Subgroups — #33084584

Order 18688.

998 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988703	C₂	1	2	yes	maximal
2	C₄×#14556613	C₂	1	1	yes	maximal, characteristic
3	#22618360	C₂	1	2	yes	maximal, complemented
4	#22609813	C₂	1	2	yes	maximal, complemented
5	#22609774	C₂	1	4	yes	maximal, complemented
6	#22609744	C₂	1	2	yes	maximal, complemented
7	#22609617	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#108×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556613	C₂²	1	1	yes	characteristic
10	C₂×#14556599	C₂²	1	4	yes	—
11	C₄×#11924028	C₂²	1	2	yes	—
12	#17988703	C₄	1	16	yes	complemented
13	C₄×Dic₂₉₂	C₄	1	2	yes	complemented
14	C₄×#11924017	C₄	1	4	yes	complemented
15	C₄×D₅₈₄	C₄	1	2	yes	complemented
16	#17988625	C₂²	1	1	yes	characteristic
17	C₂×#14556517	C₂²	1	4	yes	—
18	#17988584	C₂²	1	2	yes	—
19	#17988545	C₂²	1	4	yes	—
20	#17988541	—	2	4	no	—
21	#17988539	—	2	2	no	—
22	C₂×#14556502	C₂²	1	1	yes	characteristic
23	C₂×#14556501	C₂²	1	2	yes	—
24	#97×D₇₃	—	2	2	no	—
25	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
26	#17988433	C₂²	1	1	yes	characteristic
27	#17988348	C₂²	1	4	yes	—
28	C₂×#108×C₇₃	C₂²	1	2	yes	—
29	C₄×M₁₆×C₇₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
31	#341×C₇₃	C₂²	1	2	yes	—
32	C₂×#11924028	C₂³	1	2	yes	—
33	#14556613	C₂×C₄	1	8	yes	—
34	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
35	C₂×#11924017	C₂×C₄	1	2	yes	—
36	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic
37	#14556599	—	2	8	no	—
38	#14556599	C₂×C₄	1	16	yes	complemented
39	C₂×#11924011	—	2	4	no	—
40	C₂²×#10596831	C₂³	1	2	yes	—
41	C₂×C₈×D₇₃	—	2	4	no	—
42	C₄×Dic₁₄₆	C₂×C₄	1	4	yes	—
43	C₄×D₂₉₂	C₂×C₄	1	4	yes	—
44	C₄×C₂²⋊D₇₃	—	2	4	no	—
45	#14556550	C₂³	1	2	yes	—
46	#14556548	—	2	2	no	—
47	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
48	C₄²×D₇₃	—	2	2	no	—
49	#14556525	C₂×C₄	1	2	yes	—
50	#14556523	C₂×C₄	1	2	yes	—
51	#14556517	C₂×C₄	1	16	yes	complemented
52	#14556516	C₂×C₄	1	4	yes	—
53	#14556513	C₂×C₄	1	4	yes	—
54	#14556502	D₄	1	1	yes	characteristic
55	#14556502	D₄	1	1	yes	characteristic
56	#14556502	D₄	1	2	yes	—
57	#14556501	—	2	4	no	—
58	#14556501	—	2	4	no	—
59	C₈×Dic₇₃	—	2	4	no	—
60	C₈×Dic₇₃	C₂×C₄	1	8	yes	complemented
61	#14556498	—	2	2	no	—
62	C₂×M₁₆×C₇₃	C₂³	1	4	yes	—
63	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
66	#108×C₇₃	C₂×C₄	1	8	yes	complemented
67	#97×C₇₃	—	2	2	no	—
68	C₄×C₈×C₇₃	—	2	2	no	—
69	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
71	#11924028	—	2	4	no	—
72	#11924028	C₂²×C₄	1	8	yes	—
73	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
74	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
75	C₂×C₂²⋊D₇₃	—	2	2	no	—
76	C₂²×Dic₇₃	—	2	2	no	—
77	C₂×C₄×D₇₃	—	2	2	no	—
78	C₂×C₄×D₇₃	—	2	4	no	—
79	Dic₂₉₂	C₄²	1	4	yes	—
80	#11924017	C₄²	1	8	yes	—
81	D₅₈₄	C₄²	1	4	yes	—
82	#11924011	—	2	16	no	—
83	C₂×#10596831	C₂²×C₄	1	4	yes	—
84	C₂×#10596831	C₂×D₄	1	2	yes	—
85	C₂×#10596831	C₂×D₄	1	2	yes	—
86	C₂×#10596831	#41	1	4	yes	—
87	C₈×D₇₃	—	2	16	no	—
88	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
89	#11924007	—	2	4	no	—
90	C₂²⋊Dic₇₃	—	2	2	no	—
91	#11924005	—	2	4	no	—
92	C₄×Dic₇₃	—	2	4	no	—
93	C₄×Dic₇₃	—	2	2	no	—
94	C₄×Dic₇₃	—	2	2	no	—
95	C₄×Dic₇₃	C₂²×C₄	1	4	yes	—
96	C₂²×C₄×C₇₃	#41	1	2	yes	—
97	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
98	M₁₆×C₇₃	—	2	4	no	—
99	M₁₆×C₇₃	C₂²×C₄	1	8	yes	—
100	C₂×C₈×C₇₃	—	2	4	no	—
101	C₂×C₈×C₇₃	—	2	2	no	—
102	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
106	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
107	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
109	C₂³×C₇₃	C₂×#41	1	1	yes	characteristic
110	C₂²×D₇₃	—	2	2	no	—
111	C₂×C₄×C₇₃	—	4	2	no	—
112	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₇₃	C₂³×C₄	1	2	yes	—
115	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
118	C₂²⋊D₇₃	—	2	8	no	—
119	C₂×Dic₇₃	—	2	2	no	—
120	C₂×Dic₇₃	—	2	4	no	—
121	C₂×Dic₇₃	—	2	4	no	—
122	C₂×Dic₇₃	—	4	4	no	—
123	D₂₉₂	—	2	2	no	—
124	D₂₉₂	C₂×C₄²	1	4	yes	—
125	C₄×D₇₃	—	2	8	no	—
126	C₄×D₇₃	—	8	4	no	—
127	Dic₁₄₆	—	2	2	no	—
128	Dic₁₄₆	C₂×C₄²	1	4	yes	—
129	C₈×C₇₃	—	2	8	no	—
130	C₈×C₇₃	—	2	2	no	—
131	C₈×C₇₃	C₂×C₄²	1	2	yes	—
132	C₈×C₇₃	C₂×C₄²	1	2	yes	—
133	#10596831	C₄×D₄	1	4	yes	—
134	#10596831	C₄×D₄	1	4	yes	—
135	C₂²×C₇₃	—	2	2	no	—
136	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₇₃	C₄×#41	1	2	yes	—
138	C₂×D₇₃	—	2	4	no	—
139	C₂×D₇₃	—	4	4	no	—
140	C₄×C₇₃	—	2	2	no	—
141	C₄×C₇₃	—	2	2	no	—
142	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
143	C₄×C₇₃	C₂×C₄×D₄	1	2	yes	—
144	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
145	Dic₇₃	—	2	4	no	—
146	Dic₇₃	—	4	4	no	—
147	Dic₇₃	—	4	4	no	—
148	#20514	—	73	1	no	maximal, Sylow
149	C₂×C₇₃	—	2	2	no	—
150	C₂×C₇₃	C₄²×D₄	1	1	yes	characteristic
151	C₂×C₇₃	#2957	1	2	yes	—
152	D₇₃	—	4	4	no	—
153	C₂×#443	—	73	2	no	—
154	C₄×#135	—	73	1	no	—
155	#2875	—	73	2	no	—
156	C₄×#108	D₇₃	1	1	yes	characteristic, complemented
157	#1772	—	73	2	no	—
158	#1765	—	73	4	no	—
159	#1759	—	73	2	no	—
160	#1726	—	73	1	no	—
161	C₇₃	#20514	1	1	yes	characteristic, complemented, Sylow
162	C₂×#135	—	73	1	no	—
163	C₂×#131	—	73	4	no	—
164	C₄×#41	—	73	2	no	—
165	#443	—	73	16	no	—
166	C₄×Q₁₆	—	73	2	no	—
167	C₄×SD₁₆	—	73	4	no	—
168	C₄×D₈	—	73	2	no	—
169	C₂×#108	C₂×D₇₃	1	2	yes	—
170	#427	—	73	1	no	—
171	C₂×#104	—	73	4	no	—
172	#416	—	73	2	no	—
173	#405	—	73	4	no	—
174	#405	—	146	4	no	—
175	C₄×M₁₆	—	146	2	no	—
176	C₄×M₁₆	C₂×D₇₃	1	2	yes	—
177	C₂×#97	—	73	1	no	—
178	C₂×#97	—	73	2	no	—
179	C₂×#97	—	146	2	no	—
180	C₂×C₄×C₈	—	73	2	no	—
181	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
182	#376	—	73	1	no	—
183	#341	C₂×D₇₃	1	2	yes	—
184	#339	—	73	4	no	—
185	C₂×#41	—	73	2	no	—
186	#135	—	73	8	no	—
187	C₂×Q₁₆	—	73	1	no	—
188	C₂×SD₁₆	—	73	2	no	—
189	C₂×D₈	—	73	1	no	—
190	#131	—	73	16	no	—
191	#131	—	146	8	no	—
192	C₂×M₁₆	—	146	4	no	—
193	C₂×M₁₆	C₂²×D₇₃	1	4	yes	—
194	C₂²×C₈	—	73	2	no	—
195	C₂²×C₈	—	146	4	no	—
196	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
198	C₄×Q₈	—	73	4	no	—
199	C₄×D₄	—	73	4	no	—
200	C₄×D₄	—	146	4	no	—
201	#117	—	73	2	no	—
202	#117	—	146	2	no	—
203	C₂×C₄²	—	73	2	no	—
204	C₂×C₄²	—	146	2	no	—
205	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
206	#108	C₄×D₇₃	1	8	yes	complemented
207	#107	—	73	2	no	—
208	#106	—	73	2	no	—
209	#104	—	73	16	no	—
210	#103	—	73	4	no	—
211	#102	—	73	4	no	—
212	#97	—	2	2	no	—
213	#97	—	73	1	no	—
214	#97	—	73	1	no	—
215	#97	—	73	2	no	—
216	#97	—	146	4	no	—
217	#97	—	146	4	no	—
218	C₄×C₈	—	2	2	no	—
219	C₄×C₈	—	73	8	no	—
220	C₄×C₈	—	146	4	no	—
221	C₄×C₈	C₄×D₇₃	1	2	yes	—
222	C₄×C₈	C₄×D₇₃	1	2	yes	—
223	#95	—	146	2	no	—
224	#41	—	73	8	no	—
225	#41	—	146	4	no	—
226	C₂×Q₈	—	73	2	no	—
227	C₂×D₄	—	73	2	no	—
228	C₂×D₄	—	146	2	no	—
229	C₂²×C₄	—	146	2	no	—
230	C₂²×C₄	—	146	4	no	—
231	C₂²×C₄	—	146	2	no	—
232	C₂²×C₄	#11924029	1	2	yes	—
233	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
234	Q₁₆	—	73	4	no	—
235	SD₁₆	—	73	8	no	—
236	D₈	—	73	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	146	16	no	—
239	M₁₆	C₂×C₄×D₇₃	1	8	yes	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	73	4	no	—
243	C₂×C₈	—	73	2	no	—
244	C₂×C₈	—	73	2	no	—
245	C₂×C₈	—	73	4	no	—
246	C₂×C₈	—	146	16	no	—
247	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
248	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
249	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
251	C₄⋊C₄	—	73	4	no	—
252	C₄⋊C₄	—	146	4	no	—
253	C₂²⋊C₄	—	146	2	no	—
254	C₂²⋊C₄	—	146	4	no	—
255	C₄²	—	73	4	no	—
256	C₄²	—	146	4	no	—
257	C₄²	—	146	2	no	—
258	C₄²	—	146	2	no	—
259	C₄²	C₂×C₄×D₇₃	1	2	yes	—
260	C₄²	D₄×D₇₃	1	1	yes	characteristic
261	C₄²	D₄×D₇₃	1	1	yes	characteristic
262	C₂³	—	146	2	no	—
263	C₂³	C₂×#11924029	1	1	yes	characteristic
264	Q₈	—	73	4	no	—
265	Q₈	—	146	2	no	—
266	D₄	—	73	4	no	—
267	D₄	—	146	8	no	—
268	D₄	—	146	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	292	4	no	—
271	C₂×C₄	—	584	4	no	—
272	C₂×C₄	—	146	4	no	—
273	C₂×C₄	—	146	4	no	—
274	C₂×C₄	—	146	2	no	—
275	C₂×C₄	—	146	8	no	—
276	C₂×C₄	#14556553	1	2	yes	—
277	C₂×C₄	#14556553	1	2	yes	—
278	C₂×C₄	C₂²×C₄×D₇₃	1	2	yes	—
279	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
280	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
281	C₂×C₄	#41×D₇₃	1	2	yes	—
282	C₈	—	2	8	no	—
283	C₈	—	2	2	no	—
284	C₈	—	73	4	no	—
285	C₈	—	73	4	no	—
286	C₈	C₄²×D₇₃	1	2	yes	—
287	C₈	C₄²×D₇₃	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	292	4	no	—
290	C₂²	—	146	4	no	—
291	C₂²	C₂×#14556553	1	1	yes	characteristic
292	C₂²	C₄×#11924029	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	292	4	no	—
296	C₄	—	292	4	no	—
297	C₄	—	146	4	no	—
298	C₄	C₂×C₄²×D₇₃	1	1	yes	characteristic
299	C₄	C₄×D₄×D₇₃	1	2	yes	—
300	C₄	#17989055	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	292	4	no	—
303	C₂	C₄×#14556553	1	1	yes	characteristic
304	C₂	#22618847	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₄×C₇₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₄×#108×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂×#14556613	C₂²
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₄×#108	D₇₃
hypocenter	C₇₃	#20514
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
3. #22618360	C₂	semidirect
4. #22609813	C₂	semidirect
5. #22609774	C₂	semidirect
6. #22609744	C₂	semidirect
7. #22609617	C₂	semidirect
8. C₄×#108×C₇₃	C₂	semidirect
12. #17988703	C₄	semidirect
13. C₄×Dic₂₉₂	C₄	semidirect
14. C₄×#11924017	C₄	semidirect
15. C₄×D₅₈₄	C₄	semidirect
38. #14556599	C₂×C₄	semidirect
51. #14556517	C₂×C₄	semidirect
60. C₈×Dic₇₃	C₂×C₄	semidirect
66. #108×C₇₃	C₂×C₄	semidirect
156. C₄×#108	D₇₃	semidirect
161. C₇₃	#20514	semidirect
206. #108	C₄×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988703	2
C₄×#14556613	2
#22618360	2
#22609813	2
#22609774	2
#22609744	2
#22609617	2
C₄×#108×C₇₃	2
#20514	73