Subgroups — #33241584

Order 18688.

532 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620581	C₂	1	1	yes	maximal, characteristic, complemented
2	#22612207	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612180	C₂	1	1	yes	maximal, characteristic, complemented
4	#22612174	C₂	1	1	yes	maximal, characteristic, complemented
5	#22612172	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612032	C₂	1	2	yes	maximal, complemented
7	#22612015	C₂	1	2	yes	maximal, complemented
8	#22611805	C₂	1	2	yes	maximal, complemented
9	C₄×#14556532	C₂	1	2	yes	maximal, complemented
10	#22611528	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#3176	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989192	—	2	1	no	—
13	C₂×#14556532	C₂²	1	2	yes	—
14	C₂×#14556532	C₂²	1	2	yes	—
15	C₂×#14556531	C₂²	1	1	yes	characteristic
16	C₂²×#11924003	C₂²	1	1	yes	characteristic
17	#17988989	C₂²	1	1	yes	characteristic
18	#17988964	C₂²	1	1	yes	characteristic
19	#17988952	—	2	1	no	—
20	#17988840	—	2	1	no	—
21	#17988823	C₂²	1	2	yes	—
22	#17988700	—	2	1	no	—
23	C₄×D₅₈₄	C₂²	1	2	yes	—
24	#17988414	C₂²	1	2	yes	—
25	#17988412	C₂²	1	2	yes	—
26	#17988388	C₂²	1	2	yes	—
27	#17988388	C₂²	1	2	yes	—
28	#17988387	C₂²	1	2	yes	—
29	#17988387	C₂²	1	2	yes	—
30	#17988363	C₂²	1	1	yes	characteristic
31	#17988363	C₂²	1	1	yes	characteristic
32	C₄×#11924003	C₂²	1	1	yes	characteristic
33	C₄×#11924003	C₂²	1	1	yes	characteristic
34	#540×C₇₃	—	2	1	no	—
35	#499×C₇₃	C₂²	1	1	yes	characteristic
36	#493×C₇₃	C₂²	1	1	yes	characteristic
37	#490×C₇₃	—	2	1	no	—
38	#469×C₇₃	—	2	1	no	—
39	#466×C₇₃	C₂²	1	2	yes	—
40	#442×C₇₃	—	2	1	no	—
41	C₄×D₈×C₇₃	C₂²	1	2	yes	—
42	#405×C₇₃	C₂²	1	1	yes	characteristic
43	C₂×D₅₈₄	—	2	1	no	—
44	C₂×D₅₈₄	C₂³	1	2	yes	—
45	#14556588	—	2	1	no	—
46	#14556583	—	2	1	no	—
47	#14556576	—	4	2	no	—
48	#14556565	—	2	2	no	—
49	#14556564	—	4	2	no	—
50	C₄×D₂₉₂	—	2	2	no	—
51	#14556532	—	2	4	no	—
52	#14556532	—	2	4	no	—
53	#14556531	—	2	2	no	—
54	C₂×#11924003	—	2	2	no	—
55	C₂×#11924003	C₂³	1	1	yes	characteristic
56	C₂×#11924003	C₂³	1	1	yes	characteristic
57	C₂×#11924003	C₂³	1	1	yes	characteristic
58	C₂×#11924003	C₂³	1	1	yes	characteristic
59	#14556525	C₂³	1	2	yes	—
60	#14556523	—	2	1	no	—
61	#14556513	—	2	2	no	—
62	#14556513	—	2	2	no	—
63	C₂×D₈×C₇₃	—	2	1	no	—
64	C₂×D₈×C₇₃	C₂³	1	2	yes	—
65	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
67	#127×C₇₃	—	2	1	no	—
68	#125×C₇₃	—	2	1	no	—
69	#123×C₇₃	—	4	2	no	—
70	#121×C₇₃	—	2	2	no	—
71	#121×C₇₃	—	4	2	no	—
72	C₄×D₄×C₇₃	—	2	2	no	—
73	#117×C₇₃	D₄	1	1	yes	characteristic
74	#107×C₇₃	C₂³	1	2	yes	—
75	#106×C₇₃	—	2	1	no	—
76	#102×C₇₃	—	2	2	no	—
77	#102×C₇₃	—	2	2	no	—
78	#97×C₇₃	D₄	1	1	yes	characteristic
79	#97×C₇₃	D₄	1	1	yes	characteristic
80	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
82	C₂×D₂₉₂	—	2	2	no	—
83	C₂×D₂₉₂	—	4	2	no	—
84	C₂×C₂²⋊D₇₃	—	4	2	no	—
85	C₂×C₄×D₇₃	—	4	2	no	—
86	D₅₈₄	—	2	2	no	—
87	D₅₈₄	—	2	2	no	—
88	D₅₈₄	—	4	2	no	—
89	C₄⋊Dic₇₃	—	2	2	no	—
90	#11924007	—	4	2	no	—
91	#11924005	—	4	2	no	—
92	#11924005	—	4	2	no	—
93	#11924003	—	2	1	no	—
94	#11924003	—	2	1	no	—
95	#11924003	#41	1	2	yes	—
96	#11924003	#41	1	2	yes	—
97	C₂×D₄×C₇₃	—	2	2	no	—
98	C₂×D₄×C₇₃	—	4	2	no	—
99	C₂×D₄×C₇₃	—	4	2	no	—
100	C₂²×C₄×C₇₃	—	4	2	no	—
101	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
102	D₈×C₇₃	—	2	2	no	—
103	D₈×C₇₃	—	2	2	no	—
104	D₈×C₇₃	—	4	2	no	—
105	M₁₆×C₇₃	—	2	2	no	—
106	C₂×C₈×C₇₃	—	2	1	no	—
107	C₂×C₈×C₇₃	—	2	1	no	—
108	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₇₃	—	2	2	no	—
113	C₄⋊C₄×C₇₃	—	4	2	no	—
114	C₄⋊C₄×C₇₃	D₈	1	2	yes	—
115	C₂²⋊C₄×C₇₃	—	4	2	no	—
116	C₂²⋊C₄×C₇₃	—	4	2	no	—
117	C₂²⋊C₄×C₇₃	D₈	1	2	yes	—
118	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
120	C₂³×C₇₃	—	4	2	no	—
121	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
122	C₂²×D₇₃	—	4	2	no	—
123	D₄×C₇₃	—	4	2	no	—
124	D₄×C₇₃	—	4	2	no	—
125	D₄×C₇₃	—	16	2	no	—
126	D₄×C₇₃	—	16	2	no	—
127	C₂×C₄×C₇₃	—	2	1	no	—
128	C₂×C₄×C₇₃	—	2	1	no	—
129	C₂×C₄×C₇₃	—	4	2	no	—
130	C₂×C₄×C₇₃	—	16	2	no	—
131	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₇₃	—	16	2	no	—
136	C₂×Dic₇₃	—	4	2	no	—
137	D₂₉₂	—	4	2	no	—
138	D₂₉₂	—	4	2	no	—
139	D₂₉₂	—	16	2	no	—
140	C₄×D₇₃	—	16	2	no	—
141	C₈×C₇₃	—	2	1	no	—
142	C₈×C₇₃	—	2	1	no	—
143	C₈×C₇₃	#135	1	2	yes	—
144	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
145	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
146	C₂²×C₇₃	—	2	2	no	—
147	C₂²×C₇₃	—	2	1	no	—
148	C₂²×C₇₃	—	8	2	no	—
149	C₂²×C₇₃	—	8	2	no	—
150	C₂²×C₇₃	—	8	2	no	—
151	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
152	C₂×D₇₃	—	8	2	no	—
153	C₂×D₇₃	—	8	2	no	—
154	C₂×D₇₃	—	8	2	no	—
155	C₄×C₇₃	—	2	2	no	—
156	C₄×C₇₃	—	2	2	no	—
157	C₄×C₇₃	—	2	1	no	—
158	C₄×C₇₃	—	8	2	no	—
159	C₄×C₇₃	#532	1	1	yes	characteristic
160	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
161	Dic₇₃	—	8	2	no	—
162	#33740	—	73	1	no	maximal, Sylow
163	C₂×C₇₃	—	4	1	no	—
164	C₂×C₇₃	—	16	2	no	—
165	C₂×C₇₃	#3169	1	1	yes	characteristic
166	C₂×C₇₃	#3415	1	2	yes	—
167	D₇₃	—	16	2	no	—
168	#3176	—	73	1	no	—
169	#3176	D₇₃	1	1	yes	characteristic, complemented
170	#2253	—	73	1	no	—
171	#2246	—	73	1	no	—
172	#2244	—	73	1	no	—
173	#2244	—	73	1	no	—
174	#2218	—	73	2	no	—
175	#2215	—	73	2	no	—
176	#2177	—	73	2	no	—
177	C₄×D₁₆	—	73	2	no	—
178	#2108	—	73	1	no	—
179	C₇₃	#33740	1	1	yes	characteristic, complemented, Sylow
180	#540	—	2	1	no	—
181	#540	—	146	1	no	—
182	C₂×D₁₆	—	73	2	no	—
183	C₂×D₁₆	—	73	2	no	—
184	C₂×M₃₂	—	73	1	no	—
185	C₂²×C₁₆	—	73	1	no	—
186	#499	—	73	1	no	—
187	#499	C₂×D₇₃	1	1	yes	characteristic
188	#493	—	73	1	no	—
189	#493	C₂×D₇₃	1	1	yes	characteristic
190	#490	—	2	1	no	—
191	#490	—	146	1	no	—
192	#469	—	2	1	no	—
193	#469	—	146	1	no	—
194	#466	—	73	2	no	—
195	#466	C₂×D₇₃	1	2	yes	—
196	#442	—	2	1	no	—
197	#442	—	146	1	no	—
198	C₄×D₈	—	73	2	no	—
199	C₄×D₈	C₂×D₇₃	1	2	yes	—
200	#405	C₂×D₇₃	1	1	yes	characteristic
201	#367	—	73	2	no	—
202	#366	—	73	2	no	—
203	#357	—	73	2	no	—
204	#357	—	73	2	no	—
205	#357	—	73	2	no	—
206	#357	—	73	2	no	—
207	#346	—	73	1	no	—
208	#346	—	73	1	no	—
209	C₄×C₁₆	—	73	1	no	—
210	C₄×C₁₆	—	73	1	no	—
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	—	73	2	no	—
213	C₂×D₈	—	146	1	no	—
214	C₂×D₈	C₂²×D₇₃	1	2	yes	—
215	C₂×M₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
216	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
217	#127	—	2	1	no	—
218	#127	—	146	1	no	—
219	#125	—	2	1	no	—
220	#125	—	146	1	no	—
221	#123	—	4	2	no	—
222	#123	—	292	2	no	—
223	#121	—	2	2	no	—
224	#121	—	4	2	no	—
225	#121	—	292	2	no	—
226	#121	—	146	2	no	—
227	C₄×D₄	—	2	2	no	—
228	C₄×D₄	—	146	2	no	—
229	#117	C₂²⋊D₇₃	1	1	yes	characteristic
230	D₁₆	—	146	4	no	—
231	D₁₆	—	146	4	no	—
232	M₃₂	—	146	2	no	—
233	C₂×C₁₆	—	73	1	no	—
234	C₂×C₁₆	—	73	1	no	—
235	C₂×C₁₆	—	73	1	no	—
236	C₂×C₁₆	—	73	1	no	—
237	C₂×C₁₆	—	146	2	no	—
238	#107	—	73	2	no	—
239	#107	C₂²×D₇₃	1	2	yes	—
240	#106	—	2	1	no	—
241	#106	—	146	1	no	—
242	#102	—	2	2	no	—
243	#102	—	2	2	no	—
244	#102	—	146	2	no	—
245	#102	—	146	2	no	—
246	#97	C₂²⋊D₇₃	1	1	yes	characteristic
247	#97	C₂²⋊D₇₃	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	292	2	no	—
254	C₂×D₄	—	292	2	no	—
255	C₂×D₄	—	146	2	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	—	292	2	no	—
258	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
259	D₈	—	2	2	no	—
260	D₈	—	2	2	no	—
261	D₈	—	4	2	no	—
262	D₈	—	292	2	no	—
263	D₈	—	146	2	no	—
264	D₈	—	146	2	no	—
265	M₁₆	—	2	2	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
269	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
270	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	292	2	no	—
275	C₄⋊C₄	—	146	2	no	—
276	C₄⋊C₄	#11924013	1	2	yes	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	292	2	no	—
280	C₂²⋊C₄	—	292	2	no	—
281	C₂²⋊C₄	#11924013	1	2	yes	—
282	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
283	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
284	C₁₆	—	73	2	no	—
285	C₁₆	—	73	2	no	—
286	C₁₆	—	146	1	no	—
287	C₁₆	—	146	1	no	—
288	C₂³	—	4	2	no	—
289	C₂³	—	292	2	no	—
290	C₂³	C₂×#11924013	1	1	yes	characteristic
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	16	2	no	—
294	D₄	—	16	2	no	—
295	D₄	—	292	2	no	—
296	D₄	—	292	2	no	—
297	D₄	—	1168	2	no	—
298	D₄	—	1168	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	16	2	no	—
303	C₂×C₄	—	292	2	no	—
304	C₂×C₄	—	1168	2	no	—
305	C₂×C₄	C₂×#11924013	1	2	yes	—
306	C₂×C₄	C₂×#11924013	1	2	yes	—
307	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	#14556611	1	2	yes	—
312	C₈	#41×D₇₃	1	1	yes	characteristic
313	C₈	#41×D₇₃	1	1	yes	characteristic
314	C₂²	—	2	1	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	584	2	no	—
320	C₂²	—	584	2	no	—
321	C₂²	—	584	2	no	—
322	C₂²	C₂²×#11924013	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	8	2	no	—
327	C₄	—	584	2	no	—
328	C₄	#17989133	1	1	yes	characteristic
329	C₄	C₂×#14556611	1	1	yes	characteristic
330	C₂	—	4	1	no	—
331	C₂	—	16	2	no	—
332	C₂	—	1168	2	no	—
333	C₂	#22620506	1	1	yes	characteristic
334	C₂	#22623070	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11924013
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#3176	C₂
socle	C₂²×C₇₃	C₂²×D₈
hypercenter	#3176	D₇₃
hypocenter	C₇₃	#33740
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3176

Complements

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

N	H ≅ G/N	type
1. #22620581	C₂	semidirect
2. #22612207	C₂	semidirect
3. #22612180	C₂	semidirect
4. #22612174	C₂	semidirect
5. #22612172	C₂	semidirect
6. #22612032	C₂	semidirect
7. #22612015	C₂	semidirect
8. #22611805	C₂	semidirect
9. C₄×#14556532	C₂	semidirect
10. #22611528	C₂	semidirect
11. C₇₃×#3176	C₂	semidirect
169. #3176	D₇₃	semidirect
179. C₇₃	#33740	semidirect

Maximal subgroups

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

subgroup	index
#22620581	2
#22612207	2
#22612180	2
#22612174	2
#22612172	2
#22612032	2
#22612015	2
#22611805	2
C₄×#14556532	2
#22611528	2
C₇₃×#3176	2
#33740	73