Subgroups — #33084012

Order 18688.

1042 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988636	C₂	1	1	yes	maximal, characteristic
2	#22618350	C₂	1	2	yes	maximal
3	C₂×M₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	#22609817	C₂	1	4	yes	maximal, complemented
5	#22609812	C₂	1	4	yes	maximal, complemented
6	C₂×#17988346	C₂	1	2	yes	maximal
7	C₇₃×#2911	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11924012	C₂²	1	2	yes	—
9	C₂²×#11924012	C₂²	1	1	yes	characteristic
10	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
11	#17988636	C₄	1	8	yes	complemented
12	C₂×#14556527	C₂²	1	4	yes	—
13	#17988543	—	2	4	no	—
14	#17988543	C₂²	1	8	yes	—
15	C₄×#11924012	—	2	4	no	—
16	M₁₆×Dic₇₃	C₄	1	8	yes	complemented
17	C₂×#14556501	C₂²	1	2	yes	—
18	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
19	#17988346	C₂²	1	4	yes	—
20	#17988346	C₂²	1	4	yes	—
21	C₂²×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
22	C₂×#117×C₇₃	C₂²	1	2	yes	—
23	#428×C₇₃	C₄	1	8	yes	complemented
24	#427×C₇₃	C₂²	1	4	yes	—
25	C₂³×Dic₇₃	—	2	1	no	—
26	C₂×#11924012	—	2	2	no	—
27	C₂×#11924012	—	2	4	no	—
28	C₂×#11924012	C₂×C₄	1	8	yes	—
29	C₂×#11924012	C₂×C₄	1	4	yes	—
30	C₂×#11924012	D₄	1	4	yes	—
31	C₂×#11924012	Q₈	1	4	yes	—
32	C₂×#11924012	C₂³	1	4	yes	—
33	C₂²×#10596831	—	2	2	no	—
34	C₂²×#10596831	C₂³	1	2	yes	—
35	C₂×C₄×Dic₇₃	—	2	1	no	—
36	C₂×C₄×Dic₇₃	—	2	2	no	—
37	C₂×C₄×Dic₇₃	C₂×C₄	1	4	yes	—
38	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
39	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
40	#14556527	C₂×C₄	1	16	yes	complemented
41	#14556502	—	2	8	no	—
42	#14556501	C₂×C₄	1	8	yes	complemented
43	C₄×#10596831	—	2	8	no	—
44	C₈×Dic₇₃	C₂×C₄	1	8	yes	complemented
45	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
46	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
47	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
48	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
50	#117×C₇₃	—	2	2	no	—
51	#117×C₇₃	—	2	2	no	—
52	#117×C₇₃	C₂×C₄	1	4	yes	—
53	#117×C₇₃	C₂³	1	4	yes	—
54	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
55	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
56	C₂×C₄²×C₇₃	—	2	2	no	—
57	#107×C₇₃	C₂×C₄	1	8	yes	complemented
58	#106×C₇₃	C₂×C₄	1	8	yes	complemented
59	C₂²×Dic₇₃	—	2	1	no	—
60	C₂²×Dic₇₃	—	2	4	no	—
61	C₂²×Dic₇₃	—	2	1	no	—
62	C₂²×Dic₇₃	—	4	4	no	—
63	#11924012	—	2	8	no	—
64	#11924012	—	2	4	no	—
65	#11924012	C₄⋊C₄	1	16	yes	—
66	#11924012	C₂²×C₄	1	8	yes	—
67	C₂×#10596831	—	2	8	no	—
68	C₂×#10596831	—	2	4	no	—
69	C₂×#10596831	C₂²×C₄	1	4	yes	—
70	C₂×#10596831	C₂×D₄	1	4	yes	—
71	C₂×#10596831	C₂×Q₈	1	4	yes	—
72	C₄×Dic₇₃	—	2	4	no	—
73	C₄×Dic₇₃	—	2	1	no	—
74	C₄×Dic₇₃	—	2	1	no	—
75	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
76	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
77	C₂⁴×C₇₃	#41	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	—	2	4	no	—
79	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	#41	1	2	yes	—
86	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
87	M₁₆×C₇₃	C₄²	1	16	yes	—
88	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
91	C₄⋊C₄×C₇₃	—	2	4	no	—
92	C₄⋊C₄×C₇₃	C₂²×C₄	1	8	yes	—
93	C₂²⋊C₄×C₇₃	—	2	4	no	—
94	C₂²⋊C₄×C₇₃	—	2	4	no	—
95	C₄²×C₇₃	—	2	4	no	—
96	C₄²×C₇₃	—	2	4	no	—
97	C₂³×C₇₃	—	4	2	no	—
98	C₂³×C₇₃	#117	1	2	yes	—
99	C₂³×C₇₃	C₄×D₄	1	2	yes	—
100	C₂³×C₇₃	C₄×Q₈	1	2	yes	—
101	C₂³×C₇₃	C₂×#41	1	1	yes	characteristic
102	C₂×C₄×C₇₃	—	2	8	no	—
103	C₂×C₄×C₇₃	—	8	4	no	—
104	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
105	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄×C₇₃	#117	1	2	yes	—
107	C₂×C₄×C₇₃	C₄×D₄	1	4	yes	—
108	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₇₃	C₄×Q₈	1	4	yes	—
111	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
117	C₂×Dic₇₃	—	2	4	no	—
118	C₂×Dic₇₃	—	2	2	no	—
119	C₂×Dic₇₃	—	2	2	no	—
120	C₂×Dic₇₃	—	4	8	no	—
121	C₂×Dic₇₃	—	4	2	no	—
122	C₈×C₇₃	C₂×C₄²	1	8	yes	—
123	#10596831	—	2	8	no	—
124	#10596831	C₂×C₄⋊C₄	1	8	yes	—
125	C₂²×C₇₃	—	2	2	no	—
126	C₂²×C₇₃	—	2	4	no	—
127	C₂²×C₇₃	—	4	2	no	—
128	C₂²×C₇₃	—	4	2	no	—
129	C₂²×C₇₃	C₄×C₄⋊C₄	1	4	yes	—
130	C₂²×C₇₃	C₂×#117	1	1	yes	characteristic
131	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂²×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
133	C₄×C₇₃	—	4	8	no	—
134	C₄×C₇₃	C₄×C₄⋊C₄	1	4	yes	—
135	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
136	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
138	C₄×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
139	Dic₇₃	—	2	4	no	—
140	Dic₇₃	—	4	2	no	—
141	#20445	—	73	1	no	maximal, Sylow
142	C₂×C₇₃	—	2	1	no	—
143	C₂×C₇₃	—	2	1	no	—
144	C₂×C₇₃	—	2	4	no	—
145	C₂×C₇₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₇₃	#2956	1	2	yes	—
147	C₂×#430	—	73	1	no	—
148	#2911	D₇₃	1	1	yes	characteristic
149	#2873	—	73	2	no	—
150	C₂×C₄×M₁₆	—	73	1	no	—
151	#1773	—	73	4	no	—
152	#1772	—	73	4	no	—
153	C₂×#339	—	73	2	no	—
154	C₇₃	#20445	1	1	yes	characteristic, complemented, Sylow
155	C₂²×M₁₆	—	73	1	no	—
156	C₂²×M₁₆	—	73	2	no	—
157	C₂²×M₁₆	C₂×D₇₃	1	1	yes	characteristic
158	C₂×#117	C₂×D₇₃	1	2	yes	—
159	C₂²×C₄²	—	73	1	no	—
160	#430	—	73	8	no	—
161	C₂×#108	—	73	4	no	—
162	#428	Dic₇₃	1	8	yes	complemented
163	#427	C₂×D₇₃	1	4	yes	—
164	#405	—	73	8	no	—
165	#405	—	146	4	no	—
166	C₄×M₁₆	—	73	8	no	—
167	C₄×M₁₆	—	146	4	no	—
168	C₂×#97	—	73	2	no	—
169	C₂×C₄×C₈	—	73	2	no	—
170	#339	—	73	4	no	—
171	#339	—	73	4	no	—
172	C₂³×C₄	—	146	1	no	—
173	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
174	C₂×M₁₆	—	73	4	no	—
175	C₂×M₁₆	—	73	4	no	—
176	C₂×M₁₆	—	73	4	no	—
177	C₂×M₁₆	—	73	8	no	—
178	C₂×M₁₆	—	73	4	no	—
179	C₂×M₁₆	—	146	2	no	—
180	C₂×M₁₆	—	146	4	no	—
181	C₂×M₁₆	C₄×D₇₃	1	4	yes	—
182	C₂×M₁₆	C₄×D₇₃	1	4	yes	—
183	C₂×M₁₆	C₂×Dic₇₃	1	4	yes	—
184	C₂²×C₈	—	73	2	no	—
185	C₂²×C₈	—	146	2	no	—
186	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
187	#117	—	2	2	no	—
188	#117	—	2	2	no	—
189	#117	C₂×Dic₇₃	1	4	yes	—
190	#117	C₂²×D₇₃	1	4	yes	—
191	C₂×C₄⋊C₄	C₂×Dic₇₃	1	4	yes	—
192	C₂×C₂²⋊C₄	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	73	4	no	—
195	C₂×C₄²	—	73	1	no	—
196	C₂×C₄²	—	73	1	no	—
197	C₂×C₄²	—	146	1	no	—
198	C₂×C₄²	—	146	2	no	—
199	#108	—	73	16	no	—
200	#107	C₂×Dic₇₃	1	8	yes	complemented
201	#106	C₂×Dic₇₃	1	8	yes	complemented
202	#97	—	73	8	no	—
203	#97	—	146	8	no	—
204	C₄×C₈	—	73	8	no	—
205	C₄×C₈	—	146	8	no	—
206	C₂⁴	#11924027	1	1	yes	characteristic
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	292	4	no	—
209	C₂²×C₄	—	146	4	no	—
210	C₂²×C₄	—	146	1	no	—
211	C₂²×C₄	—	146	1	no	—
212	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
213	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
214	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
215	C₂²×C₄	D₄×D₇₃	1	2	yes	—
216	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
217	C₂²×C₄	#11924027	1	1	yes	characteristic
218	C₂²×C₄	#11924029	1	2	yes	—
219	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
220	M₁₆	—	73	16	no	—
221	M₁₆	—	73	8	no	—
222	M₁₆	—	146	8	no	—
223	M₁₆	—	146	4	no	—
224	M₁₆	C₄×Dic₇₃	1	16	yes	—
225	C₂×C₈	—	73	4	no	—
226	C₂×C₈	—	73	4	no	—
227	C₂×C₈	—	73	4	no	—
228	C₂×C₈	—	146	8	no	—
229	C₂×C₈	—	146	4	no	—
230	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
231	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
232	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	C₂²×Dic₇₃	1	8	yes	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	73	2	no	—
240	C₄²	—	73	2	no	—
241	C₄²	—	146	4	no	—
242	C₄²	—	146	1	no	—
243	C₄²	—	146	1	no	—
244	C₂³	—	4	2	no	—
245	C₂³	#14556549	1	2	yes	—
246	C₂³	D₄×Dic₇₃	1	2	yes	—
247	C₂³	#14556556	1	2	yes	—
248	C₂³	C₂×#11924027	1	1	yes	characteristic
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	—	292	2	no	—
252	C₂×C₄	—	292	8	no	—
253	C₂×C₄	—	146	2	no	—
254	C₂×C₄	—	146	4	no	—
255	C₂×C₄	—	146	2	no	—
256	C₂×C₄	C₂×C₄×Dic₇₃	1	4	yes	—
257	C₂×C₄	C₄⋊C₄×D₇₃	1	4	yes	—
258	C₂×C₄	#14556549	1	2	yes	—
259	C₂×C₄	#14556553	1	4	yes	—
260	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
261	C₂×C₄	#14556556	1	2	yes	—
262	C₂×C₄	Q₈×Dic₇₃	1	4	yes	—
263	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
265	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
266	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
267	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
268	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
269	C₈	—	73	8	no	—
270	C₈	—	146	8	no	—
271	C₈	C₂×C₄×Dic₇₃	1	8	yes	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	C₄⋊C₄×Dic₇₃	1	4	yes	—
277	C₂²	C₂×#14556549	1	1	yes	characteristic
278	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
279	C₂²	C₂×#14556556	1	1	yes	characteristic
280	C₄	—	4	8	no	—
281	C₄	—	292	2	no	—
282	C₄	—	146	4	no	—
283	C₄	C₄⋊C₄×Dic₇₃	1	4	yes	—
284	C₄	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
285	C₄	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
286	C₄	C₂×#14556553	1	1	yes	characteristic
287	C₄	C₂×Q₈×Dic₇₃	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	4	no	—
290	C₂	—	2	1	no	—
291	C₂	C₂×C₄⋊C₄×Dic₇₃	1	1	yes	characteristic
292	C₂	#22618841	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₄×C₇₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#2911	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂⁴×C₇₃	#41
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#2911	D₇₃
hypocenter	C₇₃	#20445
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2911

Complements

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

N	H ≅ G/N	type
4. #22609817	C₂	semidirect
5. #22609812	C₂	semidirect
11. #17988636	C₄	semidirect
16. M₁₆×Dic₇₃	C₄	semidirect
23. #428×C₇₃	C₄	semidirect
40. #14556527	C₂×C₄	semidirect
42. #14556501	C₂×C₄	semidirect
44. C₈×Dic₇₃	C₂×C₄	semidirect
57. #107×C₇₃	C₂×C₄	semidirect
58. #106×C₇₃	C₂×C₄	semidirect
154. C₇₃	#20445	semidirect
162. #428	Dic₇₃	semidirect
200. #107	C₂×Dic₇₃	semidirect
201. #106	C₂×Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988636	2
#22618350	2
C₂×M₁₆×Dic₇₃	2
#22609817	2
#22609812	2
C₂×#17988346	2
C₇₃×#2911	2
#20445	73