Subgroups — #33239367

Order 18688.

674 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989017	C₂	1	1	yes	maximal, characteristic
2	#22618808	C₂	1	1	yes	maximal, characteristic, complemented
3	#22611795	C₂	1	2	yes	maximal, complemented
4	#22611793	C₂	1	2	yes	maximal, complemented
5	C₄×#14556537	C₂	1	2	yes	maximal, complemented
6	C₄×#14556532	C₂	1	2	yes	maximal, complemented
7	C₂×#17988412	C₂	1	1	yes	maximal, characteristic, complemented
8	#22611643	C₂	1	1	yes	maximal, characteristic, complemented
9	#22611641	C₂	1	1	yes	maximal, characteristic, complemented
10	#22611528	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2951	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556613	C₂²	1	1	yes	characteristic
13	#17989067	—	2	1	no	—
14	#17989017	C₄	1	8	yes	complemented
15	C₂×#14556537	C₂²	1	1	yes	characteristic
16	C₂×#14556535	C₂²	1	1	yes	characteristic
17	C₂×#14556534	C₂²	1	1	yes	characteristic
18	C₂×#14556532	C₂²	1	1	yes	characteristic
19	C₂×#14556531	C₂²	1	1	yes	characteristic
20	C₂²×#11924003	C₂²	1	1	yes	characteristic
21	#17988689	—	2	2	no	—
22	C₄×Dic₂₉₂	C₂²	1	2	yes	—
23	C₄×D₅₈₄	C₂²	1	2	yes	—
24	C₂×#14556525	C₂²	1	1	yes	characteristic
25	#17988589	—	2	1	no	—
26	#17988412	C₂²	1	2	yes	—
27	#17988412	C₂²	1	2	yes	—
28	#17988391	C₂²	1	2	yes	—
29	#17988390	C₂²	1	2	yes	—
30	#17988388	C₂²	1	2	yes	—
31	#17988387	C₂²	1	2	yes	—
32	#17988363	C₂²	1	2	yes	—
33	C₄×#11924003	C₂²	1	2	yes	—
34	C₂×#135×C₇₃	C₂²	1	1	yes	characteristic
35	#520×C₇₃	—	2	1	no	—
36	#440×C₇₃	—	2	2	no	—
37	C₄×Q₁₆×C₇₃	C₂²	1	2	yes	—
38	C₄×D₈×C₇₃	C₂²	1	2	yes	—
39	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
40	#417×C₇₃	—	2	1	no	—
41	#405×C₇₃	C₂²	1	1	yes	characteristic
42	C₂×#11924028	—	2	1	no	—
43	#14556613	—	2	2	no	—
44	#14556613	C₂×C₄	1	4	yes	complemented
45	C₂×Dic₂₉₂	C₂³	1	1	yes	characteristic
46	C₂×#11924017	—	2	1	no	—
47	C₂×D₅₈₄	C₂³	1	1	yes	characteristic
48	C₄×Dic₁₄₆	—	2	2	no	—
49	C₄×D₂₉₂	—	2	2	no	—
50	#14556551	—	4	2	no	—
51	#14556549	—	4	1	no	—
52	C₂×C₄⋊Dic₇₃	—	2	1	no	—
53	C₄⋊C₄×D₇₃	—	4	1	no	—
54	#14556537	C₂×C₄	1	4	yes	complemented
55	#14556535	C₂×C₄	1	4	yes	complemented
56	#14556534	C₂×C₄	1	4	yes	complemented
57	#14556532	C₂×C₄	1	4	yes	complemented
58	#14556531	—	2	2	no	—
59	C₂×#11924003	C₂×C₄	1	4	yes	complemented
60	C₂×#11924003	C₂³	1	2	yes	—
61	C₂×#11924003	C₂³	1	1	yes	characteristic
62	C₂×#11924003	C₂³	1	1	yes	characteristic
63	#14556525	—	2	1	no	—
64	#14556525	C₂³	1	2	yes	—
65	#14556516	—	2	2	no	—
66	#14556513	—	2	2	no	—
67	C₂×#41×C₇₃	—	2	1	no	—
68	#135×C₇₃	—	2	2	no	—
69	#135×C₇₃	C₂×C₄	1	4	yes	complemented
70	C₂×Q₁₆×C₇₃	C₂³	1	1	yes	characteristic
71	C₂×SD₁₆×C₇₃	—	2	1	no	—
72	C₂×D₈×C₇₃	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
74	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
75	C₄×Q₈×C₇₃	—	2	2	no	—
76	C₄×D₄×C₇₃	—	2	2	no	—
77	C₄×D₄×C₇₃	—	4	2	no	—
78	#117×C₇₃	—	4	1	no	—
79	#117×C₇₃	D₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₇₃	—	4	1	no	—
82	#107×C₇₃	—	2	1	no	—
83	#107×C₇₃	C₂³	1	2	yes	—
84	#103×C₇₃	—	2	2	no	—
85	#102×C₇₃	—	2	2	no	—
86	#97×C₇₃	D₄	1	2	yes	—
87	C₄×C₈×C₇₃	C₂³	1	2	yes	—
88	#11924028	—	2	4	no	—
89	#11924028	—	4	2	no	—
90	C₂×Dic₁₄₆	—	2	1	no	—
91	C₂×D₂₉₂	—	2	1	no	—
92	C₂×C₂²⋊D₇₃	—	4	1	no	—
93	C₂²×Dic₇₃	—	4	1	no	—
94	C₂×C₄×D₇₃	—	4	1	no	—
95	C₂×C₄×D₇₃	—	4	2	no	—
96	Dic₂₉₂	—	2	1	no	—
97	Dic₂₉₂	C₂²×C₄	1	2	yes	—
98	#11924017	—	2	4	no	—
99	D₅₈₄	—	2	1	no	—
100	D₅₈₄	C₂²×C₄	1	2	yes	—
101	C₄⋊Dic₇₃	—	2	2	no	—
102	C₄⋊Dic₇₃	—	4	1	no	—
103	#11924007	—	4	2	no	—
104	#11924005	—	4	2	no	—
105	C₄×Dic₇₃	—	4	2	no	—
106	#11924003	—	2	2	no	—
107	#11924003	C₂²×C₄	1	2	yes	—
108	#11924003	C₂²×C₄	1	2	yes	—
109	#41×C₇₃	—	2	4	no	—
110	#41×C₇₃	—	4	2	no	—
111	C₂×Q₈×C₇₃	—	2	1	no	—
112	C₂×D₄×C₇₃	—	2	1	no	—
113	C₂×D₄×C₇₃	—	4	1	no	—
114	C₂²×C₄×C₇₃	—	4	1	no	—
115	C₂²×C₄×C₇₃	—	4	1	no	—
116	C₂²×C₄×C₇₃	—	4	2	no	—
117	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
118	Q₁₆×C₇₃	—	2	1	no	—
119	Q₁₆×C₇₃	C₂²×C₄	1	2	yes	—
120	SD₁₆×C₇₃	—	2	4	no	—
121	D₈×C₇₃	—	2	1	no	—
122	D₈×C₇₃	C₂²×C₄	1	2	yes	—
123	M₁₆×C₇₃	—	2	2	no	—
124	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
125	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₇₃	#41	1	2	yes	—
128	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
129	C₄⋊C₄×C₇₃	—	2	2	no	—
130	C₄⋊C₄×C₇₃	—	4	1	no	—
131	C₄⋊C₄×C₇₃	—	4	2	no	—
132	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
133	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
134	C₂²⋊C₄×C₇₃	—	4	2	no	—
135	C₂²⋊C₄×C₇₃	D₈	1	2	yes	—
136	C₄²×C₇₃	—	4	2	no	—
137	C₄²×C₇₃	C₂×D₄	1	2	yes	—
138	C₂³×C₇₃	—	4	1	no	—
139	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
140	C₂²×D₇₃	—	4	1	no	—
141	Q₈×C₇₃	—	2	2	no	—
142	Q₈×C₇₃	—	4	1	no	—
143	D₄×C₇₃	—	2	2	no	—
144	D₄×C₇₃	—	4	4	no	—
145	D₄×C₇₃	—	4	1	no	—
146	C₂×C₄×C₇₃	—	4	1	no	—
147	C₂×C₄×C₇₃	—	4	4	no	—
148	C₂×C₄×C₇₃	—	4	2	no	—
149	C₂×C₄×C₇₃	—	16	2	no	—
150	C₂×C₄×C₇₃	—	16	1	no	—
151	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
152	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
153	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
154	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₇₃	#135	1	2	yes	—
156	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
157	C₂²⋊D₇₃	—	4	4	no	—
158	C₂×Dic₇₃	—	4	1	no	—
159	C₂×Dic₇₃	—	4	2	no	—
160	C₂×Dic₇₃	—	16	1	no	—
161	D₂₉₂	—	2	2	no	—
162	D₂₉₂	—	4	1	no	—
163	C₄×D₇₃	—	4	4	no	—
164	C₄×D₇₃	—	16	2	no	—
165	Dic₁₄₆	—	2	2	no	—
166	Dic₁₄₆	—	4	1	no	—
167	C₈×C₇₃	—	2	2	no	—
168	C₈×C₇₃	C₄×D₄	1	2	yes	—
169	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
170	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
171	C₂²×C₇₃	—	2	1	no	—
172	C₂²×C₇₃	—	2	1	no	—
173	C₂²×C₇₃	—	4	2	no	—
174	C₂²×C₇₃	—	8	1	no	—
175	C₂²×C₇₃	—	8	1	no	—
176	C₂²×C₇₃	C₄×D₈	1	2	yes	—
177	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
178	C₂×D₇₃	—	4	2	no	—
179	C₂×D₇₃	—	8	1	no	—
180	C₂×D₇₃	—	8	1	no	—
181	C₄×C₇₃	—	2	2	no	—
182	C₄×C₇₃	—	2	2	no	—
183	C₄×C₇₃	—	4	2	no	—
184	C₄×C₇₃	—	8	2	no	—
185	C₄×C₇₃	C₄×D₈	1	2	yes	—
186	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
187	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
188	Dic₇₃	—	4	2	no	—
189	Dic₇₃	—	8	2	no	—
190	#33459	—	73	1	no	maximal, Sylow
191	C₂×C₇₃	—	2	2	no	—
192	C₂×C₇₃	—	8	2	no	—
193	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
194	C₂×C₇₃	#3415	1	1	yes	characteristic
195	C₂×C₇₃	#3417	1	1	yes	characteristic
196	D₇₃	—	8	2	no	—
197	C₂×#508	—	73	1	no	—
198	#2951	—	73	1	no	—
199	#2951	D₇₃	1	1	yes	characteristic, complemented
200	#2175	—	73	2	no	—
201	#2175	—	73	2	no	—
202	C₄×Q₃₂	—	73	2	no	—
203	C₄×D₁₆	—	73	2	no	—
204	C₂×#366	—	73	1	no	—
205	#2139	—	73	1	no	—
206	#2139	—	73	1	no	—
207	#2108	—	73	1	no	—
208	C₇₃	#33459	1	1	yes	characteristic, complemented, Sylow
209	C₂×#135	—	73	1	no	—
210	C₂×#135	C₂×D₇₃	1	1	yes	characteristic
211	#520	—	2	1	no	—
212	#520	—	146	1	no	—
213	#508	—	73	8	no	—
214	C₂×Q₃₂	—	73	1	no	—
215	C₂×SD₃₂	—	73	1	no	—
216	C₂×SD₃₂	—	73	1	no	—
217	C₂×D₁₆	—	73	1	no	—
218	C₂×M₃₂	—	73	1	no	—
219	C₂²×C₁₆	—	73	1	no	—
220	#440	—	2	2	no	—
221	#440	—	146	2	no	—
222	C₄×Q₁₆	—	73	2	no	—
223	C₄×Q₁₆	C₂×D₇₃	1	2	yes	—
224	C₄×D₈	—	73	2	no	—
225	C₄×D₈	C₂×D₇₃	1	2	yes	—
226	C₂×#107	—	73	1	no	—
227	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
228	#417	—	2	1	no	—
229	#417	—	146	1	no	—
230	#405	C₂×D₇₃	1	1	yes	characteristic
231	#366	—	73	2	no	—
232	#366	—	73	2	no	—
233	#358	—	73	2	no	—
234	#358	—	73	2	no	—
235	#357	—	73	2	no	—
236	#357	—	73	2	no	—
237	#346	—	73	2	no	—
238	C₄×C₁₆	—	73	2	no	—
239	C₂×#41	—	2	1	no	—
240	C₂×#41	—	146	1	no	—
241	#135	—	2	2	no	—
242	#135	—	73	4	no	—
243	#135	—	146	2	no	—
244	#135	C₄×D₇₃	1	4	yes	complemented
245	C₂×Q₁₆	—	73	1	no	—
246	C₂×Q₁₆	C₂²×D₇₃	1	1	yes	characteristic
247	C₂×SD₁₆	—	2	1	no	—
248	C₂×SD₁₆	—	146	1	no	—
249	C₂×D₈	—	73	1	no	—
250	C₂×D₈	C₂²×D₇₃	1	1	yes	characteristic
251	C₂×M₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
252	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
253	C₄×Q₈	—	2	2	no	—
254	C₄×Q₈	—	146	2	no	—
255	C₄×D₄	—	2	2	no	—
256	C₄×D₄	—	4	2	no	—
257	C₄×D₄	—	292	2	no	—
258	C₄×D₄	—	146	2	no	—
259	#117	—	4	1	no	—
260	#117	—	292	1	no	—
261	#117	C₂²⋊D₇₃	1	1	yes	characteristic
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₄⋊C₄	—	4	1	no	—
264	C₂×C₄⋊C₄	—	292	1	no	—
265	C₂×C₄⋊C₄	—	146	1	no	—
266	Q₃₂	—	73	4	no	—
267	SD₃₂	—	73	4	no	—
268	SD₃₂	—	73	4	no	—
269	D₁₆	—	73	4	no	—
270	M₃₂	—	146	2	no	—
271	C₂×C₁₆	—	73	4	no	—
272	C₂×C₁₆	—	73	1	no	—
273	C₂×C₁₆	—	73	1	no	—
274	C₂×C₁₆	—	73	2	no	—
275	#107	—	2	1	no	—
276	#107	—	73	2	no	—
277	#107	—	146	1	no	—
278	#107	C₂²×D₇₃	1	2	yes	—
279	#103	—	2	2	no	—
280	#103	—	146	2	no	—
281	#102	—	2	2	no	—
282	#102	—	146	2	no	—
283	#97	C₂²⋊D₇₃	1	2	yes	—
284	C₄×C₈	C₂²×D₇₃	1	2	yes	—
285	#41	—	2	4	no	—
286	#41	—	4	2	no	—
287	#41	—	292	2	no	—
288	#41	—	146	4	no	—
289	C₂×Q₈	—	2	1	no	—
290	C₂×Q₈	—	146	1	no	—
291	C₂×D₄	—	2	1	no	—
292	C₂×D₄	—	4	1	no	—
293	C₂×D₄	—	292	1	no	—
294	C₂×D₄	—	146	1	no	—
295	C₂²×C₄	—	4	1	no	—
296	C₂²×C₄	—	4	1	no	—
297	C₂²×C₄	—	4	2	no	—
298	C₂²×C₄	—	292	1	no	—
299	C₂²×C₄	—	292	1	no	—
300	C₂²×C₄	—	292	2	no	—
301	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
302	Q₁₆	—	2	1	no	—
303	Q₁₆	—	73	2	no	—
304	Q₁₆	—	146	1	no	—
305	Q₁₆	C₂×C₄×D₇₃	1	2	yes	—
306	SD₁₆	—	2	4	no	—
307	SD₁₆	—	146	4	no	—
308	D₈	—	2	1	no	—
309	D₈	—	73	2	no	—
310	D₈	—	146	1	no	—
311	D₈	C₂×C₄×D₇₃	1	2	yes	—
312	M₁₆	—	2	2	no	—
313	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
314	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
315	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
316	C₂×C₈	#11924028	1	2	yes	—
317	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	4	1	no	—
320	C₄⋊C₄	—	4	2	no	—
321	C₄⋊C₄	—	292	2	no	—
322	C₄⋊C₄	—	292	1	no	—
323	C₄⋊C₄	—	146	2	no	—
324	C₄⋊C₄	#11924013	1	1	yes	characteristic
325	C₄⋊C₄	#11924013	1	1	yes	characteristic
326	C₂²⋊C₄	—	4	2	no	—
327	C₂²⋊C₄	—	292	2	no	—
328	C₂²⋊C₄	#11924013	1	2	yes	—
329	C₄²	—	4	2	no	—
330	C₄²	—	292	2	no	—
331	C₄²	C₂×C₂²⋊D₇₃	1	2	yes	—
332	C₁₆	—	73	2	no	—
333	C₁₆	—	73	2	no	—
334	C₁₆	—	146	2	no	—
335	C₂³	—	4	1	no	—
336	C₂³	—	292	1	no	—
337	C₂³	C₂×#11924013	1	1	yes	characteristic
338	Q₈	—	2	2	no	—
339	Q₈	—	4	1	no	—
340	Q₈	—	292	1	no	—
341	Q₈	—	146	2	no	—
342	D₄	—	2	2	no	—
343	D₄	—	4	4	no	—
344	D₄	—	4	1	no	—
345	D₄	—	292	4	no	—
346	D₄	—	292	1	no	—
347	D₄	—	146	2	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	4	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	16	2	no	—
352	C₂×C₄	—	16	1	no	—
353	C₂×C₄	—	292	1	no	—
354	C₂×C₄	—	292	4	no	—
355	C₂×C₄	—	292	2	no	—
356	C₂×C₄	—	1168	2	no	—
357	C₂×C₄	—	1168	1	no	—
358	C₂×C₄	C₄×C₂²⋊D₇₃	1	2	yes	—
359	C₂×C₄	C₂×#11924013	1	2	yes	—
360	C₂×C₄	C₂×#11924013	1	2	yes	—
361	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
362	C₂×C₄	#14556611	1	2	yes	—
363	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
364	C₈	—	2	2	no	—
365	C₈	C₄×C₂²⋊D₇₃	1	2	yes	—
366	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
367	C₈	C₂×#11924028	1	1	yes	characteristic
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	4	2	no	—
371	C₂²	—	8	1	no	—
372	C₂²	—	8	1	no	—
373	C₂²	—	292	2	no	—
374	C₂²	—	584	1	no	—
375	C₂²	—	584	1	no	—
376	C₂²	C₄×#11924013	1	2	yes	—
377	C₂²	C₂²×#11924013	1	1	yes	characteristic
378	C₄	—	2	2	no	—
379	C₄	—	2	2	no	—
380	C₄	—	4	2	no	—
381	C₄	—	8	2	no	—
382	C₄	—	292	2	no	—
383	C₄	—	584	2	no	—
384	C₄	C₄×#11924013	1	2	yes	—
385	C₄	C₂×C₄×C₂²⋊D₇₃	1	1	yes	characteristic
386	C₄	C₂×#14556611	1	1	yes	characteristic
387	C₂	—	2	2	no	—
388	C₂	—	8	2	no	—
389	C₂	—	584	2	no	—
390	C₂	C₂×C₄×#11924013	1	1	yes	characteristic
391	C₂	#22623070	1	1	yes	characteristic
392	C₂	#22623082	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11924013
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2951	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×#17989017	C₂
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#2951	D₇₃
hypocenter	C₇₃	#33459
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2951

Complements

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

N	H ≅ G/N	type
2. #22618808	C₂	semidirect
3. #22611795	C₂	semidirect
4. #22611793	C₂	semidirect
5. C₄×#14556537	C₂	semidirect
6. C₄×#14556532	C₂	semidirect
7. C₂×#17988412	C₂	semidirect
8. #22611643	C₂	semidirect
9. #22611641	C₂	semidirect
10. #22611528	C₂	semidirect
11. C₇₃×#2951	C₂	semidirect
14. #17989017	C₄	semidirect
44. #14556613	C₂×C₄	semidirect
54. #14556537	C₂×C₄	semidirect
55. #14556535	C₂×C₄	semidirect
56. #14556534	C₂×C₄	semidirect
57. #14556532	C₂×C₄	semidirect
59. C₂×#11924003	C₂×C₄	semidirect
69. #135×C₇₃	C₂×C₄	semidirect
199. #2951	D₇₃	semidirect
208. C₇₃	#33459	semidirect
244. #135	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×#17989017	2
#22618808	2
#22611795	2
#22611793	2
C₄×#14556537	2
C₄×#14556532	2
C₂×#17988412	2
#22611643	2
#22611641	2
#22611528	2
C₇₃×#2951	2
#33459	73