Subgroups — #33243648

Order 18688.

486 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622921	C₂	1	1	yes	maximal, characteristic, complemented
2	#22622800	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612164	C₂	1	2	yes	maximal, complemented
4	#22612097	C₂	1	2	yes	maximal, complemented
5	#22612075	C₂	1	1	yes	maximal, characteristic, complemented
6	#22611989	C₂	1	2	yes	maximal
7	#22611973	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₆×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic, complemented
9	#22611780	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₁₆₈	C₂	1	1	yes	maximal, characteristic
11	C₇₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989311	—	2	1	no	—
13	#17989298	—	2	1	no	—
14	C₂×#14556534	C₂²	1	2	yes	—
15	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
16	#17988988	C₂²	1	2	yes	—
17	#17988972	C₂²	1	2	yes	—
18	#17988890	—	2	2	no	—
19	#17988874	—	2	1	no	—
20	#17988811	—	2	2	no	—
21	#17988789	—	2	1	no	—
22	C₈×Dic₁₄₆	C₂²	1	1	yes	characteristic
23	Q₁₆×Dic₇₃	C₂²	1	2	yes	—
24	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
25	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
26	#17988669	C₂²	1	2	yes	—
27	#17988414	C₂²	1	2	yes	—
28	#17988413	C₂²	1	1	yes	characteristic
29	C₄⋊#11924003	C₂²	1	1	yes	characteristic
30	#17988405	C₂²	1	2	yes	—
31	#17988391	C₂²	1	2	yes	—
32	#17988389	C₂²	1	1	yes	characteristic
33	#17988389	C₂²	1	1	yes	characteristic
34	#17988387	C₂²	1	2	yes	—
35	C₁₆×Dic₇₃	C₂²	1	2	yes	—
36	#500×C₇₃	C₂²	1	1	yes	characteristic
37	#494×C₇₃	C₂²	1	1	yes	characteristic
38	#358×C₇₃	C₂²	1	2	yes	—
39	#358×C₇₃	C₂²	1	2	yes	—
40	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
41	C₂×#11924015	—	2	2	no	—
42	C₂×D₅₈₄	C₂³	1	1	yes	characteristic
43	C₂×D₅₈₄	C₂³	1	1	yes	characteristic
44	#14556586	—	4	2	no	—
45	#14556582	—	2	2	no	—
46	#14556580	—	2	2	no	—
47	#14556571	—	4	1	no	—
48	#14556571	—	4	1	no	—
49	#14556567	—	4	2	no	—
50	Q₈×Dic₇₃	—	2	2	no	—
51	C₄×Dic₁₄₆	D₄	1	1	yes	characteristic
52	C₄×D₂₉₂	—	2	1	no	—
53	C₄×D₂₉₂	—	2	1	no	—
54	#14556553	—	2	2	no	—
55	#14556549	—	4	1	no	—
56	C₄⋊C₄×D₇₃	—	4	1	no	—
57	#14556534	—	2	4	no	—
58	D₁₁₆₈	—	2	2	no	—
59	C₂×#11924003	C₂³	1	2	yes	—
60	#14556525	C₂³	1	1	yes	characteristic
61	#14556525	C₂³	1	1	yes	characteristic
62	#14556522	—	2	2	no	—
63	C₄⋊#10596831	D₄	1	1	yes	characteristic
64	#14556520	D₄	1	2	yes	—
65	#14556515	—	2	2	no	—
66	#14556513	—	2	1	no	—
67	#14556513	—	2	1	no	—
68	#14556511	—	2	2	no	—
69	C₈×Dic₇₃	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₇₃	—	2	1	no	—
71	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
72	C₄⋊Q₈×C₇₃	—	2	1	no	—
73	C₄⋊Q₈×C₇₃	—	2	1	no	—
74	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
75	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
76	#107×C₇₃	—	2	1	no	—
77	#107×C₇₃	C₂³	1	2	yes	—
78	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₄₆	D₈	1	1	yes	characteristic
80	C₂×D₂₉₂	—	2	1	no	—
81	C₂×D₂₉₂	—	2	1	no	—
82	C₂×C₄×D₇₃	—	4	1	no	—
83	C₂×C₄×D₇₃	—	4	1	no	—
84	C₂×C₄×D₇₃	—	4	2	no	—
85	#11924015	—	8	2	no	—
86	D₅₈₄	—	2	1	no	—
87	D₅₈₄	—	2	1	no	—
88	D₅₈₄	—	2	1	no	—
89	D₅₈₄	#41	1	2	yes	—
90	C₂×#10596831	C₂×D₄	1	2	yes	—
91	C₄⋊Dic₇₃	—	2	1	no	—
92	C₄⋊Dic₇₃	—	2	1	no	—
93	C₄⋊Dic₇₃	—	4	2	no	—
94	C₄⋊Dic₇₃	—	4	2	no	—
95	C₄⋊Dic₇₃	D₈	1	1	yes	characteristic
96	#11924007	—	4	2	no	—
97	#11924007	D₈	1	2	yes	—
98	#11924005	—	4	1	no	—
99	#11924005	—	4	1	no	—
100	#11924005	—	4	2	no	—
101	#11924005	—	4	2	no	—
102	#11924005	—	4	2	no	—
103	C₄×Dic₇₃	—	4	2	no	—
104	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
105	#11924003	—	2	2	no	—
106	C₂×Q₈×C₇₃	—	2	2	no	—
107	C₂×Q₈×C₇₃	—	4	1	no	—
108	Q₁₆×C₇₃	—	2	2	no	—
109	Q₁₆×C₇₃	—	2	2	no	—
110	Q₁₆×C₇₃	—	8	1	no	—
111	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₇₃	—	2	2	no	—
114	C₄⋊C₄×C₇₃	—	4	1	no	—
115	C₄⋊C₄×C₇₃	—	4	1	no	—
116	C₄⋊C₄×C₇₃	—	4	1	no	—
117	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₇₃	—	2	1	no	—
119	C₁₆×C₇₃	—	2	1	no	—
120	C₂²×D₇₃	—	4	1	no	—
121	C₂²×D₇₃	—	4	1	no	—
122	Q₈×C₇₃	—	4	2	no	—
123	Q₈×C₇₃	—	4	2	no	—
124	Q₈×C₇₃	—	16	1	no	—
125	C₂×C₄×C₇₃	—	4	2	no	—
126	C₂×C₄×C₇₃	—	4	2	no	—
127	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₇₃	—	4	1	no	—
131	C₂×Dic₇₃	—	4	1	no	—
132	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
133	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
134	D₂₉₂	—	2	2	no	—
135	D₂₉₂	—	4	1	no	—
136	D₂₉₂	—	4	1	no	—
137	D₂₉₂	—	4	1	no	—
138	C₄×D₇₃	—	8	2	no	—
139	C₄×D₇₃	—	8	2	no	—
140	C₄×D₇₃	—	16	2	no	—
141	Dic₁₄₆	—	2	2	no	—
142	C₈×C₇₃	—	2	1	no	—
143	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
144	C₈×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
145	#10596831	—	2	2	no	—
146	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₇₃	—	4	2	no	—
148	C₂×D₇₃	—	8	1	no	—
149	C₂×D₇₃	—	8	1	no	—
150	C₂×D₇₃	—	8	1	no	—
151	C₂×D₇₃	—	8	1	no	—
152	C₂×D₇₃	—	8	1	no	—
153	C₄×C₇₃	—	2	1	no	—
154	C₄×C₇₃	—	8	2	no	—
155	C₄×C₇₃	—	8	2	no	—
156	C₄×C₇₃	#508	1	2	yes	—
157	C₄×C₇₃	#548	1	1	yes	characteristic
158	C₄×C₇₃	#577	1	1	yes	characteristic
159	Dic₇₃	—	2	2	no	—
160	Dic₇₃	—	2	2	no	—
161	Dic₇₃	—	8	1	no	—
162	Dic₇₃	—	8	1	no	—
163	#33927	—	73	1	no	maximal, Sylow
164	C₂×C₇₃	#3358	1	1	yes	characteristic
165	C₂×C₇₃	C₂×#508	1	1	yes	characteristic
166	C₂×C₇₃	#3416	1	1	yes	characteristic
167	D₇₃	—	8	2	no	—
168	D₇₃	—	16	1	no	—
169	#3393	—	73	1	no	—
170	#3380	—	73	1	no	—
171	#2242	—	73	2	no	—
172	#2241	D₇₃	1	1	yes	characteristic, complemented
173	#2229	—	73	2	no	—
174	#2226	—	73	1	no	—
175	#2210	—	73	2	no	—
176	#2208	—	73	1	no	—
177	Q₈×C₁₆	—	73	1	no	—
178	C₄×SD₃₂	—	73	2	no	—
179	C₄×D₁₆	—	73	1	no	—
180	C₇₃	#33927	1	1	yes	characteristic, complemented, Sylow
181	#556	—	146	1	no	—
182	#554	—	146	1	no	—
183	C₂×SD₃₂	—	73	2	no	—
184	C₂×D₁₆	—	73	1	no	—
185	#500	C₂×D₇₃	1	1	yes	characteristic
186	#499	—	73	2	no	—
187	#495	—	73	2	no	—
188	#494	C₂×D₇₃	1	1	yes	characteristic
189	#478	—	146	2	no	—
190	#476	—	146	1	no	—
191	#464	—	146	2	no	—
192	#461	—	146	1	no	—
193	C₈×Q₈	—	73	1	no	—
194	C₄×Q₁₆	—	73	2	no	—
195	C₄×D₈	—	73	2	no	—
196	C₄×D₈	—	73	1	no	—
197	C₄×D₈	—	73	1	no	—
198	#367	—	73	2	no	—
199	#366	—	73	1	no	—
200	C₄⋊C₁₆	—	73	2	no	—
201	C₄⋊C₁₆	—	73	1	no	—
202	#358	—	73	2	no	—
203	#358	C₂×D₇₃	1	2	yes	—
204	#358	C₂×D₇₃	1	2	yes	—
205	#357	—	73	1	no	—
206	#357	—	73	1	no	—
207	#357	—	73	2	no	—
208	C₄×C₁₆	—	73	2	no	—
209	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
210	C₂×Q₁₆	—	2	1	no	—
211	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
212	C₂×SD₁₆	—	146	2	no	—
213	C₂×D₈	—	73	1	no	—
214	C₂×D₈	—	73	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	#126	—	292	2	no	—
218	#125	—	146	2	no	—
219	#124	—	146	2	no	—
220	C₂²⋊Q₈	—	292	1	no	—
221	C₂²⋊Q₈	—	292	1	no	—
222	C₂²⋊Q₈	—	292	2	no	—
223	C₄×Q₈	—	73	1	no	—
224	C₄×Q₈	—	146	2	no	—
225	C₄×D₄	—	146	2	no	—
226	C₄×D₄	—	146	1	no	—
227	C₄×D₄	—	146	1	no	—
228	#117	—	292	1	no	—
229	C₂×C₄⋊C₄	—	292	1	no	—
230	SD₃₂	—	146	4	no	—
231	D₁₆	—	146	2	no	—
232	C₂×C₁₆	—	73	2	no	—
233	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
234	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
235	#107	—	2	1	no	—
236	#107	—	73	1	no	—
237	#107	—	73	1	no	—
238	#107	C₂²×D₇₃	1	2	yes	—
239	#106	—	146	2	no	—
240	C₄⋊C₈	—	73	2	no	—
241	C₄⋊C₈	—	73	1	no	—
242	#103	—	146	2	no	—
243	#102	—	146	2	no	—
244	#102	—	146	1	no	—
245	#102	—	146	1	no	—
246	C₄×C₈	—	73	2	no	—
247	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	73	1	no	—
251	C₂×D₄	—	146	1	no	—
252	C₂×D₄	—	146	1	no	—
253	C₂²×C₄	—	292	2	no	—
254	C₂²×C₄	—	292	1	no	—
255	C₂²×C₄	—	292	1	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	2	2	no	—
258	Q₁₆	—	8	1	no	—
259	SD₁₆	—	584	2	no	—
260	D₈	—	73	2	no	—
261	D₈	—	146	1	no	—
262	D₈	—	146	1	no	—
263	D₈	—	146	1	no	—
264	C₂×C₈	—	73	2	no	—
265	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
266	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	292	2	no	—
272	C₄⋊C₄	—	292	2	no	—
273	C₄⋊C₄	—	292	2	no	—
274	C₄⋊C₄	—	73	2	no	—
275	C₄⋊C₄	—	73	1	no	—
276	C₄⋊C₄	—	146	1	no	—
277	C₄⋊C₄	—	146	1	no	—
278	C₂²⋊C₄	—	292	2	no	—
279	C₂²⋊C₄	—	292	2	no	—
280	C₂²⋊C₄	—	292	2	no	—
281	C₂²⋊C₄	—	292	1	no	—
282	C₂²⋊C₄	—	292	1	no	—
283	C₄²	—	292	2	no	—
284	C₄²	—	73	2	no	—
285	C₄²	D₄×D₇₃	1	1	yes	characteristic
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	—	146	2	no	—
289	C₂³	—	292	1	no	—
290	C₂³	—	292	1	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	16	1	no	—
294	Q₈	—	146	2	no	—
295	D₄	—	292	1	no	—
296	D₄	—	292	1	no	—
297	D₄	—	292	1	no	—
298	D₄	—	146	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	292	1	no	—
302	C₂×C₄	—	292	1	no	—
303	C₂×C₄	—	584	2	no	—
304	C₂×C₄	—	584	2	no	—
305	C₂×C₄	—	73	2	no	—
306	C₂×C₄	—	73	2	no	—
307	C₂×C₄	—	1168	2	no	—
308	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
309	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	146	2	no	—
313	C₈	#41×D₇₃	1	1	yes	characteristic
314	C₈	#14556638	1	1	yes	characteristic
315	C₂²	—	292	2	no	—
316	C₂²	—	584	1	no	—
317	C₂²	—	584	1	no	—
318	C₂²	—	584	1	no	—
319	C₂²	—	584	1	no	—
320	C₂²	—	584	1	no	—
321	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	584	1	no	—
326	C₄	—	584	1	no	—
327	C₄	—	146	2	no	—
328	C₄	—	146	2	no	—
329	C₄	#17989020	1	2	yes	—
330	C₄	#17989245	1	1	yes	characteristic
331	C₄	#17989393	1	1	yes	characteristic
332	C₂	—	584	2	no	—
333	C₂	—	1168	1	no	—
334	C₂	#22622563	1	1	yes	characteristic
335	C₂	C₂×#17989020	1	1	yes	characteristic
336	C₂	#22623073	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2241	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×D₁₁₆₈	C₂²
hypercenter	#2241	D₇₃
hypocenter	C₇₃	#33927
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
1. #22622921	C₂	semidirect
2. #22622800	C₂	semidirect
3. #22612164	C₂	semidirect
4. #22612097	C₂	semidirect
5. #22612075	C₂	semidirect
7. #22611973	C₂	semidirect
8. C₁₆×Dic₁₄₆	C₂	semidirect
9. #22611780	C₂	semidirect
11. C₇₃×#2241	C₂	semidirect
172. #2241	D₇₃	semidirect
180. C₇₃	#33927	semidirect

Maximal subgroups

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

subgroup	index
#22622921	2
#22622800	2
#22612164	2
#22612097	2
#22612075	2
#22611989	2
#22611973	2
C₁₆×Dic₁₄₆	2
#22611780	2
C₄×D₁₁₆₈	2
C₇₃×#2241	2
#33927	73