Subgroups — #33078040

Order 18688.

644 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622937	C₂	1	1	yes	maximal, characteristic, complemented
2	#22622658	C₂	1	2	yes	maximal, complemented
3	#22621691	C₂	1	1	yes	maximal, characteristic
4	#22609537	C₂	1	1	yes	maximal, characteristic, complemented
5	#22609500	C₂	1	2	yes	maximal, complemented
6	#22609296	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609291	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609127	C₂	1	2	yes	maximal, complemented
9	#22609030	C₂	1	2	yes	maximal, complemented
10	#22608829	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556613	—	2	1	no	—
13	#17989316	C₂²	1	2	yes	—
14	#17989258	C₂²	1	2	yes	—
15	#17988967	—	2	1	no	—
16	#17988967	C₂²	1	1	yes	characteristic
17	#17988964	C₂²	1	1	yes	characteristic
18	#17988950	—	2	2	no	—
19	#17988934	C₂²	1	2	yes	—
20	#17988934	C₂²	1	2	yes	—
21	#17988886	—	2	2	no	—
22	#17988867	C₂²	1	2	yes	—
23	#17988823	—	2	1	no	—
24	#17988798	—	2	2	no	—
25	#17988798	C₂²	1	2	yes	—
26	#17988788	C₂²	1	2	yes	—
27	#17988787	C₂²	1	2	yes	—
28	#17988785	—	2	2	no	—
29	#17988734	—	2	2	no	—
30	#17988717	C₂²	1	2	yes	—
31	#17988690	—	2	2	no	—
32	C₄×Dic₂₉₂	—	2	1	no	—
33	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
34	#17988340	C₂²	1	1	yes	characteristic
35	#17988334	C₂²	1	2	yes	—
36	#17988330	C₂²	1	2	yes	—
37	#17988319	C₂²	1	2	yes	—
38	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
39	C₄×SD₁₆×C₇₃	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₇₃	C₂²	1	1	yes	characteristic
41	#334×C₇₃	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
44	C₈²×C₇₃	C₂²	1	1	yes	characteristic
45	C₂×#11924028	—	2	2	no	—
46	#14556613	—	4	4	no	—
47	C₂×Dic₂₉₂	—	2	2	no	—
48	C₂×Dic₂₉₂	—	2	1	no	—
49	C₂×Dic₂₉₂	—	2	1	no	—
50	C₂×Dic₂₉₂	D₄	1	1	yes	characteristic
51	C₂×#11924017	—	2	2	no	—
52	C₂×#11924015	—	2	2	no	—
53	C₂×D₅₈₄	—	2	1	no	—
54	C₂×D₅₈₄	—	2	1	no	—
55	#14556591	—	2	2	no	—
56	#14556591	C₂³	1	2	yes	—
57	#14556588	C₂³	1	2	yes	—
58	#14556587	—	2	2	no	—
59	#14556582	—	2	2	no	—
60	#14556582	—	4	2	no	—
61	#14556565	—	2	2	no	—
62	#14556564	—	4	2	no	—
63	Q₈×Dic₇₃	—	2	2	no	—
64	C₄×Dic₁₄₆	—	2	2	no	—
65	C₄×Dic₁₄₆	C₂³	1	2	yes	—
66	#14556553	—	2	2	no	—
67	#14556525	D₄	1	1	yes	characteristic
68	#14556524	—	2	2	no	—
69	C₄⋊#10596831	C₂³	1	2	yes	—
70	#14556520	—	2	2	no	—
71	#14556516	—	2	2	no	—
72	#14556516	D₄	1	2	yes	—
73	#14556513	—	2	2	no	—
74	#14556513	—	2	2	no	—
75	#14556513	—	2	2	no	—
76	#14556513	—	4	2	no	—
77	#14556512	—	2	2	no	—
78	#14556511	—	2	2	no	—
79	#14556501	—	2	2	no	—
80	C₂×SD₁₆×C₇₃	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₇₃	—	2	1	no	—
82	C₄×Q₈×C₇₃	C₂³	1	1	yes	characteristic
83	C₄×D₄×C₇₃	C₂³	1	1	yes	characteristic
84	#106×C₇₃	D₄	1	1	yes	characteristic
85	C₄⋊C₈×C₇₃	—	2	1	no	—
86	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
88	#103×C₇₃	D₄	1	1	yes	characteristic
89	#102×C₇₃	D₄	1	1	yes	characteristic
90	C₂²⋊C₈×C₇₃	—	2	1	no	—
91	C₄×C₈×C₇₃	—	2	1	no	—
92	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
95	#11924028	—	4	4	no	—
96	#11924028	—	8	2	no	—
97	C₂×Dic₁₄₆	—	2	2	no	—
98	C₂×Dic₁₄₆	C₂×D₄	1	2	yes	—
99	C₂×D₂₉₂	—	2	2	no	—
100	C₂×D₂₉₂	—	2	2	no	—
101	C₂×D₂₉₂	—	2	2	no	—
102	C₂×C₂²⋊D₇₃	—	4	2	no	—
103	C₂×C₂²⋊D₇₃	—	4	2	no	—
104	C₂×C₄×D₇₃	—	4	2	no	—
105	Dic₂₉₂	—	2	2	no	—
106	Dic₂₉₂	—	4	2	no	—
107	Dic₂₉₂	—	8	1	no	—
108	Dic₂₉₂	—	8	2	no	—
109	#11924017	—	4	4	no	—
110	#11924015	—	4	4	no	—
111	D₅₈₄	—	4	2	no	—
112	D₅₈₄	—	8	1	no	—
113	C₂×#10596831	—	2	2	no	—
114	C₄⋊Dic₇₃	—	2	2	no	—
115	C₄⋊Dic₇₃	—	2	2	no	—
116	C₄⋊Dic₇₃	—	4	2	no	—
117	C₄⋊Dic₇₃	—	4	2	no	—
118	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
119	#11924007	—	2	2	no	—
120	#11924007	—	4	2	no	—
121	#11924005	—	4	2	no	—
122	C₄×Dic₇₃	—	2	2	no	—
123	C₄×Dic₇₃	—	4	2	no	—
124	C₂×Q₈×C₇₃	C₂×D₄	1	1	yes	characteristic
125	C₂×D₄×C₇₃	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₇₃	—	2	1	no	—
127	SD₁₆×C₇₃	—	2	2	no	—
128	C₂×C₈×C₇₃	—	2	1	no	—
129	C₂×C₈×C₇₃	—	2	1	no	—
130	C₂×C₈×C₇₃	—	2	1	no	—
131	C₂×C₈×C₇₃	—	4	2	no	—
132	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
136	C₄⋊C₄×C₇₃	—	2	1	no	—
137	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
138	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
139	C₂²⋊C₄×C₇₃	—	2	1	no	—
140	C₄²×C₇₃	—	2	1	no	—
141	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
142	C₂³×C₇₃	—	2	1	no	—
143	C₂²×D₇₃	—	4	2	no	—
144	Q₈×C₇₃	—	2	1	no	—
145	Q₈×C₇₃	#135	1	2	yes	complemented
146	D₄×C₇₃	—	2	1	no	—
147	D₄×C₇₃	—	2	1	no	—
148	C₂×C₄×C₇₃	—	2	1	no	—
149	C₂×C₄×C₇₃	—	2	1	no	—
150	C₂×C₄×C₇₃	—	2	1	no	—
151	C₂×C₄×C₇₃	—	4	1	no	—
152	C₂×C₄×C₇₃	—	4	1	no	—
153	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²⋊D₇₃	—	8	4	no	—
157	C₂²⋊D₇₃	—	16	2	no	—
158	C₂×Dic₇₃	—	2	2	no	—
159	C₂×Dic₇₃	—	2	2	no	—
160	C₂×Dic₇₃	—	4	2	no	—
161	D₂₉₂	—	4	2	no	—
162	D₂₉₂	—	4	2	no	—
163	D₂₉₂	—	4	2	no	—
164	D₂₉₂	—	4	2	no	—
165	D₂₉₂	—	8	2	no	—
166	C₄×D₇₃	—	8	4	no	—
167	Dic₁₄₆	—	2	2	no	—
168	Dic₁₄₆	—	2	2	no	—
169	Dic₁₄₆	—	4	2	no	—
170	Dic₁₄₆	—	4	2	no	—
171	C₈×C₇₃	—	2	1	no	—
172	C₈×C₇₃	—	2	1	no	—
173	C₈×C₇₃	—	2	1	no	—
174	C₈×C₇₃	—	4	1	no	—
175	C₈×C₇₃	—	4	1	no	—
176	C₈×C₇₃	—	4	1	no	—
177	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
178	#10596831	—	4	2	no	—
179	C₂²×C₇₃	—	4	1	no	—
180	C₂²×C₇₃	—	4	1	no	—
181	C₂²×C₇₃	—	4	1	no	—
182	C₂²×C₇₃	D₄²	1	1	yes	characteristic
183	C₂×D₇₃	—	8	2	no	—
184	C₂×D₇₃	—	8	2	no	—
185	C₂×D₇₃	—	8	2	no	—
186	C₄×C₇₃	—	2	2	no	—
187	C₄×C₇₃	—	2	1	no	—
188	C₄×C₇₃	—	4	1	no	—
189	C₄×C₇₃	—	4	1	no	—
190	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
191	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
192	C₄×C₇₃	#576	1	1	yes	characteristic
193	C₄×C₇₃	#578	1	1	yes	characteristic
194	Dic₇₃	—	4	2	no	—
195	Dic₇₃	—	4	2	no	—
196	Dic₇₃	—	8	2	no	—
197	#19943	—	73	1	no	maximal, Sylow
198	C₂×C₇₃	—	8	1	no	—
199	C₂×C₇₃	#3274	1	1	yes	characteristic
200	C₂×C₇₃	#3285	1	1	yes	characteristic
201	C₂×C₇₃	#3290	1	1	yes	characteristic
202	D₇₃	—	16	2	no	—
203	#3395	—	73	1	no	—
204	#3368	—	73	2	no	—
205	#3287	—	73	1	no	—
206	#1709	—	73	1	no	—
207	#1702	—	73	2	no	—
208	#1669	—	73	1	no	—
209	#1668	—	73	1	no	—
210	#1641	—	73	2	no	—
211	#1626	—	73	2	no	—
212	#1591	—	73	1	no	—
213	C₈×SD₁₆	D₇₃	1	1	yes	characteristic, complemented
214	C₇₃	#19943	1	1	yes	characteristic, complemented, Sylow
215	C₂×#135	—	146	1	no	—
216	#557	—	73	2	no	—
217	#550	—	73	2	no	—
218	#494	—	73	1	no	—
219	#494	—	146	1	no	—
220	#493	—	73	1	no	—
221	#490	—	146	2	no	—
222	#486	—	73	2	no	—
223	#486	—	73	2	no	—
224	#478	—	146	2	no	—
225	#474	—	73	2	no	—
226	#466	—	146	1	no	—
227	#462	—	73	2	no	—
228	#462	—	146	2	no	—
229	#460	—	73	2	no	—
230	#460	—	73	2	no	—
231	#460	—	146	2	no	—
232	#449	—	146	2	no	—
233	#446	—	73	2	no	—
234	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
235	#440	—	146	2	no	—
236	C₄×Q₁₆	—	73	1	no	—
237	C₄×Q₁₆	—	146	1	no	—
238	C₄×SD₁₆	C₂×D₇₃	1	1	yes	characteristic
239	C₈×D₄	C₂×D₇₃	1	1	yes	characteristic
240	#335	—	73	1	no	—
241	#334	C₂×D₇₃	1	1	yes	characteristic
242	#332	—	73	2	no	—
243	#331	—	73	2	no	—
244	Q₈⋊C₈	—	73	2	no	—
245	Q₈⋊C₈	C₂×D₇₃	1	1	yes	characteristic
246	D₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
247	C₈²	C₂×D₇₃	1	1	yes	characteristic
248	C₂×#41	—	146	2	no	—
249	#135	—	292	4	no	—
250	C₂×Q₁₆	—	73	1	no	—
251	C₂×Q₁₆	—	146	2	no	—
252	C₂×Q₁₆	—	146	1	no	—
253	C₂×Q₁₆	—	146	1	no	—
254	C₂×SD₁₆	—	146	2	no	—
255	C₂×SD₁₆	—	146	2	no	—
256	C₂×SD₁₆	D₂₉₂	1	1	yes	characteristic
257	C₂×D₈	—	146	1	no	—
258	C₂×D₈	—	146	1	no	—
259	C₂²×C₈	—	2	1	no	—
260	C₄⋊Q₈	—	73	2	no	—
261	C₄⋊Q₈	—	146	2	no	—
262	#127	—	73	2	no	—
263	#127	—	146	2	no	—
264	#125	—	292	2	no	—
265	#125	—	146	2	no	—
266	#121	—	292	2	no	—
267	#121	—	146	2	no	—
268	C₄×Q₈	—	73	2	no	—
269	C₄×Q₈	—	146	2	no	—
270	C₄×Q₈	—	146	2	no	—
271	C₄×Q₈	C₂²×D₇₃	1	1	yes	characteristic
272	C₄×D₄	—	146	2	no	—
273	C₄×D₄	C₂²×D₇₃	1	1	yes	characteristic
274	#107	—	73	1	no	—
275	#107	—	146	2	no	—
276	#106	D₂₉₂	1	1	yes	characteristic
277	C₄⋊C₈	—	2	1	no	—
278	C₄⋊C₈	—	73	2	no	—
279	C₄⋊C₈	—	146	2	no	—
280	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
281	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
282	#103	—	73	2	no	—
283	#103	—	146	2	no	—
284	#103	D₂₉₂	1	1	yes	characteristic
285	#102	—	292	2	no	—
286	#102	—	146	2	no	—
287	#102	—	146	2	no	—
288	#102	—	146	2	no	—
289	#102	—	146	2	no	—
290	#102	—	146	2	no	—
291	#102	D₂₉₂	1	1	yes	characteristic
292	C₂²⋊C₈	—	2	1	no	—
293	#97	—	146	2	no	—
294	C₄×C₈	—	2	1	no	—
295	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
296	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
297	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
298	#41	—	292	4	no	—
299	#41	—	584	2	no	—
300	C₂×Q₈	—	73	2	no	—
301	C₂×Q₈	—	146	2	no	—
302	C₂×Q₈	C₂×D₂₉₂	1	1	yes	characteristic
303	C₂×D₄	—	292	2	no	—
304	C₂×D₄	—	292	2	no	—
305	C₂×D₄	—	146	2	no	—
306	C₂×D₄	—	146	2	no	—
307	C₂×D₄	—	146	2	no	—
308	C₂×D₄	C₂×D₂₉₂	1	1	yes	characteristic
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	292	2	no	—
311	Q₁₆	—	292	2	no	—
312	Q₁₆	—	584	1	no	—
313	Q₁₆	—	584	2	no	—
314	Q₁₆	—	146	2	no	—
315	SD₁₆	—	2	2	no	—
316	SD₁₆	—	292	4	no	—
317	SD₁₆	—	292	4	no	—
318	D₈	—	292	2	no	—
319	D₈	—	584	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	4	2	no	—
324	C₂×C₈	—	146	2	no	—
325	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
326	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
327	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
328	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	292	2	no	—
331	C₄⋊C₄	—	292	2	no	—
332	C₄⋊C₄	—	292	2	no	—
333	C₄⋊C₄	—	73	2	no	—
334	C₄⋊C₄	—	146	2	no	—
335	C₄⋊C₄	—	146	2	no	—
336	C₄⋊C₄	—	146	2	no	—
337	C₄⋊C₄	C₂×D₂₉₂	1	1	yes	characteristic
338	C₄⋊C₄	C₂×D₂₉₂	1	1	yes	characteristic
339	C₂²⋊C₄	—	2	1	no	—
340	C₂²⋊C₄	—	292	2	no	—
341	C₄²	—	2	1	no	—
342	C₄²	—	292	2	no	—
343	C₄²	—	146	2	no	—
344	C₄²	C₂³×D₇₃	1	1	yes	characteristic
345	C₂³	—	2	1	no	—
346	C₂³	—	292	2	no	—
347	Q₈	—	2	1	no	—
348	Q₈	—	292	2	no	—
349	Q₈	—	292	2	no	—
350	Q₈	—	146	2	no	—
351	Q₈	—	146	2	no	—
352	Q₈	#14556613	1	2	yes	complemented
353	D₄	—	2	1	no	—
354	D₄	—	2	1	no	—
355	D₄	—	292	2	no	—
356	D₄	—	292	2	no	—
357	D₄	—	292	2	no	—
358	D₄	—	292	2	no	—
359	D₄	—	584	2	no	—
360	D₄	—	584	4	no	—
361	D₄	—	1168	2	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	292	2	no	—
368	C₂×C₄	—	584	4	no	—
369	C₂×C₄	—	146	2	no	—
370	C₂×C₄	—	146	2	no	—
371	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
372	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
373	C₂×C₄	#14556637	1	1	yes	characteristic
374	C₈	—	2	1	no	—
375	C₈	—	2	1	no	—
376	C₈	—	2	1	no	—
377	C₈	—	4	1	no	—
378	C₈	—	4	1	no	—
379	C₈	—	4	1	no	—
380	C₈	—	292	2	no	—
381	C₈	#14556618	1	2	yes	—
382	C₂²	—	4	1	no	—
383	C₂²	—	4	1	no	—
384	C₂²	—	4	1	no	—
385	C₂²	—	584	2	no	—
386	C₂²	—	584	2	no	—
387	C₂²	—	584	2	no	—
388	C₂²	D₄×D₂₉₂	1	1	yes	characteristic
389	C₄	—	2	1	no	—
390	C₄	—	2	2	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	292	2	no	—
394	C₄	—	292	2	no	—
395	C₄	—	584	2	no	—
396	C₄	C₂×#14556613	1	1	yes	characteristic
397	C₄	C₂×#14556618	1	1	yes	characteristic
398	C₄	#17989389	1	1	yes	characteristic
399	C₄	#17989404	1	1	yes	characteristic
400	C₂	—	8	1	no	—
401	C₂	—	1168	2	no	—
402	C₂	#22621552	1	1	yes	characteristic
403	C₂	#22621674	1	1	yes	characteristic
404	C₂	#22621718	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₉₂
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₈×SD₁₆×C₇₃	C₂
socle	C₂²×C₇₃	D₄²
prime radical	#22621691	C₂
hypercenter	C₈×SD₁₆	D₇₃
hypocenter	C₇₃	#19943
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #22622937	C₂	semidirect
2. #22622658	C₂	semidirect
4. #22609537	C₂	semidirect
5. #22609500	C₂	semidirect
6. #22609296	C₂	semidirect
7. #22609291	C₂	semidirect
8. #22609127	C₂	semidirect
9. #22609030	C₂	semidirect
10. #22608829	C₂	semidirect
11. C₈×SD₁₆×C₇₃	C₂	semidirect
145. Q₈×C₇₃	#135	semidirect
213. C₈×SD₁₆	D₇₃	semidirect
214. C₇₃	#19943	semidirect
352. Q₈	#14556613	semidirect

Maximal subgroups

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

subgroup	index
#22622937	2
#22622658	2
#22621691	2
#22609537	2
#22609500	2
#22609296	2
#22609291	2
#22609127	2
#22609030	2
#22608829	2
C₈×SD₁₆×C₇₃	2
#19943	73