Subgroups — #33239329

Order 18688.

696 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989022	C₂	1	1	yes	maximal, characteristic
2	#22618807	C₂	1	1	yes	maximal, characteristic, complemented
3	#22618782	C₂	1	1	yes	maximal, characteristic, complemented
4	#22611804	C₂	1	2	yes	maximal, complemented
5	#22611798	C₂	1	2	yes	maximal, complemented
6	#22611780	C₂	1	2	yes	maximal, complemented
7	#22611775	C₂	1	2	yes	maximal, complemented
8	#22611720	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#17988389	C₂	1	1	yes	maximal, characteristic, complemented
10	#22611527	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2138	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₅₈₄	C₂²	1	1	yes	characteristic
13	#17989070	—	2	1	no	—
14	#17989061	—	2	1	no	—
15	#17989022	C₄	1	8	yes	complemented
16	C₂×#14556534	C₂²	1	2	yes	—
17	C₂×#14556532	C₂²	1	2	yes	—
18	C₂×#14556531	C₂²	1	1	yes	characteristic
19	#17988698	—	2	2	no	—
20	#17988688	—	2	2	no	—
21	Q₁₆×Dic₇₃	C₂²	1	2	yes	complemented
22	D₈×Dic₇₃	C₂²	1	2	yes	complemented
23	#17988669	C₂²	1	2	yes	—
24	#17988669	C₂²	1	2	yes	—
25	C₂×#14556525	C₂²	1	1	yes	characteristic
26	#17988591	—	2	1	no	—
27	#17988588	—	2	1	no	—
28	#17988542	C₂²	1	1	yes	characteristic
29	#17988414	C₂²	1	2	yes	complemented
30	#17988412	C₂²	1	2	yes	complemented
31	#17988389	C₂²	1	2	yes	—
32	#17988389	C₂²	1	2	yes	—
33	#17988362	C₂²	1	2	yes	complemented
34	C₁₆×Dic₇₃	C₂²	1	2	yes	complemented
35	C₂×#135×C₇₃	C₂²	1	1	yes	characteristic
36	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
37	#427×C₇₃	C₂²	1	1	yes	characteristic
38	#358×C₇₃	C₂²	1	1	yes	characteristic, complemented
39	#358×C₇₃	C₂²	1	1	yes	characteristic, complemented
40	#357×C₇₃	C₂²	1	1	yes	characteristic, complemented
41	#357×C₇₃	C₂²	1	1	yes	characteristic, complemented
42	C₂²×D₂₉₂	—	2	1	no	—
43	C₂×D₅₈₄	—	2	1	no	—
44	C₂×D₅₈₄	—	2	2	no	—
45	C₂×D₅₈₄	C₂×C₄	1	4	yes	complemented
46	C₂×D₅₈₄	C₂³	1	2	yes	—
47	C₂×#11924012	D₄	1	1	yes	characteristic
48	Q₈×Dic₇₃	—	2	2	no	—
49	C₄×D₂₉₂	—	4	1	no	—
50	C₄×D₂₉₂	—	4	1	no	—
51	D₄×Dic₇₃	—	2	2	no	—
52	D₄×Dic₇₃	—	4	2	no	—
53	#14556553	—	2	2	no	—
54	#14556553	—	2	2	no	—
55	#14556550	—	4	1	no	—
56	#14556550	D₄	1	1	yes	characteristic
57	C₂×C₄⋊Dic₇₃	—	2	1	no	—
58	C₂×C₄⋊Dic₇₃	—	4	1	no	—
59	C₂²⋊C₄×D₇₃	—	4	2	no	—
60	#14556534	C₂×C₄	1	8	yes	complemented
61	#14556532	C₂×C₄	1	8	yes	complemented
62	#14556531	C₂×C₄	1	4	yes	complemented
63	C₂×#11924003	C₂³	1	2	yes	—
64	#14556525	—	2	1	no	—
65	#14556525	C₂³	1	2	yes	—
66	#14556515	—	2	2	no	—
67	#14556512	—	2	2	no	—
68	#14556511	—	2	2	no	—
69	#14556511	—	2	2	no	—
70	#14556501	D₄	1	2	yes	—
71	C₈×Dic₇₃	C₂³	1	2	yes	—
72	C₂×#41×C₇₃	—	2	1	no	—
73	#135×C₇₃	—	2	2	no	—
74	#135×C₇₃	C₂×C₄	1	4	yes	complemented
75	C₂×Q₁₆×C₇₃	C₂³	1	1	yes	characteristic
76	C₂×SD₁₆×C₇₃	—	2	1	no	—
77	C₂×D₈×C₇₃	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
79	#117×C₇₃	—	2	1	no	—
80	C₂×C₁₆×C₇₃	—	2	2	no	—
81	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
83	#107×C₇₃	C₂³	1	1	yes	characteristic
84	#107×C₇₃	C₂³	1	1	yes	characteristic
85	#106×C₇₃	—	2	1	no	—
86	C₂³×D₇₃	—	4	1	no	—
87	C₂×D₂₉₂	—	2	4	no	—
88	C₂×D₂₉₂	—	2	2	no	—
89	C₂×D₂₉₂	—	4	1	no	—
90	C₂×D₂₉₂	—	4	2	no	—
91	C₂²×Dic₇₃	—	4	1	no	—
92	C₂²×Dic₇₃	—	4	2	no	—
93	C₂×C₄×D₇₃	—	4	2	no	—
94	C₂×C₄×D₇₃	—	4	2	no	—
95	D₅₈₄	—	2	4	no	—
96	D₅₈₄	—	2	2	no	—
97	D₅₈₄	C₂²×C₄	1	4	yes	—
98	#11924012	—	2	2	no	—
99	C₂×#10596831	C₂×D₄	1	2	yes	—
100	C₄⋊Dic₇₃	—	2	2	no	—
101	C₄⋊Dic₇₃	—	4	1	no	—
102	C₄⋊Dic₇₃	—	4	2	no	—
103	C₄⋊Dic₇₃	D₈	1	1	yes	characteristic
104	C₄⋊Dic₇₃	D₈	1	1	yes	characteristic
105	C₂²⋊Dic₇₃	—	4	2	no	—
106	C₂²⋊Dic₇₃	D₈	1	2	yes	—
107	#11924005	—	4	2	no	—
108	#11924005	—	4	2	no	—
109	C₄×Dic₇₃	—	4	2	no	—
110	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
111	#11924003	C₂²×C₄	1	4	yes	—
112	#41×C₇₃	—	2	4	no	—
113	#41×C₇₃	—	4	2	no	—
114	C₂×Q₈×C₇₃	—	2	1	no	—
115	C₂×D₄×C₇₃	—	2	1	no	—
116	C₂×D₄×C₇₃	—	4	1	no	—
117	C₂²×C₄×C₇₃	—	4	1	no	—
118	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
119	Q₁₆×C₇₃	—	2	1	no	—
120	Q₁₆×C₇₃	C₂²×C₄	1	2	yes	—
121	SD₁₆×C₇₃	—	2	4	no	—
122	D₈×C₇₃	—	2	1	no	—
123	D₈×C₇₃	C₂²×C₄	1	2	yes	—
124	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
125	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₇₃	#41	1	2	yes	—
127	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
128	C₄⋊C₄×C₇₃	—	2	1	no	—
129	C₄⋊C₄×C₇₃	—	2	1	no	—
130	C₂²⋊C₄×C₇₃	—	4	1	no	—
131	C₄²×C₇₃	—	4	1	no	—
132	C₁₆×C₇₃	—	2	1	no	—
133	C₁₆×C₇₃	—	2	1	no	—
134	C₂³×C₇₃	—	4	1	no	—
135	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
136	C₂²×D₇₃	—	4	2	no	—
137	C₂²×D₇₃	—	4	4	no	—
138	C₂²×D₇₃	—	16	1	no	—
139	C₂²×D₇₃	—	16	1	no	—
140	Q₈×C₇₃	—	2	2	no	—
141	Q₈×C₇₃	—	4	1	no	—
142	D₄×C₇₃	—	2	2	no	—
143	D₄×C₇₃	—	4	4	no	—
144	D₄×C₇₃	—	4	1	no	—
145	C₂×C₄×C₇₃	—	4	1	no	—
146	C₂×C₄×C₇₃	—	4	4	no	—
147	C₂×C₄×C₇₃	—	4	1	no	—
148	C₂×C₄×C₇₃	—	4	1	no	—
149	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
150	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₇₃	#135	1	2	yes	—
152	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×Dic₇₃	—	4	2	no	—
154	C₂×Dic₇₃	—	16	2	no	—
155	C₂×Dic₇₃	—	16	1	no	—
156	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
157	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
158	D₂₉₂	—	2	4	no	—
159	D₂₉₂	—	4	2	no	—
160	D₂₉₂	—	4	4	no	—
161	C₄×D₇₃	—	16	2	no	—
162	C₄×D₇₃	—	16	2	no	—
163	C₈×C₇₃	C₄×D₄	1	2	yes	—
164	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
165	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
166	#10596831	—	2	2	no	—
167	C₂²×C₇₃	—	2	1	no	—
168	C₂²×C₇₃	—	2	1	no	—
169	C₂²×C₇₃	—	4	2	no	—
170	C₂²×C₇₃	—	8	1	no	—
171	C₂²×C₇₃	—	8	1	no	—
172	C₂²×C₇₃	C₄×D₈	1	2	yes	—
173	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
174	C₂×D₇₃	—	4	4	no	—
175	C₂×D₇₃	—	8	2	no	—
176	C₂×D₇₃	—	8	2	no	—
177	C₂×D₇₃	—	16	4	no	—
178	C₄×C₇₃	—	4	2	no	—
179	C₄×C₇₃	—	8	1	no	—
180	C₄×C₇₃	—	8	1	no	—
181	C₄×C₇₃	C₄×D₈	1	2	yes	—
182	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
183	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
184	Dic₇₃	—	2	2	no	—
185	Dic₇₃	—	2	2	no	—
186	Dic₇₃	—	8	2	no	—
187	#33454	—	73	1	no	maximal, Sylow
188	C₂×C₇₃	—	2	2	no	—
189	C₂×C₇₃	—	8	2	no	—
190	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
191	C₂×C₇₃	#3415	1	1	yes	characteristic
192	C₂×C₇₃	#3416	1	1	yes	characteristic
193	D₇₃	—	8	4	no	—
194	C₂×#509	—	73	1	no	—
195	#2951	—	73	1	no	—
196	#2948	—	73	1	no	—
197	#2177	—	73	2	no	—
198	#2175	—	73	2	no	—
199	C₄×SD₃₂	—	73	2	no	—
200	C₄×D₁₆	—	73	2	no	—
201	#2159	—	73	1	no	—
202	#2138	D₇₃	1	1	yes	characteristic, complemented
203	C₂×#357	—	73	1	no	—
204	#2108	—	73	1	no	—
205	C₇₃	#33454	1	1	yes	characteristic, complemented, Sylow
206	C₂×#135	C₂×D₇₃	1	1	yes	characteristic
207	C₂²×D₈	—	73	1	no	—
208	#520	—	146	1	no	—
209	#518	—	146	1	no	—
210	#509	—	73	8	no	—
211	C₂×SD₃₂	—	73	2	no	—
212	C₂×D₁₆	—	73	2	no	—
213	C₂×M₃₂	—	73	1	no	—
214	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
215	#442	—	146	2	no	—
216	#440	—	146	2	no	—
217	C₄×Q₁₆	—	73	2	no	—
218	C₄×D₈	—	73	2	no	—
219	C₄×D₈	—	73	2	no	—
220	C₄×D₈	—	73	2	no	—
221	#427	C₂×D₇₃	1	1	yes	characteristic
222	C₂×#107	—	73	1	no	—
223	#418	—	146	1	no	—
224	#417	—	146	1	no	—
225	#405	—	73	1	no	—
226	#367	—	73	2	no	—
227	#366	—	73	2	no	—
228	#358	C₂×D₇₃	1	1	yes	characteristic, complemented
229	#358	C₂×D₇₃	1	1	yes	characteristic, complemented
230	#357	—	73	2	no	—
231	#357	—	73	2	no	—
232	#357	C₂×D₇₃	1	1	yes	characteristic, complemented
233	#357	C₂×D₇₃	1	1	yes	characteristic, complemented
234	#346	—	73	2	no	—
235	C₄×C₁₆	—	73	2	no	—
236	C₂×#41	—	2	1	no	—
237	C₂²×D₄	—	146	1	no	—
238	#135	—	2	2	no	—
239	#135	C₄×D₇₃	1	4	yes	complemented
240	C₂×Q₁₆	C₂²×D₇₃	1	1	yes	characteristic
241	C₂×SD₁₆	—	2	1	no	—
242	C₂×D₈	—	73	4	no	—
243	C₂×D₈	—	73	2	no	—
244	C₂×D₈	—	146	1	no	—
245	C₂×D₈	—	146	2	no	—
246	C₂×D₈	C₂²×D₇₃	1	1	yes	characteristic
247	C₂×M₁₆	—	73	1	no	—
248	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
249	C₄×Q₈	—	146	2	no	—
250	C₄×D₄	—	292	1	no	—
251	C₄×D₄	—	292	1	no	—
252	C₄×D₄	—	292	2	no	—
253	C₄×D₄	—	146	2	no	—
254	C₄×D₄	—	146	2	no	—
255	C₄×D₄	—	146	2	no	—
256	#117	—	2	1	no	—
257	#117	—	292	1	no	—
258	#117	—	73	1	no	—
259	C₂×C₄⋊C₄	—	292	1	no	—
260	C₂×C₄⋊C₄	—	146	1	no	—
261	C₂×C₂²⋊C₄	—	292	2	no	—
262	SD₃₂	—	73	8	no	—
263	D₁₆	—	73	8	no	—
264	M₃₂	—	73	4	no	—
265	C₂×C₁₆	—	2	2	no	—
266	C₂×C₁₆	—	73	2	no	—
267	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
268	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
269	#107	—	73	2	no	—
270	#107	—	146	1	no	—
271	#107	C₂²×D₇₃	1	1	yes	characteristic
272	#107	C₂²×D₇₃	1	1	yes	characteristic
273	#106	—	2	1	no	—
274	#103	—	146	2	no	—
275	#102	—	146	2	no	—
276	#102	—	146	2	no	—
277	#102	—	146	2	no	—
278	#97	—	73	2	no	—
279	C₄×C₈	—	73	2	no	—
280	C₂⁴	—	292	1	no	—
281	#41	—	2	4	no	—
282	#41	—	4	2	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	4	1	no	—
286	C₂×D₄	—	292	1	no	—
287	C₂×D₄	—	292	2	no	—
288	C₂×D₄	—	146	4	no	—
289	C₂×D₄	—	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	C₂²×C₄	—	292	2	no	—
295	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
296	Q₁₆	—	2	1	no	—
297	Q₁₆	C₂×C₄×D₇₃	1	2	yes	—
298	SD₁₆	—	2	4	no	—
299	D₈	—	2	1	no	—
300	D₈	—	73	4	no	—
301	D₈	—	146	4	no	—
302	D₈	—	146	2	no	—
303	D₈	C₂×C₄×D₇₃	1	2	yes	—
304	M₁₆	—	146	2	no	—
305	C₂×C₈	—	73	2	no	—
306	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
307	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
308	C₂×C₈	#11924027	1	2	yes	—
309	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	292	2	no	—
313	C₄⋊C₄	—	292	1	no	—
314	C₄⋊C₄	—	73	1	no	—
315	C₄⋊C₄	—	73	1	no	—
316	C₄⋊C₄	—	146	2	no	—
317	C₂²⋊C₄	—	4	1	no	—
318	C₂²⋊C₄	—	292	2	no	—
319	C₂²⋊C₄	—	292	2	no	—
320	C₂²⋊C₄	—	292	2	no	—
321	C₂²⋊C₄	—	73	2	no	—
322	C₄²	—	4	1	no	—
323	C₄²	—	292	2	no	—
324	C₄²	—	73	2	no	—
325	C₁₆	—	2	1	no	—
326	C₁₆	—	2	1	no	—
327	C₁₆	—	73	4	no	—
328	C₂³	—	4	1	no	—
329	C₂³	—	292	2	no	—
330	C₂³	—	292	4	no	—
331	C₂³	—	1168	1	no	—
332	C₂³	—	1168	1	no	—
333	C₂³	D₈×D₇₃	1	1	yes	characteristic
334	Q₈	—	2	2	no	—
335	Q₈	—	4	1	no	—
336	D₄	—	2	2	no	—
337	D₄	—	4	4	no	—
338	D₄	—	4	1	no	—
339	D₄	—	292	4	no	—
340	D₄	—	292	2	no	—
341	D₄	—	146	4	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	4	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	292	2	no	—
347	C₂×C₄	—	73	2	no	—
348	C₂×C₄	—	73	2	no	—
349	C₂×C₄	—	1168	2	no	—
350	C₂×C₄	—	1168	2	no	—
351	C₂×C₄	—	1168	1	no	—
352	C₂×C₄	—	1168	2	no	—
353	C₂×C₄	#14556551	1	2	yes	—
354	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
355	C₂×C₄	#14556612	1	2	yes	—
356	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
357	C₈	—	146	2	no	—
358	C₈	#14556551	1	2	yes	—
359	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
360	C₈	C₂×#11924027	1	1	yes	characteristic
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	292	4	no	—
367	C₂²	—	584	2	no	—
368	C₂²	—	584	2	no	—
369	C₂²	—	1168	4	no	—
370	C₂²	#17988667	1	2	yes	—
371	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
372	C₄	—	4	2	no	—
373	C₄	—	8	1	no	—
374	C₄	—	8	1	no	—
375	C₄	—	584	2	no	—
376	C₄	—	146	2	no	—
377	C₄	—	146	2	no	—
378	C₄	#17988667	1	2	yes	—
379	C₄	C₂×#14556551	1	1	yes	characteristic
380	C₄	C₂×#14556612	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	8	2	no	—
383	C₂	—	584	4	no	—
384	C₂	C₂×#17988667	1	1	yes	characteristic
385	C₂	#22623069	1	1	yes	characteristic
386	C₂	#22623074	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2138	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×#17989022	C₂
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#2138	D₇₃
hypocenter	C₇₃	#33454
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2138

Complements

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

N	H ≅ G/N	type
2. #22618807	C₂	semidirect
3. #22618782	C₂	semidirect
4. #22611804	C₂	semidirect
5. #22611798	C₂	semidirect
6. #22611780	C₂	semidirect
7. #22611775	C₂	semidirect
8. #22611720	C₂	semidirect
9. C₂×#17988389	C₂	semidirect
10. #22611527	C₂	semidirect
11. C₇₃×#2138	C₂	semidirect
15. #17989022	C₄	semidirect
21. Q₁₆×Dic₇₃	C₂²	semidirect
22. D₈×Dic₇₃	C₂²	semidirect
29. #17988414	C₂²	semidirect
30. #17988412	C₂²	semidirect
33. #17988362	C₂²	semidirect
34. C₁₆×Dic₇₃	C₂²	semidirect
38. #358×C₇₃	C₂²	semidirect
39. #358×C₇₃	C₂²	semidirect
40. #357×C₇₃	C₂²	semidirect
41. #357×C₇₃	C₂²	semidirect
45. C₂×D₅₈₄	C₂×C₄	semidirect
60. #14556534	C₂×C₄	semidirect
61. #14556532	C₂×C₄	semidirect
62. #14556531	C₂×C₄	semidirect
74. #135×C₇₃	C₂×C₄	semidirect
202. #2138	D₇₃	semidirect
205. C₇₃	#33454	semidirect
228. #358	C₂×D₇₃	semidirect
229. #358	C₂×D₇₃	semidirect
232. #357	C₂×D₇₃	semidirect
233. #357	C₂×D₇₃	semidirect
239. #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×#17989022	2
#22618807	2
#22618782	2
#22611804	2
#22611798	2
#22611780	2
#22611775	2
#22611720	2
C₂×#17988389	2
#22611527	2
C₇₃×#2138	2
#33454	73