Subgroups — #33243705

Order 18688.

532 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622904	C₂	1	1	yes	maximal, characteristic, complemented
2	#22622796	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612171	C₂	1	2	yes	maximal, complemented
4	#22612154	C₂	1	1	yes	maximal, characteristic, complemented
5	#22612090	C₂	1	2	yes	maximal, complemented
6	#22612079	C₂	1	1	yes	maximal, characteristic, complemented
7	#22611964	C₂	1	2	yes	maximal, complemented
8	#22611836	C₂	1	1	yes	maximal, characteristic, complemented
9	#22611807	C₂	1	2	yes	maximal, complemented
10	C₄×#14556532	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2206	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989294	—	2	1	no	—
13	#17989257	—	2	1	no	—
14	C₂×#14556532	C₂²	1	1	yes	characteristic
15	#17988992	C₂²	1	1	yes	characteristic
16	#17988988	C₂²	1	2	yes	—
17	#17988962	C₂²	1	2	yes	—
18	#17988885	—	2	2	no	—
19	#17988877	—	2	1	no	—
20	#17988780	—	2	2	no	—
21	#17988779	—	2	1	no	—
22	C₈×Dic₁₄₆	C₂²	1	1	yes	characteristic
23	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
24	D₈×Dic₇₃	C₂²	1	2	yes	—
25	D₈×Dic₇₃	C₂²	1	2	yes	—
26	#17988415	C₂²	1	2	yes	—
27	#17988412	C₂²	1	1	yes	characteristic
28	#17988405	C₂²	1	2	yes	—
29	#17988389	C₂²	1	2	yes	—
30	#17988388	C₂²	1	1	yes	characteristic
31	#17988388	C₂²	1	2	yes	—
32	#17988388	C₂²	1	2	yes	—
33	#17988387	C₂²	1	1	yes	characteristic
34	#17988362	C₂²	1	2	yes	—
35	C₄×#11924003	C₂²	1	1	yes	characteristic
36	C₂×D₁₆×C₇₃	C₂²	1	2	yes	—
37	#493×C₇₃	C₂²	1	1	yes	characteristic
38	C₄×D₈×C₇₃	C₂²	1	1	yes	characteristic
39	C₄⋊C₁₆×C₇₃	C₂²	1	1	yes	characteristic
40	#357×C₇₃	C₂²	1	2	yes	—
41	C₂×#11924027	—	4	1	no	—
42	C₂×D₅₈₄	C₂³	1	1	yes	characteristic
43	C₂×#11924013	—	2	2	no	—
44	#14556591	—	2	1	no	—
45	#14556587	—	2	2	no	—
46	#14556582	—	2	2	no	—
47	#14556570	—	4	1	no	—
48	#14556568	—	4	2	no	—
49	#14556563	—	4	2	no	—
50	#14556562	—	4	1	no	—
51	C₄×Dic₁₄₆	D₄	1	1	yes	characteristic
52	C₄×D₂₉₂	—	2	1	no	—
53	D₄×Dic₇₃	—	2	2	no	—
54	D₄×Dic₇₃	—	2	2	no	—
55	C₂×C₄⋊Dic₇₃	—	4	1	no	—
56	#14556532	—	2	2	no	—
57	C₂×#11924003	C₂³	1	1	yes	characteristic
58	C₂×#11924003	C₂³	1	1	yes	characteristic
59	#14556525	—	2	1	no	—
60	#14556525	C₂³	1	2	yes	—
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	#14556513	—	2	1	no	—
66	#14556512	—	2	2	no	—
67	#14556512	—	2	2	no	—
68	C₈×Dic₇₃	C₂³	1	2	yes	—
69	C₂×D₈×C₇₃	—	2	1	no	—
70	C₂×D₈×C₇₃	C₂³	1	2	yes	—
71	C₂×D₈×C₇₃	C₂³	1	1	yes	characteristic
72	#127×C₇₃	—	2	1	no	—
73	C₄×D₄×C₇₃	—	2	1	no	—
74	D₁₆×C₇₃	—	2	4	no	—
75	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
76	#107×C₇₃	C₂³	1	1	yes	characteristic
77	#102×C₇₃	—	2	1	no	—
78	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
79	#11924027	—	8	4	no	—
80	C₂×Dic₁₄₆	—	4	1	no	—
81	C₂×Dic₁₄₆	D₈	1	1	yes	characteristic
82	C₂×D₂₉₂	—	2	1	no	—
83	C₂×C₂²⋊D₇₃	—	4	2	no	—
84	C₂×C₂²⋊D₇₃	—	4	2	no	—
85	C₂²×Dic₇₃	—	4	2	no	—
86	C₂²×Dic₇₃	—	4	2	no	—
87	C₂×C₄×D₇₃	—	4	1	no	—
88	D₅₈₄	—	2	1	no	—
89	D₅₈₄	—	2	1	no	—
90	#11924013	—	8	2	no	—
91	C₂×#10596831	C₂×D₄	1	2	yes	—
92	C₄⋊Dic₇₃	—	2	1	no	—
93	C₄⋊Dic₇₃	—	2	2	no	—
94	C₄⋊Dic₇₃	—	4	1	no	—
95	C₄⋊Dic₇₃	—	4	2	no	—
96	C₄⋊Dic₇₃	D₈	1	1	yes	characteristic
97	#11924007	—	4	2	no	—
98	#11924007	D₈	1	2	yes	—
99	C₂²⋊Dic₇₃	—	4	2	no	—
100	C₂²⋊Dic₇₃	—	4	2	no	—
101	#11924005	—	4	1	no	—
102	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
103	#11924003	—	2	1	no	—
104	#11924003	—	2	1	no	—
105	C₂×D₄×C₇₃	—	2	1	no	—
106	C₂×D₄×C₇₃	—	2	2	no	—
107	C₂×D₄×C₇₃	—	4	1	no	—
108	C₂×D₄×C₇₃	—	4	1	no	—
109	C₂²×C₄×C₇₃	—	4	1	no	—
110	D₈×C₇₃	—	2	2	no	—
111	D₈×C₇₃	—	2	1	no	—
112	D₈×C₇₃	—	2	2	no	—
113	D₈×C₇₃	—	8	1	no	—
114	D₈×C₇₃	#41	1	2	yes	complemented
115	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₇₃	—	2	1	no	—
118	C₂²⋊C₄×C₇₃	—	4	1	no	—
119	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
120	C₁₆×C₇₃	—	2	2	no	—
121	C₂³×C₇₃	—	4	1	no	—
122	C₂³×C₇₃	—	4	2	no	—
123	C₂²×D₇₃	—	4	1	no	—
124	D₄×C₇₃	—	2	2	no	—
125	D₄×C₇₃	—	4	2	no	—
126	D₄×C₇₃	—	4	1	no	—
127	D₄×C₇₃	—	4	2	no	—
128	D₄×C₇₃	—	8	2	no	—
129	D₄×C₇₃	—	16	1	no	—
130	C₂×C₄×C₇₃	—	4	1	no	—
131	C₂×C₄×C₇₃	—	8	2	no	—
132	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₇₃	—	8	4	no	—
136	C₂²⋊D₇₃	—	16	2	no	—
137	C₂×Dic₇₃	—	4	1	no	—
138	C₂×Dic₇₃	—	4	2	no	—
139	C₂×Dic₇₃	—	8	4	no	—
140	C₂×Dic₇₃	—	16	2	no	—
141	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
142	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
143	D₂₉₂	—	4	1	no	—
144	D₂₉₂	—	4	1	no	—
145	C₄×D₇₃	—	8	2	no	—
146	Dic₁₄₆	—	2	2	no	—
147	Dic₁₄₆	—	16	1	no	—
148	C₈×C₇₃	—	2	1	no	—
149	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
150	C₈×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
151	#10596831	—	2	2	no	—
152	C₂²×C₇₃	—	4	2	no	—
153	C₂²×C₇₃	—	8	1	no	—
154	C₂²×C₇₃	—	8	1	no	—
155	C₂²×C₇₃	—	8	2	no	—
156	C₂²×C₇₃	—	8	2	no	—
157	C₂²×C₇₃	—	8	2	no	—
158	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
159	C₂×D₇₃	—	8	1	no	—
160	C₂×D₇₃	—	8	1	no	—
161	C₂×D₇₃	—	8	1	no	—
162	C₄×C₇₃	—	2	1	no	—
163	C₄×C₇₃	—	8	1	no	—
164	C₄×C₇₃	#509	1	2	yes	—
165	C₄×C₇₃	#548	1	1	yes	characteristic
166	C₄×C₇₃	#577	1	1	yes	characteristic
167	Dic₇₃	—	2	2	no	—
168	Dic₇₃	—	2	2	no	—
169	Dic₇₃	—	8	1	no	—
170	Dic₇₃	—	8	2	no	—
171	#33933	—	73	1	no	maximal, Sylow
172	C₂×C₇₃	—	8	2	no	—
173	C₂×C₇₃	—	16	2	no	—
174	C₂×C₇₃	#3358	1	1	yes	characteristic
175	C₂×C₇₃	C₂×#509	1	1	yes	characteristic
176	C₂×C₇₃	#3415	1	1	yes	characteristic
177	D₇₃	—	16	1	no	—
178	#3391	—	73	1	no	—
179	#3380	—	73	1	no	—
180	#2244	—	73	2	no	—
181	#2240	—	73	1	no	—
182	#2228	—	73	2	no	—
183	#2226	—	73	1	no	—
184	#2206	—	73	2	no	—
185	#2206	D₇₃	1	1	yes	characteristic, complemented
186	#2183	—	73	1	no	—
187	#2177	—	73	2	no	—
188	C₄×D₁₆	—	73	1	no	—
189	C₇₃	#33933	1	1	yes	characteristic, complemented, Sylow
190	#554	—	146	1	no	—
191	#550	—	146	1	no	—
192	C₂×D₁₆	—	73	1	no	—
193	C₂×D₁₆	C₂×D₇₃	1	2	yes	—
194	#500	—	73	1	no	—
195	#499	—	73	2	no	—
196	#493	—	73	2	no	—
197	#493	C₂×D₇₃	1	1	yes	characteristic
198	#478	—	146	2	no	—
199	#476	—	146	1	no	—
200	#459	—	146	1	no	—
201	#459	—	146	2	no	—
202	C₈×Q₈	—	73	1	no	—
203	C₄×D₈	—	73	2	no	—
204	C₄×D₈	—	73	2	no	—
205	C₄×D₈	—	73	1	no	—
206	C₄×D₈	C₂×D₇₃	1	1	yes	characteristic
207	#367	—	73	2	no	—
208	#366	—	73	1	no	—
209	C₄⋊C₁₆	—	73	2	no	—
210	C₄⋊C₁₆	C₂×D₇₃	1	1	yes	characteristic
211	#357	—	73	1	no	—
212	#357	—	73	2	no	—
213	#357	—	73	2	no	—
214	#357	—	73	2	no	—
215	#357	—	73	1	no	—
216	#357	C₂×D₇₃	1	2	yes	—
217	#346	—	73	2	no	—
218	C₄×C₁₆	—	73	1	no	—
219	C₂×#41	—	292	1	no	—
220	C₂×D₈	—	2	1	no	—
221	C₂×D₈	—	73	1	no	—
222	C₂×D₈	—	146	2	no	—
223	C₂×D₈	C₂²×D₇₃	1	2	yes	—
224	C₂×D₈	C₂²×D₇₃	1	1	yes	characteristic
225	C₄⋊Q₈	—	146	1	no	—
226	#127	—	2	1	no	—
227	#127	—	146	2	no	—
228	#125	—	146	2	no	—
229	C₂²⋊Q₈	—	292	2	no	—
230	C₂²⋊Q₈	—	292	1	no	—
231	#121	—	292	1	no	—
232	#121	—	292	2	no	—
233	C₄×Q₈	—	73	1	no	—
234	C₄×D₄	—	2	1	no	—
235	C₄×D₄	—	146	2	no	—
236	C₄×D₄	—	146	2	no	—
237	C₄×D₄	—	146	1	no	—
238	C₂×C₄⋊C₄	—	292	1	no	—
239	D₁₆	—	2	4	no	—
240	D₁₆	—	146	2	no	—
241	C₂×C₁₆	—	73	1	no	—
242	C₂×C₁₆	—	73	1	no	—
243	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
244	#107	—	73	2	no	—
245	#107	—	73	1	no	—
246	#107	—	146	1	no	—
247	#107	C₂²×D₇₃	1	1	yes	characteristic
248	#106	—	146	2	no	—
249	C₄⋊C₈	—	73	2	no	—
250	C₄⋊C₈	—	73	1	no	—
251	#102	—	2	1	no	—
252	#102	—	146	2	no	—
253	#102	—	146	2	no	—
254	#102	—	146	1	no	—
255	C₄×C₈	—	73	2	no	—
256	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
257	#41	—	584	4	no	—
258	C₂×Q₈	—	292	1	no	—
259	C₂×Q₈	—	73	1	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	292	2	no	—
265	C₂×D₄	—	292	2	no	—
266	C₂×D₄	—	146	1	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	292	2	no	—
269	C₂²×C₄	—	292	1	no	—
270	C₂²×C₄	—	292	2	no	—
271	D₈	—	2	2	no	—
272	D₈	—	2	1	no	—
273	D₈	—	2	2	no	—
274	D₈	—	8	1	no	—
275	D₈	—	584	2	no	—
276	D₈	—	146	1	no	—
277	D₈	—	146	1	no	—
278	D₈	#11924027	1	2	yes	complemented
279	C₂×C₈	—	73	2	no	—
280	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
281	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	292	2	no	—
284	C₄⋊C₄	—	292	2	no	—
285	C₄⋊C₄	—	292	1	no	—
286	C₄⋊C₄	—	73	2	no	—
287	C₄⋊C₄	—	73	1	no	—
288	C₄⋊C₄	—	146	1	no	—
289	C₄⋊C₄	—	146	2	no	—
290	C₂²⋊C₄	—	4	1	no	—
291	C₂²⋊C₄	—	292	2	no	—
292	C₂²⋊C₄	—	292	2	no	—
293	C₂²⋊C₄	—	292	1	no	—
294	C₄²	—	73	2	no	—
295	C₄²	D₄×D₇₃	1	1	yes	characteristic
296	C₁₆	—	2	2	no	—
297	C₁₆	—	146	1	no	—
298	C₁₆	—	146	1	no	—
299	C₂³	—	4	1	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	292	1	no	—
302	Q₈	—	1168	1	no	—
303	Q₈	—	146	2	no	—
304	D₄	—	2	2	no	—
305	D₄	—	4	1	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	2	no	—
309	D₄	—	16	1	no	—
310	D₄	—	292	1	no	—
311	D₄	—	292	1	no	—
312	D₄	—	584	4	no	—
313	D₄	—	1168	2	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	—	292	1	no	—
317	C₂×C₄	—	292	2	no	—
318	C₂×C₄	—	584	4	no	—
319	C₂×C₄	—	584	2	no	—
320	C₂×C₄	—	73	2	no	—
321	C₂×C₄	—	73	2	no	—
322	C₂×C₄	—	1168	2	no	—
323	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
324	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
325	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	146	2	no	—
328	C₈	C₂×#11924027	1	1	yes	characteristic
329	C₈	#14556638	1	1	yes	characteristic
330	C₂²	—	4	2	no	—
331	C₂²	—	8	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	584	1	no	—
337	C₂²	—	584	1	no	—
338	C₂²	—	584	1	no	—
339	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	8	1	no	—
342	C₄	—	584	1	no	—
343	C₄	—	584	2	no	—
344	C₄	—	146	2	no	—
345	C₄	—	146	2	no	—
346	C₄	#17989021	1	2	yes	—
347	C₄	#17989243	1	1	yes	characteristic
348	C₄	#17989397	1	1	yes	characteristic
349	C₂	—	8	2	no	—
350	C₂	—	16	2	no	—
351	C₂	—	1168	1	no	—
352	C₂	#22622561	1	1	yes	characteristic
353	C₂	C₂×#17989021	1	1	yes	characteristic
354	C₂	#22623071	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×#2206	C₂
socle	C₂²×C₇₃	C₂²×D₈
hypercenter	#2206	D₇₃
hypocenter	C₇₃	#33933
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2206

Complements

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

N	H ≅ G/N	type
1. #22622904	C₂	semidirect
2. #22622796	C₂	semidirect
3. #22612171	C₂	semidirect
4. #22612154	C₂	semidirect
5. #22612090	C₂	semidirect
6. #22612079	C₂	semidirect
7. #22611964	C₂	semidirect
8. #22611836	C₂	semidirect
9. #22611807	C₂	semidirect
10. C₄×#14556532	C₂	semidirect
11. C₇₃×#2206	C₂	semidirect
114. D₈×C₇₃	#41	semidirect
185. #2206	D₇₃	semidirect
189. C₇₃	#33933	semidirect
278. D₈	#11924027	semidirect

Maximal subgroups

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

subgroup	index
#22622904	2
#22622796	2
#22612171	2
#22612154	2
#22612090	2
#22612079	2
#22611964	2
#22611836	2
#22611807	2
C₄×#14556532	2
C₇₃×#2206	2
#33933	73