Subgroups — #33242809

Order 18688.

654 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622280	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×D₂₉₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612155	C₂	1	1	yes	maximal, characteristic, complemented
4	#22612113	C₂	1	2	yes	maximal, complemented
5	#22612068	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612016	C₂	1	2	yes	maximal, complemented
7	#22611966	C₂	1	1	yes	maximal, characteristic, complemented
8	#22611841	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₂₉₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#357×D₇₃	C₂	1	2	yes	maximal, complemented
11	C₄×D₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₇₃	C₂²	1	2	yes	—
13	#17989242	—	2	1	no	—
14	D₄×D₂₉₂	—	2	1	no	—
15	C₂×#14556532	C₂²	1	2	yes	complemented
16	C₂×C₁₆×D₇₃	C₂²	1	2	yes	complemented
17	#17988992	C₂²	1	1	yes	characteristic, complemented
18	#17988964	C₂²	1	1	yes	characteristic
19	#17988899	—	2	2	no	—
20	#17988867	—	2	1	no	—
21	#17988824	C₂²	1	2	yes	—
22	#17988820	C₂²	1	2	yes	complemented
23	#17988782	—	2	1	no	—
24	#17988722	—	2	2	no	—
25	C₈×D₂₉₂	C₂²	1	1	yes	characteristic
26	#107×D₇₃	C₂²	1	2	yes	complemented
27	#17988412	C₂²	1	2	yes	complemented
28	C₄⋊#11924003	C₂²	1	1	yes	characteristic, complemented
29	#17988389	C₂²	1	2	yes	complemented
30	#17988389	C₂²	1	2	yes	complemented
31	#17988388	C₂²	1	2	yes	—
32	#17988387	C₂²	1	2	yes	complemented
33	#17988366	C₂²	1	2	yes	complemented
34	C₂×D₁₆×C₇₃	C₂²	1	1	yes	characteristic
35	C₄×D₈×C₇₃	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₇₃	C₂²	1	1	yes	characteristic, complemented
37	#366×C₇₃	C₂²	1	1	yes	characteristic, complemented
38	#357×C₇₃	C₂²	1	1	yes	characteristic
39	#357×C₇₃	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic, complemented
41	C₂×#11924028	—	4	1	no	—
42	C₂²×D₂₉₂	—	4	1	no	—
43	C₂×D₄×D₇₃	—	2	2	no	—
44	C₂×D₅₈₄	—	2	1	no	—
45	C₂×D₅₈₄	C₂³	1	2	yes	—
46	C₂×#11924013	—	2	2	no	—
47	D₈×D₇₃	—	2	4	no	—
48	D₈×D₇₃	—	4	2	no	—
49	C₂×C₈×D₇₃	C₂³	1	2	yes	—
50	#14556591	—	2	1	no	—
51	#14556588	—	2	1	no	—
52	#14556573	—	4	2	no	—
53	#14556570	—	4	1	no	—
54	#14556566	—	2	2	no	—
55	#14556565	—	4	1	no	—
56	#14556562	—	2	2	no	—
57	#14556561	—	4	2	no	—
58	C₄×D₂₉₂	D₄	1	1	yes	characteristic
59	C₄⋊C₄×D₇₃	—	2	2	no	—
60	#14556532	—	4	2	no	—
61	C₂×#11924003	C₂³	1	2	yes	—
62	C₁₆×D₇₃	—	4	2	no	—
63	#14556525	—	2	1	no	—
64	#14556525	C₂³	1	2	yes	—
65	#14556524	—	2	2	no	—
66	C₄⋊#10596831	D₄	1	1	yes	characteristic
67	#14556512	—	2	2	no	—
68	#14556511	—	2	2	no	—
69	#14556503	D₄	1	2	yes	—
70	C₂×D₈×C₇₃	C₂³	1	1	yes	characteristic
71	C₂×D₈×C₇₃	C₂³	1	1	yes	characteristic
72	C₄×D₄×C₇₃	—	2	1	no	—
73	C₄×D₄×C₇₃	—	2	1	no	—
74	D₁₆×C₇₃	—	2	2	no	—
75	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
77	#107×C₇₃	C₂³	1	1	yes	characteristic
78	#107×C₇₃	C₂³	1	1	yes	characteristic
79	#102×C₇₃	—	2	1	no	—
80	#102×C₇₃	—	2	1	no	—
81	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
82	C₂³×D₇₃	—	4	2	no	—
83	#11924028	—	8	4	no	—
84	C₂×Dic₁₄₆	—	4	1	no	—
85	C₂×D₂₉₂	—	2	2	no	—
86	C₂×D₂₉₂	—	4	2	no	—
87	C₂×D₂₉₂	—	4	1	no	—
88	C₂×D₂₉₂	—	4	1	no	—
89	C₂×D₂₉₂	—	8	4	no	—
90	C₂×D₂₉₂	D₈	1	1	yes	characteristic
91	C₂×C₂²⋊D₇₃	—	4	2	no	—
92	C₂×C₂²⋊D₇₃	—	4	2	no	—
93	D₄×D₇₃	—	4	4	no	—
94	D₄×D₇₃	—	8	2	no	—
95	C₂×C₄×D₇₃	—	4	2	no	—
96	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
97	D₅₈₄	—	2	2	no	—
98	D₅₈₄	—	2	2	no	—
99	D₅₈₄	—	8	1	no	—
100	#11924013	—	4	4	no	—
101	C₂×#10596831	C₂×D₄	1	2	yes	—
102	C₈×D₇₃	—	2	2	no	—
103	C₈×D₇₃	—	2	2	no	—
104	C₄⋊Dic₇₃	—	2	2	no	—
105	C₄⋊Dic₇₃	—	4	1	no	—
106	C₄⋊Dic₇₃	D₈	1	1	yes	characteristic
107	#11924007	—	4	2	no	—
108	C₂²⋊Dic₇₃	—	4	2	no	—
109	#11924005	—	4	2	no	—
110	#11924005	D₈	1	2	yes	—
111	#11924003	—	2	2	no	—
112	C₂×D₄×C₇₃	—	2	1	no	—
113	C₂×D₄×C₇₃	—	2	1	no	—
114	C₂²×C₄×C₇₃	—	4	1	no	—
115	C₂²×C₄×C₇₃	—	4	1	no	—
116	D₈×C₇₃	—	2	1	no	—
117	D₈×C₇₃	—	2	1	no	—
118	D₈×C₇₃	—	2	1	no	—
119	D₈×C₇₃	#41	1	2	yes	complemented
120	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
122	C₄⋊C₄×C₇₃	—	2	1	no	—
123	C₄⋊C₄×C₇₃	—	2	1	no	—
124	C₂²⋊C₄×C₇₃	—	4	1	no	—
125	C₂²⋊C₄×C₇₃	—	4	1	no	—
126	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
127	C₁₆×C₇₃	—	2	1	no	—
128	C₁₆×C₇₃	—	2	1	no	—
129	C₂³×C₇₃	—	4	1	no	—
130	C₂³×C₇₃	—	4	1	no	—
131	C₂²×D₇₃	—	4	2	no	—
132	C₂²×D₇₃	—	8	4	no	—
133	C₂²×D₇₃	—	8	4	no	—
134	C₂²×D₇₃	—	16	2	no	—
135	C₂²×D₇₃	C₂×D₈	1	2	yes	—
136	D₄×C₇₃	—	2	2	no	—
137	D₄×C₇₃	—	4	1	no	—
138	D₄×C₇₃	—	4	1	no	—
139	D₄×C₇₃	—	4	1	no	—
140	C₂×C₄×C₇₃	—	4	1	no	—
141	C₂×C₄×C₇₃	—	4	1	no	—
142	C₂×C₄×C₇₃	—	8	2	no	—
143	C₂×C₄×C₇₃	—	8	2	no	—
144	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
147	C₂²⋊D₇₃	—	8	4	no	—
148	C₂²⋊D₇₃	—	8	4	no	—
149	C₂×Dic₇₃	—	4	2	no	—
150	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
151	D₂₉₂	—	4	2	no	—
152	D₂₉₂	—	4	2	no	—
153	D₂₉₂	—	8	4	no	—
154	D₂₉₂	—	8	2	no	—
155	D₂₉₂	—	16	1	no	—
156	D₂₉₂	D₁₆	1	4	yes	complemented
157	C₄×D₇₃	—	2	2	no	—
158	C₄×D₇₃	—	2	2	no	—
159	C₄×D₇₃	—	8	4	no	—
160	Dic₁₄₆	—	8	2	no	—
161	C₈×C₇₃	—	2	1	no	—
162	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
163	C₈×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
164	#10596831	—	2	2	no	—
165	C₂²×C₇₃	—	4	2	no	—
166	C₂²×C₇₃	—	8	1	no	—
167	C₂²×C₇₃	—	8	1	no	—
168	C₂²×C₇₃	—	8	1	no	—
169	C₂²×C₇₃	—	8	1	no	—
170	C₂²×C₇₃	—	8	1	no	—
171	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
172	C₂×D₇₃	—	2	2	no	—
173	C₂×D₇₃	—	2	2	no	—
174	C₂×D₇₃	—	8	2	no	—
175	C₂×D₇₃	—	8	2	no	—
176	C₂×D₇₃	—	8	2	no	—
177	C₂×D₇₃	—	16	8	no	—
178	C₂×D₇₃	C₂×D₁₆	1	4	yes	—
179	C₄×C₇₃	—	2	1	no	—
180	C₄×C₇₃	—	8	1	no	—
181	C₄×C₇₃	—	8	1	no	—
182	C₄×C₇₃	C₂×D₁₆	1	2	yes	—
183	C₄×C₇₃	#546	1	1	yes	characteristic
184	C₄×C₇₃	#577	1	1	yes	characteristic
185	Dic₇₃	—	2	2	no	—
186	Dic₇₃	—	8	2	no	—
187	#33861	—	73	1	no	maximal, Sylow
188	C₂×C₇₃	—	8	2	no	—
189	C₂×C₇₃	—	16	1	no	—
190	C₂×C₇₃	#3294	1	1	yes	characteristic
191	C₂×C₇₃	C₂²×D₁₆	1	1	yes	characteristic
192	C₂×C₇₃	#3416	1	1	yes	characteristic
193	D₇₃	—	2	4	no	—
194	D₇₃	—	16	2	no	—
195	#3334	—	73	1	no	—
196	D₄×D₈	—	73	1	no	—
197	#2240	—	73	1	no	—
198	#2232	—	73	2	no	—
199	#2224	—	73	1	no	—
200	#2215	—	73	2	no	—
201	#2206	—	73	1	no	—
202	#2184	—	73	2	no	—
203	C₄×D₁₆	D₇₃	1	1	yes	characteristic, complemented
204	D₄×C₁₆	—	73	1	no	—
205	C₂×#357	—	73	2	no	—
206	C₇₃	#33861	1	1	yes	characteristic, complemented, Sylow
207	C₂²×D₈	—	73	2	no	—
208	#547	—	146	1	no	—
209	D₄²	—	146	1	no	—
210	C₂×D₁₆	—	73	2	no	—
211	C₂×D₁₆	C₂×D₇₃	1	1	yes	characteristic
212	C₂²×C₁₆	—	73	2	no	—
213	#500	—	73	1	no	—
214	#493	—	73	1	no	—
215	#480	—	146	2	no	—
216	#474	—	146	1	no	—
217	#466	—	73	2	no	—
218	#466	—	73	2	no	—
219	#459	—	146	1	no	—
220	#447	—	146	2	no	—
221	C₄×D₈	C₂×D₇₃	1	1	yes	characteristic
222	C₄×D₈	C₂×D₇₃	1	1	yes	characteristic, complemented
223	C₈×D₄	—	73	1	no	—
224	C₂×#107	—	73	2	no	—
225	#366	—	73	2	no	—
226	#366	C₂×D₇₃	1	1	yes	characteristic, complemented
227	C₄⋊C₁₆	—	73	1	no	—
228	#357	—	73	2	no	—
229	#357	—	73	2	no	—
230	#357	—	73	2	no	—
231	#357	—	73	2	no	—
232	#357	C₂×D₇₃	1	1	yes	characteristic
233	#357	C₂×D₇₃	1	1	yes	characteristic, complemented
234	C₂²⋊C₁₆	—	73	2	no	—
235	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic, complemented
236	C₂×#41	—	292	1	no	—
237	C₂²×D₄	—	292	1	no	—
238	C₂²×D₄	—	146	2	no	—
239	C₂×D₈	—	292	2	no	—
240	C₂×D₈	—	73	2	no	—
241	C₂×D₈	—	146	2	no	—
242	C₂×D₈	—	146	4	no	—
243	C₂×D₈	—	146	1	no	—
244	C₂×D₈	C₂²×D₇₃	1	1	yes	characteristic
245	C₂×D₈	C₂²×D₇₃	1	1	yes	characteristic
246	C₂²×C₈	—	73	2	no	—
247	C₄⋊Q₈	—	146	1	no	—
248	#127	—	146	1	no	—
249	#123	—	292	2	no	—
250	C₂²⋊Q₈	—	292	1	no	—
251	#121	—	292	1	no	—
252	#121	—	146	2	no	—
253	#121	—	146	2	no	—
254	#120	—	292	2	no	—
255	C₄×D₄	—	2	1	no	—
256	C₄×D₄	—	2	1	no	—
257	C₄×D₄	—	73	1	no	—
258	C₂×C₄⋊C₄	—	146	2	no	—
259	D₁₆	—	2	2	no	—
260	D₁₆	—	292	2	no	—
261	C₂×C₁₆	—	292	2	no	—
262	C₂×C₁₆	—	73	2	no	—
263	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
264	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
265	#107	—	73	2	no	—
266	#107	—	146	2	no	—
267	#107	—	146	1	no	—
268	#107	C₂²×D₇₃	1	1	yes	characteristic
269	#107	C₂²×D₇₃	1	1	yes	characteristic
270	C₄⋊C₈	—	73	1	no	—
271	#102	—	2	1	no	—
272	#102	—	2	1	no	—
273	#102	—	146	2	no	—
274	#102	—	146	2	no	—
275	C₂²⋊C₈	—	73	2	no	—
276	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
277	C₂⁴	—	292	2	no	—
278	#41	—	584	4	no	—
279	C₂×Q₈	—	292	1	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	292	2	no	—
283	C₂×D₄	—	292	2	no	—
284	C₂×D₄	—	292	2	no	—
285	C₂×D₄	—	292	4	no	—
286	C₂×D₄	—	292	1	no	—
287	C₂×D₄	—	292	1	no	—
288	C₂×D₄	—	584	4	no	—
289	C₂×D₄	—	584	2	no	—
290	C₂×D₄	—	73	1	no	—
291	C₂×D₄	—	146	2	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	4	1	no	—
294	C₂²×C₄	—	292	2	no	—
295	C₂²×C₄	—	73	2	no	—
296	D₈	—	2	1	no	—
297	D₈	—	2	1	no	—
298	D₈	—	2	1	no	—
299	D₈	—	292	4	no	—
300	D₈	—	584	1	no	—
301	D₈	—	146	2	no	—
302	D₈	—	146	2	no	—
303	D₈	#11924028	1	2	yes	complemented
304	C₂×C₈	—	73	2	no	—
305	C₂×C₈	—	146	2	no	—
306	C₂×C₈	—	146	2	no	—
307	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	292	2	no	—
312	C₄⋊C₄	—	292	1	no	—
313	C₄⋊C₄	—	73	1	no	—
314	C₄⋊C₄	—	146	2	no	—
315	C₂²⋊C₄	—	4	1	no	—
316	C₂²⋊C₄	—	4	1	no	—
317	C₂²⋊C₄	—	292	2	no	—
318	C₂²⋊C₄	—	292	2	no	—
319	C₂²⋊C₄	—	73	2	no	—
320	C₄²	D₄×D₇₃	1	1	yes	characteristic
321	C₁₆	—	2	1	no	—
322	C₁₆	—	2	1	no	—
323	C₁₆	—	146	2	no	—
324	C₂³	—	4	1	no	—
325	C₂³	—	4	1	no	—
326	C₂³	—	292	2	no	—
327	C₂³	—	584	4	no	—
328	C₂³	—	584	4	no	—
329	C₂³	—	73	2	no	—
330	C₂³	—	1168	2	no	—
331	Q₈	—	584	2	no	—
332	D₄	—	2	2	no	—
333	D₄	—	4	1	no	—
334	D₄	—	4	1	no	—
335	D₄	—	4	1	no	—
336	D₄	—	292	2	no	—
337	D₄	—	292	2	no	—
338	D₄	—	584	4	no	—
339	D₄	—	584	4	no	—
340	D₄	—	584	4	no	—
341	D₄	—	584	2	no	—
342	D₄	—	73	4	no	—
343	D₄	—	1168	1	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	8	2	no	—
347	C₂×C₄	—	8	2	no	—
348	C₂×C₄	—	292	2	no	—
349	C₂×C₄	—	584	4	no	—
350	C₂×C₄	—	73	2	no	—
351	C₂×C₄	—	146	2	no	—
352	C₂×C₄	—	146	2	no	—
353	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
354	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
355	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	146	2	no	—
358	C₈	C₂×#11924028	1	1	yes	characteristic
359	C₈	#14556637	1	1	yes	characteristic
360	C₂²	—	4	2	no	—
361	C₂²	—	8	1	no	—
362	C₂²	—	8	1	no	—
363	C₂²	—	8	1	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	584	2	no	—
367	C₂²	—	584	2	no	—
368	C₂²	—	584	2	no	—
369	C₂²	—	73	4	no	—
370	C₂²	—	1168	8	no	—
371	C₂²	—	146	2	no	—
372	C₂²	—	146	2	no	—
373	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	8	1	no	—
376	C₄	—	8	1	no	—
377	C₄	—	584	2	no	—
378	C₄	—	146	2	no	—
379	C₄	D₁₆×D₇₃	1	2	yes	—
380	C₄	#17989233	1	1	yes	characteristic
381	C₄	#17989394	1	1	yes	characteristic
382	C₂	—	8	2	no	—
383	C₂	—	16	1	no	—
384	C₂	—	1168	2	no	—
385	C₂	—	146	4	no	—
386	C₂	#22621746	1	1	yes	characteristic
387	C₂	C₂×D₁₆×D₇₃	1	1	yes	characteristic
388	C₂	#22623074	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₄×D₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₈
hypercenter	C₄×D₁₆	D₇₃
hypocenter	C₇₃	#33861
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
1. #22622280	C₂	semidirect
2. D₈×D₂₉₂	C₂	semidirect
3. #22612155	C₂	semidirect
4. #22612113	C₂	semidirect
5. #22612068	C₂	semidirect
6. #22612016	C₂	semidirect
7. #22611966	C₂	semidirect
8. #22611841	C₂	semidirect
9. C₁₆×D₂₉₂	C₂	semidirect
10. #357×D₇₃	C₂	semidirect
11. C₄×D₁₆×C₇₃	C₂	semidirect
15. C₂×#14556532	C₂²	semidirect
16. C₂×C₁₆×D₇₃	C₂²	semidirect
17. #17988992	C₂²	semidirect
22. #17988820	C₂²	semidirect
26. #107×D₇₃	C₂²	semidirect
27. #17988412	C₂²	semidirect
28. C₄⋊#11924003	C₂²	semidirect
29. #17988389	C₂²	semidirect
30. #17988389	C₂²	semidirect
32. #17988387	C₂²	semidirect
33. #17988366	C₂²	semidirect
36. C₄×D₈×C₇₃	C₂²	semidirect
37. #366×C₇₃	C₂²	semidirect
39. #357×C₇₃	C₂²	semidirect
40. C₄×C₁₆×C₇₃	C₂²	semidirect
119. D₈×C₇₃	#41	semidirect
156. D₂₉₂	D₁₆	semidirect
203. C₄×D₁₆	D₇₃	semidirect
206. C₇₃	#33861	semidirect
222. C₄×D₈	C₂×D₇₃	semidirect
226. #366	C₂×D₇₃	semidirect
233. #357	C₂×D₇₃	semidirect
235. C₄×C₁₆	C₂×D₇₃	semidirect
303. D₈	#11924028	semidirect

Maximal subgroups

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

subgroup	index
#22622280	2
D₈×D₂₉₂	2
#22612155	2
#22612113	2
#22612068	2
#22612016	2
#22611966	2
#22611841	2
C₁₆×D₂₉₂	2
#357×D₇₃	2
C₄×D₁₆×C₇₃	2
#33861	73