Subgroups — #33239374

Order 18688.

674 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989020	C₂	1	1	yes	maximal, characteristic
2	#22618806	C₂	1	2	yes	maximal, complemented
3	#22611808	C₂	1	1	yes	maximal, characteristic, complemented
4	#22611806	C₂	1	1	yes	maximal, characteristic, complemented
5	#22611802	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14556534	C₂	1	2	yes	maximal, complemented
7	#22611780	C₂	1	2	yes	maximal, complemented
8	#367×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#22611642	C₂	1	2	yes	maximal, complemented
10	#22611529	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2176	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556614	C₂²	1	2	yes	—
13	#17989069	—	2	2	no	—
14	#17989020	C₄	1	8	yes	complemented
15	C₂×#14556534	C₂²	1	2	yes	—
16	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
17	C₂×#14556530	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
19	#17988690	—	2	2	no	—
20	#17988685	—	2	2	no	—
21	Q₁₆×Dic₇₃	C₂²	1	2	yes	—
22	C₄×D₅₈₄	C₂²	1	2	yes	—
23	#17988669	C₂²	1	2	yes	—
24	#107×D₇₃	C₂²	1	2	yes	—
25	#17988587	—	2	2	no	—
26	#17988541	C₂²	1	1	yes	characteristic
27	#17988415	C₂²	1	1	yes	characteristic
28	#17988414	C₂²	1	2	yes	—
29	#17988391	C₂²	1	2	yes	—
30	#17988389	C₂²	1	2	yes	—
31	#17988387	C₂²	1	2	yes	—
32	#17988362	C₂²	1	1	yes	characteristic
33	C₄×#11924003	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₇₃	C₂²	1	2	yes	—
37	#367×C₇₃	C₂²	1	1	yes	characteristic
38	#358×C₇₃	C₂²	1	2	yes	—
39	#346×C₇₃	C₂²	1	1	yes	characteristic
40	C₂×#11924029	—	2	2	no	—
41	#14556614	—	2	4	no	—
42	#14556614	C₂×C₄	1	8	yes	complemented
43	C₂×#11924015	—	2	2	no	—
44	C₂×D₅₈₄	C₂³	1	2	yes	—
45	C₂×#11924011	D₄	1	1	yes	characteristic
46	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
47	Q₈×Dic₇₃	—	2	2	no	—
48	C₄×D₂₉₂	—	2	2	no	—
49	C₄×D₂₉₂	—	4	2	no	—
50	#14556553	—	2	2	no	—
51	#14556553	—	4	2	no	—
52	#14556548	—	4	2	no	—
53	#14556548	D₄	1	1	yes	characteristic
54	C₄⋊C₄×D₇₃	—	2	2	no	—
55	C₄⋊C₄×D₇₃	—	4	2	no	—
56	#14556534	C₂×C₄	1	8	yes	complemented
57	D₁₁₆₈	C₂×C₄	1	4	yes	complemented
58	#14556530	—	2	2	no	—
59	C₂×#11924003	C₂³	1	1	yes	characteristic
60	C₂×#11924003	C₂³	1	1	yes	characteristic
61	C₁₆×D₇₃	C₂×C₄	1	4	yes	complemented
62	#14556525	C₂³	1	2	yes	—
63	#14556524	—	2	2	no	—
64	#14556515	—	2	2	no	—
65	#14556513	—	2	2	no	—
66	#14556511	—	2	2	no	—
67	#14556502	D₄	1	1	yes	characteristic
68	#14556501	D₄	1	1	yes	characteristic
69	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
71	C₄×Q₈×C₇₃	—	2	2	no	—
72	Q₃₂×C₇₃	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
75	#107×C₇₃	C₂³	1	2	yes	—
76	#103×C₇₃	—	2	2	no	—
77	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
78	#11924029	—	2	8	no	—
79	#11924029	—	4	4	no	—
80	C₂×D₂₉₂	—	2	2	no	—
81	C₂×D₂₉₂	—	4	2	no	—
82	C₂×C₄×D₇₃	—	4	2	no	—
83	C₂×C₄×D₇₃	—	4	2	no	—
84	C₂×C₄×D₇₃	—	4	2	no	—
85	C₂×C₄×D₇₃	—	4	2	no	—
86	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
87	#11924015	—	2	8	no	—
88	D₅₈₄	—	2	2	no	—
89	D₅₈₄	C₂²×C₄	1	4	yes	—
90	#11924011	—	2	2	no	—
91	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
92	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
93	C₈×D₇₃	C₂²×C₄	1	2	yes	—
94	C₈×D₇₃	#41	1	2	yes	—
95	C₄⋊Dic₇₃	—	2	2	no	—
96	C₄⋊Dic₇₃	—	4	2	no	—
97	#11924007	—	4	2	no	—
98	#11924007	D₈	1	2	yes	—
99	#11924005	—	4	2	no	—
100	#11924005	—	4	2	no	—
101	#11924005	D₈	1	2	yes	—
102	C₄×Dic₇₃	—	4	2	no	—
103	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
104	#11924003	—	2	1	no	—
105	#11924003	C₂²×C₄	1	2	yes	—
106	C₂×Q₈×C₇₃	—	2	2	no	—
107	Q₁₆×C₇₃	—	2	2	no	—
108	Q₁₆×C₇₃	C₂²×C₄	1	4	yes	—
109	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₇₃	—	2	2	no	—
112	C₄⋊C₄×C₇₃	—	4	2	no	—
113	C₄²×C₇₃	—	4	2	no	—
114	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₇₃	—	2	1	no	—
116	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
117	C₂²×D₇₃	—	4	2	no	—
118	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
119	Q₈×C₇₃	—	2	4	no	—
120	Q₈×C₇₃	—	4	2	no	—
121	C₂×C₄×C₇₃	—	4	2	no	—
122	C₂×C₄×C₇₃	—	4	2	no	—
123	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×Dic₇₃	—	4	2	no	—
126	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
127	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
128	D₂₉₂	—	2	4	no	—
129	D₂₉₂	—	4	8	no	—
130	D₂₉₂	—	4	2	no	—
131	C₄×D₇₃	—	4	8	no	—
132	C₄×D₇₃	—	16	2	no	—
133	C₄×D₇₃	—	16	2	no	—
134	C₄×D₇₃	—	16	2	no	—
135	C₄×D₇₃	C₄×D₄	1	2	yes	—
136	C₄×D₇₃	#135	1	2	yes	—
137	C₈×C₇₃	—	2	1	no	—
138	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
139	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
140	#10596831	—	2	1	no	—
141	#10596831	C₄×D₄	1	2	yes	—
142	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
143	C₂×D₇₃	—	2	2	no	—
144	C₂×D₇₃	—	4	4	no	—
145	C₂×D₇₃	—	8	2	no	—
146	C₂×D₇₃	—	8	2	no	—
147	C₂×D₇₃	C₄×D₈	1	2	yes	—
148	C₄×C₇₃	—	2	2	no	—
149	C₄×C₇₃	—	4	4	no	—
150	C₄×C₇₃	—	8	2	no	—
151	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
153	Dic₇₃	—	2	2	no	—
154	Dic₇₃	—	8	2	no	—
155	Dic₇₃	C₄×D₈	1	2	yes	—
156	#33460	—	73	1	no	maximal, Sylow
157	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
158	C₂×C₇₃	#3416	1	2	yes	—
159	D₇₃	—	2	2	no	—
160	D₇₃	—	8	4	no	—
161	C₂×#508	—	73	1	no	—
162	#2951	—	73	2	no	—
163	#2177	—	73	1	no	—
164	#2177	—	73	1	no	—
165	#2176	—	73	1	no	—
166	#2176	D₇₃	1	1	yes	characteristic, complemented
167	C₄×SD₃₂	—	73	2	no	—
168	C₄×SD₃₂	—	73	2	no	—
169	C₂×#367	—	73	1	no	—
170	#2139	—	73	2	no	—
171	#2108	—	73	1	no	—
172	C₇₃	#33460	1	1	yes	characteristic, complemented, Sylow
173	C₂×#135	—	73	2	no	—
174	#520	—	146	2	no	—
175	#508	—	73	8	no	—
176	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic
177	C₂×SD₃₂	—	73	2	no	—
178	C₂×D₁₆	—	73	1	no	—
179	C₂×M₃₂	—	73	1	no	—
180	C₂²×C₁₆	—	73	1	no	—
181	#440	—	146	2	no	—
182	#440	—	146	2	no	—
183	C₄×Q₁₆	—	73	2	no	—
184	C₄×Q₁₆	C₂×D₇₃	1	2	yes	—
185	C₄×D₈	—	73	2	no	—
186	C₄×D₈	—	73	2	no	—
187	C₂×#107	—	73	2	no	—
188	#417	—	146	2	no	—
189	#405	—	73	1	no	—
190	#367	—	73	1	no	—
191	#367	—	73	2	no	—
192	#367	C₂×D₇₃	1	1	yes	characteristic
193	#358	—	73	2	no	—
194	#358	C₂×D₇₃	1	2	yes	—
195	#357	—	73	2	no	—
196	#357	—	73	2	no	—
197	#346	—	73	1	no	—
198	#346	C₂×D₇₃	1	1	yes	characteristic
199	C₄×C₁₆	—	73	1	no	—
200	C₄×C₁₆	—	73	1	no	—
201	C₂×#41	—	146	2	no	—
202	#135	—	73	8	no	—
203	#135	—	146	4	no	—
204	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
205	C₂×SD₁₆	—	146	2	no	—
206	C₂×D₈	—	73	2	no	—
207	C₂×M₁₆	—	73	1	no	—
208	C₂²×C₈	—	73	1	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	146	2	no	—
211	C₄×D₄	—	292	2	no	—
212	C₄×D₄	—	292	2	no	—
213	C₄×D₄	—	146	2	no	—
214	C₄×D₄	—	146	2	no	—
215	#117	—	292	2	no	—
216	#117	—	73	1	no	—
217	C₂×C₄⋊C₄	—	292	2	no	—
218	C₂×C₄⋊C₄	—	146	2	no	—
219	Q₃₂	C₄×D₇₃	1	4	yes	complemented
220	SD₃₂	—	73	8	no	—
221	D₁₆	—	73	4	no	—
222	M₃₂	—	146	2	no	—
223	C₂×C₁₆	—	73	1	no	—
224	C₂×C₁₆	—	73	4	no	—
225	C₂×C₁₆	—	73	1	no	—
226	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
227	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
228	#107	—	73	2	no	—
229	#107	—	146	2	no	—
230	#107	C₂²×D₇₃	1	2	yes	—
231	#103	—	2	2	no	—
232	#103	—	146	2	no	—
233	#102	—	146	2	no	—
234	#102	—	146	2	no	—
235	#97	—	73	1	no	—
236	#97	—	73	1	no	—
237	C₄×C₈	—	73	1	no	—
238	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
239	#41	—	292	4	no	—
240	#41	—	146	8	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×D₄	—	292	2	no	—
243	C₂×D₄	—	146	2	no	—
244	C₂²×C₄	—	292	2	no	—
245	C₂²×C₄	—	292	2	no	—
246	C₂²×C₄	—	292	2	no	—
247	C₂²×C₄	—	292	2	no	—
248	C₂²×C₄	—	73	1	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	C₂×C₄×D₇₃	1	4	yes	—
251	SD₁₆	—	146	8	no	—
252	D₈	—	73	4	no	—
253	D₈	—	146	2	no	—
254	M₁₆	—	146	2	no	—
255	C₂×C₈	—	73	1	no	—
256	C₂×C₈	—	73	2	no	—
257	C₂×C₈	—	73	2	no	—
258	C₂×C₈	—	73	1	no	—
259	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
260	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	292	2	no	—
264	C₄⋊C₄	—	292	2	no	—
265	C₄⋊C₄	—	73	2	no	—
266	C₄⋊C₄	—	146	2	no	—
267	C₂²⋊C₄	—	292	2	no	—
268	C₂²⋊C₄	—	292	2	no	—
269	C₂²⋊C₄	—	73	2	no	—
270	C₄²	—	4	2	no	—
271	C₄²	—	292	2	no	—
272	C₄²	—	73	1	no	—
273	C₄²	D₄×D₇₃	1	1	yes	characteristic
274	C₁₆	—	2	1	no	—
275	C₁₆	—	73	2	no	—
276	C₁₆	—	146	1	no	—
277	C₁₆	C₂×C₄×D₇₃	1	2	yes	—
278	C₂³	—	292	2	no	—
279	C₂³	—	73	1	no	—
280	Q₈	—	2	4	no	—
281	Q₈	—	4	2	no	—
282	D₄	—	292	8	no	—
283	D₄	—	292	2	no	—
284	D₄	—	146	4	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	292	8	no	—
288	C₂×C₄	—	292	2	no	—
289	C₂×C₄	—	73	1	no	—
290	C₂×C₄	—	73	2	no	—
291	C₂×C₄	—	73	2	no	—
292	C₂×C₄	—	73	2	no	—
293	C₂×C₄	—	1168	2	no	—
294	C₂×C₄	—	1168	2	no	—
295	C₂×C₄	—	1168	2	no	—
296	C₂×C₄	D₈×D₇₃	1	2	yes	—
297	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	73	2	no	—
300	C₈	—	146	1	no	—
301	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
302	C₈	#41×D₇₃	1	1	yes	characteristic
303	C₂²	—	292	4	no	—
304	C₂²	—	584	2	no	—
305	C₂²	—	584	2	no	—
306	C₂²	—	73	2	no	—
307	C₂²	—	146	2	no	—
308	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	4	4	no	—
311	C₄	—	8	2	no	—
312	C₄	—	584	2	no	—
313	C₄	—	73	2	no	—
314	C₄	—	146	2	no	—
315	C₄	C₄×D₄×D₇₃	1	1	yes	characteristic
316	C₄	#135×D₇₃	1	1	yes	characteristic
317	C₂	—	584	4	no	—
318	C₂	—	146	2	no	—
319	C₂	C₄×D₈×D₇₃	1	1	yes	characteristic
320	C₂	#22623076	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2176	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×#17989020	C₂
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#2176	D₇₃
hypocenter	C₇₃	#33460
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

Complements

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

N	H ≅ G/N	type
2. #22618806	C₂	semidirect
3. #22611808	C₂	semidirect
4. #22611806	C₂	semidirect
5. #22611802	C₂	semidirect
6. C₄×#14556534	C₂	semidirect
7. #22611780	C₂	semidirect
8. #367×D₇₃	C₂	semidirect
9. #22611642	C₂	semidirect
10. #22611529	C₂	semidirect
11. C₇₃×#2176	C₂	semidirect
14. #17989020	C₄	semidirect
42. #14556614	C₂×C₄	semidirect
56. #14556534	C₂×C₄	semidirect
57. D₁₁₆₈	C₂×C₄	semidirect
61. C₁₆×D₇₃	C₂×C₄	semidirect
72. Q₃₂×C₇₃	C₂×C₄	semidirect
166. #2176	D₇₃	semidirect
172. C₇₃	#33460	semidirect
219. Q₃₂	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×#17989020	2
#22618806	2
#22611808	2
#22611806	2
#22611802	2
C₄×#14556534	2
#22611780	2
#367×D₇₃	2
#22611642	2
#22611529	2
C₇₃×#2176	2
#33460	73