24	yes	complemented
19	C₂³×D₄×C₇₃	C₂²	1	2	yes	—
20	C₂²×D₈×C₇₃	—	2	8	no	—
21	C₂²×D₈×C₇₃	C₂²	1	12	yes	complemented
22	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂⁴×D₇₃	D₄	1	1	yes	characteristic
24	C₂²×C₂²⋊D₇₃	—	2	24	no	—
25	C₂²×C₂²⋊D₇₃	—	4	16	no	—
26	C₂³×Dic₇₃	—	2	2	no	—
27	C₂³×Dic₇₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×D₇₃	—	2	4	no	—
29	C₂²×C₄×D₇₃	—	2	4	no	—
30	C₂²×C₄×D₇₃	D₄	1	6	yes	—
31	C₂²×C₄×D₇₃	C₂³	1	6	yes	—
32	C₂²×#10596831	—	2	4	no	—
33	C₂²×#10596831	C₂³	1	6	yes	—
34	C₂×C₈×D₇₃	—	2	48	no	—
35	C₂×C₈×D₇₃	C₂³	1	16	yes	complemented
36	#14556562	—	2	48	no	—
37	#14556562	C₂³	1	32	yes	complemented
38	C₂×C₄⋊Dic₇₃	—	2	6	no	—
39	C₂×C₄⋊Dic₇₃	C₂³	1	12	yes	—
40	C₂×C₂²⋊Dic₇₃	—	2	24	no	—
41	#14556525	C₂³	1	16	yes	complemented
42	#14556512	C₂³	1	32	yes	complemented
43	C₂⁵×C₇₃	—	2	2	no	—
44	C₂²×D₄×C₇₃	—	2	8	no	—
45	C₂²×D₄×C₇₃	—	2	6	no	—
46	C₂²×D₄×C₇₃	—	2	8	no	—
47	C₂²×D₄×C₇₃	C₂³	1	12	yes	—
48	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
49	C₂×D₈×C₇₃	—	2	48	no	—
50	C₂×D₈×C₇₃	C₂³	1	16	yes	complemented
51	C₂²×C₈×C₇₃	D₄	1	8	yes	complemented
52	C₂²×C₈×C₇₃	C₂³	1	6	yes	—
53	C₂³×D₇₃	—	2	4	no	—
54	C₂³×D₇₃	—	2	4	no	—
55	C₂³×D₇₃	D₈	1	8	yes	complemented
56	C₂³×D₇₃	C₂×D₄	1	6	yes	—
57	C₂×C₂²⋊D₇₃	—	2	32	no	—
58	C₂×C₂²⋊D₇₃	—	4	96	no	—
59	C₂²×Dic₇₃	—	2	4	no	—
60	C₂²×Dic₇₃	—	2	12	no	—
61	C₂²×Dic₇₃	—	4	8	no	—
62	C₂²×Dic₇₃	C₂×D₄	1	6	yes	—
63	C₂×C₄×D₇₃	—	2	24	no	—
64	C₂×C₄×D₇₃	—	2	12	no	—
65	C₂×C₄×D₇₃	—	2	12	no	—
66	C₂×C₄×D₇₃	C₂×D₄	1	12	yes	—
67	C₂×C₄×D₇₃	C₂⁴	1	4	yes	—
68	C₂×#10596831	—	2	12	no	—
69	C₂×#10596831	C₂⁴	1	4	yes	—
70	C₈×D₇₃	—	2	32	no	—
71	C₄⋊Dic₇₃	—	2	12	no	—
72	C₄⋊Dic₇₃	C₂⁴	1	8	yes	—
73	C₂²⋊Dic₇₃	—	2	32	no	—
74	C₂⁴×C₇₃	—	2	12	no	—
75	C₂⁴×C₇₃	—	4	8	no	—
76	C₂⁴×C₇₃	—	4	8	no	—
77	C₂⁴×C₇₃	—	4	8	no	—
78	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
79	C₂×D₄×C₇₃	—	2	24	no	—
80	C₂×D₄×C₇₃	—	2	12	no	—
81	C₂×D₄×C₇₃	—	2	24	no	—
82	C₂×D₄×C₇₃	—	4	24	no	—
83	C₂×D₄×C₇₃	C₂⁴	1	8	yes	—
84	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
85	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₇₃	#41	1	4	yes	—
87	C₂²×C₄×C₇₃	C₂⁴	1	3	yes	—
88	D₈×C₇₃	—	2	32	no	—
89	C₂×C₈×C₇₃	C₂×D₄	1	24	yes	complemented
90	C₂×C₈×C₇₃	C₂⁴	1	4	yes	—
91	C₂³×C₇₃	—	2	8	no	—
92	C₂³×C₇₃	—	4	24	no	—
93	C₂³×C₇₃	—	4	24	no	—
94	C₂³×C₇₃	—	4	24	no	—
95	C₂³×C₇₃	—	8	32	no	—
96	C₂³×C₇₃	#121	1	4	yes	—
97	C₂³×C₇₃	C₂×D₈	1	4	yes	—
98	C₂³×C₇₃	#135	1	4	yes	—
99	C₂³×C₇₃	C₂²×D₄	1	3	yes	—
100	C₂²×D₇₃	—	2	12	no	—
101	C₂²×D₇₃	—	2	12	no	—
102	C₂²×D₇₃	—	2	32	no	—
103	C₂²×D₇₃	C₂×D₈	1	24	yes	complemented
104	C₂²×D₇₃	C₂²×D₄	1	4	yes	—
105	D₄×C₇₃	—	2	8	no	—
106	D₄×C₇₃	—	2	8	no	—
107	D₄×C₇₃	—	4	24	no	—
108	C₂×C₄×C₇₃	#121	1	12	yes	—
109	C₂×C₄×C₇₃	C₂²×D₄	1	3	yes	—
110	C₂×C₄×C₇₃	C₂²×D₄	1	6	yes	—
111	C₂×C₄×C₇₃	C₂×#41	1	6	yes	—
112	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
113	C₂²⋊D₇₃	—	4	64	no	—
114	C₂×Dic₇₃	—	2	12	no	—
115	C₂×Dic₇₃	—	2	8	no	—
116	C₂×Dic₇₃	—	4	24	no	—
117	C₂×Dic₇₃	C₂²×D₄	1	4	yes	—
118	C₄×D₇₃	—	2	24	no	—
119	C₄×D₇₃	—	2	4	no	—
120	C₄×D₇₃	—	2	4	no	—
121	C₈×C₇₃	C₂²×D₄	1	8	yes	complemented
122	#10596831	—	2	4	no	—
123	C₂²×C₇₃	—	4	8	no	—
124	C₂²×C₇₃	—	4	8	no	—
125	C₂²×C₇₃	—	4	8	no	—
126	C₂²×C₇₃	—	8	48	no	—
127	C₂²×C₇₃	#466	1	16	yes	complemented
128	C₂²×C₇₃	C₂×#121	1	6	yes	—
129	C₂²×C₇₃	C₂²×D₈	1	6	yes	—
130	C₂²×C₇₃	C₂×#135	1	6	yes	—
131	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
132	C₂×D₇₃	—	2	4	no	—
133	C₂×D₇₃	—	2	48	no	—
134	C₂×D₇₃	—	2	4	no	—
135	C₂×D₇₃	C₂²×D₈	1	8	yes	complemented
136	C₄×C₇₃	C₂×#121	1	6	yes	—
137	C₄×C₇₃	C₂³×D₄	1	1	yes	characteristic
138	C₄×C₇₃	C₂²×#41	1	1	yes	characteristic
139	Dic₇₃	—	2	4	no	—
140	Dic₇₃	—	4	8	no	—
141	C₂²×#466	—	73	1	no	maximal, Sylow
142	C₂×C₇₃	—	8	8	no	—
143	C₂×C₇₃	C₂×#466	1	12	yes	complemented
144	C₂×C₇₃	C₂²×#121	1	1	yes	characteristic
145	C₂×C₇₃	C₂³×D₈	1	1	yes	characteristic
146	C₂×C₇₃	C₂²×#135	1	1	yes	characteristic
147	D₇₃	—	2	8	no	—
148	C₂³×D₈	D₇₃	1	1	yes	characteristic, complemented
149	C₂⁴×C₈	—	73	1	no	—
150	C₂²×#121	—	73	2	no	—
151	C₂×#466	—	73	24	no	—
152	C₂²×#107	—	73	1	no	—
153	C₂²×#102	—	73	2	no	—
154	C₇₃	C₂²×#466	1	1	yes	characteristic, complemented, Sylow
155	C₂³×D₄	—	146	2	no	—
156	C₂³×D₄	C₂×D₇₃	1	2	yes	—
157	C₂⁴×C₄	—	73	1	no	—
158	C₂²×D₈	—	2	8	no	—
159	C₂²×D₈	C₂×D₇₃	1	12	yes	complemented
160	C₂³×C₈	—	73	1	no	—
161	C₂³×C₈	—	73	12	no	—
162	C₂³×C₈	—	146	8	no	—
163	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
164	C₂×#121	—	73	24	no	—
165	C₂×#121	—	146	12	no	—
166	C₂²×C₄⋊C₄	—	73	2	no	—
167	C₂²×C₂²⋊C₄	—	146	2	no	—
168	#466	—	73	64	no	—
169	C₂×#107	—	73	12	no	—
170	C₂×#102	—	73	24	no	—
171	C₂⁵	—	2	2	no	—
172	C₂⁵	—	73	1	no	—
173	C₂²×D₄	—	2	6	no	—
174	C₂²×D₄	—	2	8	no	—
175	C₂²×D₄	—	2	8	no	—
176	C₂²×D₄	—	292	16	no	—
177	C₂²×D₄	—	146	24	no	—
178	C₂²×D₄	C₂²×D₇₃	1	12	yes	—
179	C₂³×C₄	—	73	1	no	—
180	C₂³×C₄	—	73	6	no	—
181	C₂³×C₄	—	73	6	no	—
182	C₂³×C₄	—	146	2	no	—
183	C₂³×C₄	—	146	4	no	—
184	C₂³×C₄	—	146	4	no	—
185	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
186	C₂×D₈	—	2	48	no	—
187	C₂×D₈	C₂²×D₇₃	1	16	yes	complemented
188	C₂²×C₈	—	73	6	no	—
189	C₂²×C₈	—	73	16	no	—
190	C₂²×C₈	—	146	4	no	—
191	C₂²×C₈	—	146	48	no	—
192	C₂²×C₈	C₂²⋊D₇₃	1	8	yes	complemented
193	C₂²×C₈	C₂²×D₇₃	1	6	yes	—
194	#121	—	73	32	no	—
195	#121	—	146	48	no	—
196	C₂×C₄⋊C₄	—	73	12	no	—
197	C₂×C₄⋊C₄	—	146	6	no	—
198	C₂×C₂²⋊C₄	—	146	24	no	—
199	#107	—	73	16	no	—
200	#102	—	73	32	no	—
201	C₂⁴	—	2	12	no	—
202	C₂⁴	—	4	8	no	—
203	C₂⁴	—	4	8	no	—
204	C₂⁴	—	4	8	no	—
205	C₂⁴	—	73	8	no	—
206	C₂⁴	—	73	6	no	—
207	C₂⁴	—	146	4	no	—
208	C₂⁴	—	146	4	no	—
209	C₂⁴	D₄×D₇₃	1	1	yes	characteristic
210	C₂×D₄	—	2	12	no	—
211	C₂×D₄	—	2	24	no	—
212	C₂×D₄	—	2	24	no	—
213	C₂×D₄	—	4	24	no	—
214	C₂×D₄	—	292	96	no	—
215	C₂×D₄	—	146	32	no	—
216	C₂×D₄	C₂³×D₇₃	1	8	yes	—
217	C₂²×C₄	—	292	8	no	—
218	C₂²×C₄	—	73	6	no	—
219	C₂²×C₄	—	73	12	no	—
220	C₂²×C₄	—	73	4	no	—
221	C₂²×C₄	—	146	12	no	—
222	C₂²×C₄	—	146	4	no	—
223	C₂²×C₄	—	146	24	no	—
224	C₂²×C₄	—	146	12	no	—
225	C₂²×C₄	—	146	12	no	—
226	C₂²×C₄	D₄×D₇₃	1	3	yes	—
227	C₂²×C₄	C₂×C₂²⋊D₇₃	1	4	yes	—
228	C₂²×C₄	#11924027	1	4	yes	—
229	C₂²×C₄	C₂³×D₇₃	1	3	yes	—
230	D₈	—	2	32	no	—
231	C₂×C₈	—	73	4	no	—
232	C₂×C₈	—	146	12	no	—
233	C₂×C₈	—	146	32	no	—
234	C₂×C₈	C₂×C₂²⋊D₇₃	1	24	yes	complemented
235	C₂×C₈	C₂³×D₇₃	1	4	yes	—
236	C₄⋊C₄	—	73	8	no	—
237	C₄⋊C₄	—	146	12	no	—
238	C₂²⋊C₄	—	146	32	no	—
239	C₂³	—	2	8	no	—
240	C₂³	—	4	24	no	—
241	C₂³	—	4	24	no	—
242	C₂³	—	4	24	no	—
243	C₂³	—	8	32	no	—
244	C₂³	—	73	4	no	—
245	C₂³	—	73	24	no	—
246	C₂³	—	146	32	no	—
247	C₂³	—	146	12	no	—
248	C₂³	—	146	12	no	—
249	C₂³	#14556562	1	4	yes	—
250	C₂³	D₈×D₇₃	1	4	yes	—
251	C₂³	#14556610	1	4	yes	—
252	C₂³	C₂×D₄×D₇₃	1	3	yes	—
253	D₄	—	2	8	no	—
254	D₄	—	2	8	no	—
255	D₄	—	4	24	no	—
256	D₄	—	292	64	no	—
257	C₂×C₄	—	292	24	no	—
258	C₂×C₄	—	73	4	no	—
259	C₂×C₄	—	146	8	no	—
260	C₂×C₄	—	146	12	no	—
261	C₂×C₄	—	146	4	no	—
262	C₂×C₄	—	146	4	no	—
263	C₂×C₄	—	146	24	no	—
264	C₂×C₄	#14556562	1	12	yes	—
265	C₂×C₄	C₂×D₄×D₇₃	1	3	yes	—
266	C₂×C₄	C₂²×C₂²⋊D₇₃	1	6	yes	—
267	C₂×C₄	C₂×#11924027	1	6	yes	—
268	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
269	C₈	—	146	4	no	—
270	C₈	C₂²×C₂²⋊D₇₃	1	8	yes	complemented
271	C₂²	—	4	8	no	—
272	C₂²	—	4	8	no	—
273	C₂²	—	4	8	no	—
274	C₂²	—	8	48	no	—
275	C₂²	—	73	8	no	—
276	C₂²	—	146	48	no	—
277	C₂²	—	146	4	no	—
278	C₂²	—	146	4	no	—
279	C₂²	#17988820	1	16	yes	direct product factor
280	C₂²	C₂×#14556562	1	6	yes	—
281	C₂²	C₂×D₈×D₇₃	1	6	yes	—
282	C₂²	C₂×#14556610	1	6	yes	—
283	C₂²	C₂²×D₄×D₇₃	1	1	yes	characteristic
284	C₄	—	292	8	no	—
285	C₄	—	146	4	no	—
286	C₄	C₂×#14556562	1	6	yes	—
287	C₄	C₂³×C₂²⋊D₇₃	1	1	yes	characteristic
288	C₄	C₂²×#11924027	1	1	yes	characteristic
289	C₂	—	8	8	no	—
290	C₂	—	146	8	no	—
291	C₂	C₂×#17988820	1	12	yes	direct product factor
292	C₂	C₂²×#14556562	1	1	yes	characteristic
293	C₂	C₂²×D₈×D₇₃	1	1	yes	characteristic
294	C₂	C₂²×#14556610	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₇₃
commutator, cosocle	C₂×C₄×C₇₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂³×D₈×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂×D₄
hypercenter	C₂³×D₈	D₇₃
hypocenter	C₇₃	C₂²×#466
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×D₈

Complements

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

N	H ≅ G/N	type
1. C₂³×C₈×D₇₃	C₂	semidirect
2. C₂²×#14556562	C₂	semidirect
3. C₂×#17988820	C₂	direct (N × H)
4. C₂²×#14556525	C₂	semidirect
5. C₂²×#14556512	C₂	semidirect
6. C₂³×D₈×C₇₃	C₂	semidirect
11. C₂²×C₈×D₇₃	C₂²	semidirect
13. C₂×#14556562	C₂²	semidirect
16. #17988820	C₂²	direct (N × H)
17. C₂×#14556525	C₂²	semidirect
18. C₂×#14556512	C₂²	semidirect
21. C₂²×D₈×C₇₃	C₂²	semidirect
35. C₂×C₈×D₇₃	C₂³	semidirect
37. #14556562	C₂³	semidirect
41. #14556525	C₂³	semidirect
42. #14556512	C₂³	semidirect
50. C₂×D₈×C₇₃	C₂³	semidirect
51. C₂²×C₈×C₇₃	D₄	semidirect
55. C₂³×D₇₃	D₈	semidirect
89. C₂×C₈×C₇₃	C₂×D₄	semidirect
103. C₂²×D₇₃	C₂×D₈	semidirect
121. C₈×C₇₃	C₂²×D₄	semidirect
127. C₂²×C₇₃	#466	semidirect
135. C₂×D₇₃	C₂²×D₈	semidirect
143. C₂×C₇₃	C₂×#466	semidirect
148. C₂³×D₈	D₇₃	semidirect
154. C₇₃	C₂²×#466	semidirect
159. C₂²×D₈	C₂×D₇₃	semidirect
187. C₂×D₈	C₂²×D₇₃	semidirect
192. C₂²×C₈	C₂²⋊D₇₃	semidirect
234. C₂×C₈	C₂×C₂²⋊D₇₃	semidirect
270. C₈	C₂²×C₂²⋊D₇₃	semidirect
279. C₂²	#17988820	direct (N × H)
291. C₂	C₂×#17988820	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×D₇₃	2
C₂²×#14556562	2
C₂×#17988820	2
C₂²×#14556525	2
C₂²×#14556512	2
C₂³×D₈×C₇₃	2
C₂²×#466	73

Subgroups — C22×#17988820 #33953073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17988820 #33953073