Subgroups — #33077966

Order 18688.

660 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622636	C₂	1	2	yes	maximal, complemented
2	#22622324	C₂	1	1	yes	maximal, characteristic, complemented
3	#22621691	C₂	1	1	yes	maximal, characteristic, complemented
4	#22609523	C₂	1	1	yes	maximal, characteristic, complemented
5	#22609472	C₂	1	2	yes	maximal, complemented
6	#22609330	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609286	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609061	C₂	1	2	yes	maximal, complemented
9	#22609005	C₂	1	2	yes	maximal, complemented
10	#22608829	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556613	—	2	1	no	—
13	#17989294	C₂²	1	2	yes	—
14	#17989258	C₂²	1	2	yes	—
15	#17988992	C₂²	1	1	yes	characteristic
16	#17988964	C₂²	1	1	yes	characteristic
17	#17988960	C₂²	1	2	yes	—
18	#17988950	—	2	2	no	—
19	#17988934	C₂²	1	2	yes	—
20	#17988923	—	2	2	no	—
21	#17988890	—	2	2	no	—
22	#17988883	C₂²	1	2	yes	—
23	#17988877	C₂²	1	2	yes	—
24	#17988828	—	2	1	no	—
25	#17988823	—	2	1	no	—
26	#17988788	C₂²	1	2	yes	—
27	#17988782	C₂²	1	2	yes	—
28	#17988780	—	2	2	no	—
29	#17988734	—	2	2	no	—
30	#17988717	C₂²	1	2	yes	—
31	#17988696	—	2	2	no	—
32	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
33	C₂×#14556525	—	2	1	no	—
34	#17988340	C₂²	1	1	yes	characteristic
35	#17988335	C₂²	1	2	yes	—
36	#17988330	C₂²	1	2	yes	—
37	#17988319	C₂²	1	2	yes	—
38	C₄×D₈×C₇₃	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₇₃	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₇₃	C₂²	1	1	yes	characteristic
41	#335×C₇₃	C₂²	1	1	yes	characteristic
42	D₄⋊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	D₄	1	1	yes	characteristic
48	C₂×#11924017	—	2	2	no	—
49	C₂×#11924017	—	4	2	no	—
50	C₂×D₅₈₄	—	2	1	no	—
51	C₂×D₅₈₄	—	2	1	no	—
52	C₂×#11924013	—	2	2	no	—
53	#14556591	C₂³	1	2	yes	—
54	#14556588	C₂³	1	2	yes	—
55	#14556587	—	2	2	no	—
56	#14556582	—	2	2	no	—
57	#14556570	—	2	2	no	—
58	#14556568	—	4	2	no	—
59	#14556565	—	2	2	no	—
60	#14556564	—	4	2	no	—
61	C₄×Dic₁₄₆	C₂³	1	2	yes	—
62	D₄×Dic₇₃	—	2	2	no	—
63	#14556553	—	2	2	no	—
64	C₂×C₄⋊Dic₇₃	—	2	2	no	—
65	#14556525	—	2	2	no	—
66	#14556525	—	2	1	no	—
67	#14556525	—	2	1	no	—
68	#14556525	D₄	1	1	yes	characteristic
69	#14556522	—	2	2	no	—
70	C₄⋊#10596831	C₂³	1	2	yes	—
71	#14556520	—	2	2	no	—
72	#14556516	—	2	2	no	—
73	#14556516	D₄	1	2	yes	—
74	#14556513	—	2	2	no	—
75	#14556513	—	2	2	no	—
76	#14556513	—	2	2	no	—
77	#14556512	—	2	2	no	—
78	#14556511	—	2	2	no	—
79	#14556501	—	2	2	no	—
80	C₂×D₈×C₇₃	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₇₃	—	2	1	no	—
82	C₂²×C₈×C₇₃	—	2	1	no	—
83	C₄×D₄×C₇₃	C₂³	1	1	yes	characteristic
84	C₄×D₄×C₇₃	C₂³	1	1	yes	characteristic
85	#107×C₇₃	D₄	1	1	yes	characteristic
86	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
88	#102×C₇₃	D₄	1	1	yes	characteristic
89	#102×C₇₃	D₄	1	1	yes	characteristic
90	C₂²⋊C₈×C₇₃	—	2	1	no	—
91	C₂²⋊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	C₂²×Dic₇₃	—	4	2	no	—
105	C₂×C₄×D₇₃	—	4	2	no	—
106	Dic₂₉₂	—	2	2	no	—
107	#11924017	—	4	4	no	—
108	#11924017	—	8	4	no	—
109	D₅₈₄	—	4	2	no	—
110	D₅₈₄	—	8	1	no	—
111	#11924013	—	4	4	no	—
112	C₂×#10596831	—	2	2	no	—
113	C₄⋊Dic₇₃	—	2	2	no	—
114	C₄⋊Dic₇₃	—	2	2	no	—
115	C₄⋊Dic₇₃	—	4	2	no	—
116	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
117	#11924007	—	2	2	no	—
118	#11924007	—	4	2	no	—
119	C₂²⋊Dic₇₃	—	4	2	no	—
120	#11924005	—	4	2	no	—
121	C₄×Dic₇₃	—	2	2	no	—
122	C₂×D₄×C₇₃	C₂×D₄	1	1	yes	characteristic
123	C₂×D₄×C₇₃	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₇₃	—	2	1	no	—
125	C₂²×C₄×C₇₃	—	2	1	no	—
126	D₈×C₇₃	—	2	2	no	—
127	C₂×C₈×C₇₃	—	2	1	no	—
128	C₂×C₈×C₇₃	—	2	1	no	—
129	C₂×C₈×C₇₃	—	2	1	no	—
130	C₂×C₈×C₇₃	—	4	2	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	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
138	C₂²⋊C₄×C₇₃	—	2	1	no	—
139	C₂²⋊C₄×C₇₃	—	2	1	no	—
140	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
141	C₂³×C₇₃	—	2	1	no	—
142	C₂³×C₇₃	—	2	1	no	—
143	C₂²×D₇₃	—	4	2	no	—
144	D₄×C₇₃	—	2	1	no	—
145	D₄×C₇₃	—	2	1	no	—
146	D₄×C₇₃	—	2	1	no	—
147	D₄×C₇₃	#135	1	2	yes	complemented
148	C₂×C₄×C₇₃	—	2	1	no	—
149	C₂×C₄×C₇₃	—	2	1	no	—
150	C₂×C₄×C₇₃	—	4	1	no	—
151	C₂×C₄×C₇₃	—	4	1	no	—
152	C₂×C₄×C₇₃	—	4	1	no	—
153	C₂×C₄×C₇₃	—	4	1	no	—
154	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂²⋊D₇₃	—	8	4	no	—
158	C₂²⋊D₇₃	—	16	2	no	—
159	C₂×Dic₇₃	—	2	2	no	—
160	C₂×Dic₇₃	—	2	2	no	—
161	C₂×Dic₇₃	—	4	2	no	—
162	C₂×Dic₇₃	—	16	2	no	—
163	D₂₉₂	—	4	2	no	—
164	D₂₉₂	—	4	2	no	—
165	D₂₉₂	—	4	2	no	—
166	D₂₉₂	—	4	2	no	—
167	D₂₉₂	—	8	2	no	—
168	C₄×D₇₃	—	8	4	no	—
169	Dic₁₄₆	—	2	2	no	—
170	Dic₁₄₆	—	2	2	no	—
171	Dic₁₄₆	—	8	2	no	—
172	C₈×C₇₃	—	2	2	no	—
173	C₈×C₇₃	—	2	1	no	—
174	C₈×C₇₃	—	2	1	no	—
175	C₈×C₇₃	—	2	1	no	—
176	C₈×C₇₃	—	4	1	no	—
177	C₈×C₇₃	—	4	1	no	—
178	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
179	#10596831	—	4	2	no	—
180	C₂²×C₇₃	—	2	2	no	—
181	C₂²×C₇₃	—	4	1	no	—
182	C₂²×C₇₃	—	4	1	no	—
183	C₂²×C₇₃	—	4	1	no	—
184	C₂²×C₇₃	—	4	1	no	—
185	C₂²×C₇₃	—	4	1	no	—
186	C₂²×C₇₃	D₄²	1	1	yes	characteristic
187	C₂×D₇₃	—	8	2	no	—
188	C₂×D₇₃	—	8	2	no	—
189	C₂×D₇₃	—	8	2	no	—
190	C₄×C₇₃	—	2	1	no	—
191	C₄×C₇₃	—	4	1	no	—
192	C₄×C₇₃	—	4	1	no	—
193	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
194	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
195	C₄×C₇₃	#577	1	1	yes	characteristic
196	C₄×C₇₃	#578	1	1	yes	characteristic
197	Dic₇₃	—	4	2	no	—
198	Dic₇₃	—	4	2	no	—
199	Dic₇₃	—	8	2	no	—
200	#19937	—	73	1	no	maximal, Sylow
201	C₂×C₇₃	—	4	2	no	—
202	C₂×C₇₃	—	8	1	no	—
203	C₂×C₇₃	#3272	1	1	yes	characteristic
204	C₂×C₇₃	#3285	1	1	yes	characteristic
205	C₂×C₇₃	#3292	1	1	yes	characteristic
206	D₇₃	—	16	2	no	—
207	#3366	—	73	2	no	—
208	#3338	—	73	1	no	—
209	#3287	—	73	1	no	—
210	#1706	—	73	1	no	—
211	#1698	—	73	2	no	—
212	#1675	—	73	1	no	—
213	#1667	—	73	1	no	—
214	#1631	—	73	2	no	—
215	#1622	—	73	2	no	—
216	#1591	—	73	1	no	—
217	C₈×D₈	D₇₃	1	1	yes	characteristic, complemented
218	C₇₃	#19937	1	1	yes	characteristic, complemented, Sylow
219	C₂×#135	—	146	1	no	—
220	#554	—	73	2	no	—
221	#550	—	73	2	no	—
222	#500	—	73	1	no	—
223	#493	—	73	1	no	—
224	#492	—	73	2	no	—
225	#490	—	146	2	no	—
226	#486	—	73	2	no	—
227	#484	—	146	2	no	—
228	#478	—	146	2	no	—
229	#477	—	73	2	no	—
230	#476	—	73	2	no	—
231	#467	—	146	1	no	—
232	#466	—	146	1	no	—
233	#460	—	73	2	no	—
234	#459	—	73	2	no	—
235	#459	—	146	2	no	—
236	#449	—	146	2	no	—
237	#446	—	73	2	no	—
238	#442	—	146	2	no	—
239	C₄×Q₁₆	—	73	1	no	—
240	C₄×D₈	C₂×D₇₃	1	1	yes	characteristic
241	C₈×D₄	C₂×D₇₃	1	1	yes	characteristic
242	C₈×D₄	C₂×D₇₃	1	1	yes	characteristic
243	C₂×#107	—	146	1	no	—
244	#335	—	73	1	no	—
245	#335	C₂×D₇₃	1	1	yes	characteristic
246	#333	—	73	2	no	—
247	#331	—	73	2	no	—
248	Q₈⋊C₈	—	73	2	no	—
249	D₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
250	D₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
251	C₈²	C₂×D₇₃	1	1	yes	characteristic
252	C₂×#41	—	146	2	no	—
253	#135	—	292	4	no	—
254	C₂×Q₁₆	—	73	1	no	—
255	C₂×SD₁₆	—	292	2	no	—
256	C₂×SD₁₆	—	146	2	no	—
257	C₂×D₈	—	146	2	no	—
258	C₂×D₈	—	146	1	no	—
259	C₂×D₈	—	146	1	no	—
260	C₂×D₈	D₂₉₂	1	1	yes	characteristic
261	C₂²×C₈	—	2	1	no	—
262	C₂²×C₈	—	2	1	no	—
263	C₄⋊Q₈	—	73	2	no	—
264	#127	—	73	2	no	—
265	#127	—	146	2	no	—
266	#125	—	146	2	no	—
267	C₂²⋊Q₈	—	292	2	no	—
268	C₂²⋊Q₈	—	146	2	no	—
269	#121	—	292	2	no	—
270	#121	—	146	2	no	—
271	C₄×Q₈	—	73	2	no	—
272	C₄×D₄	—	146	2	no	—
273	C₄×D₄	—	146	2	no	—
274	C₄×D₄	C₂²×D₇₃	1	1	yes	characteristic
275	C₄×D₄	C₂²×D₇₃	1	1	yes	characteristic
276	C₂×C₄⋊C₄	—	146	2	no	—
277	#107	—	73	1	no	—
278	#107	—	146	2	no	—
279	#107	—	146	1	no	—
280	#107	—	146	1	no	—
281	#107	D₂₉₂	1	1	yes	characteristic
282	#106	—	146	2	no	—
283	C₄⋊C₈	—	73	2	no	—
284	C₄⋊C₈	—	146	2	no	—
285	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
286	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
287	#103	—	73	2	no	—
288	#103	—	146	2	no	—
289	#102	—	146	2	no	—
290	#102	—	146	2	no	—
291	#102	—	146	2	no	—
292	#102	—	146	2	no	—
293	#102	—	146	2	no	—
294	#102	D₂₉₂	1	1	yes	characteristic
295	#102	D₂₉₂	1	1	yes	characteristic
296	C₂²⋊C₈	—	2	1	no	—
297	C₂²⋊C₈	—	2	1	no	—
298	#97	—	146	2	no	—
299	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
300	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
301	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
302	#41	—	292	4	no	—
303	#41	—	584	2	no	—
304	C₂×Q₈	—	73	2	no	—
305	C₂×Q₈	—	146	2	no	—
306	C₂×D₄	—	292	2	no	—
307	C₂×D₄	—	292	2	no	—
308	C₂×D₄	—	146	2	no	—
309	C₂×D₄	—	146	2	no	—
310	C₂×D₄	—	146	2	no	—
311	C₂×D₄	C₂×D₂₉₂	1	1	yes	characteristic
312	C₂×D₄	C₂×D₂₉₂	1	1	yes	characteristic
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	292	2	no	—
316	C₂²×C₄	—	292	2	no	—
317	Q₁₆	—	146	2	no	—
318	SD₁₆	—	292	4	no	—
319	SD₁₆	—	584	4	no	—
320	D₈	—	2	2	no	—
321	D₈	—	292	4	no	—
322	D₈	—	292	2	no	—
323	D₈	—	584	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	4	2	no	—
328	C₂×C₈	—	4	2	no	—
329	C₂×C₈	—	146	2	no	—
330	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
331	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
332	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
333	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
334	C₄⋊C₄	—	292	2	no	—
335	C₄⋊C₄	—	292	2	no	—
336	C₄⋊C₄	—	73	2	no	—
337	C₄⋊C₄	—	146	2	no	—
338	C₄⋊C₄	—	146	2	no	—
339	C₄⋊C₄	—	146	2	no	—
340	C₄⋊C₄	C₂×D₂₉₂	1	1	yes	characteristic
341	C₄⋊C₄	C₂×D₂₉₂	1	1	yes	characteristic
342	C₂²⋊C₄	—	2	1	no	—
343	C₂²⋊C₄	—	2	1	no	—
344	C₂²⋊C₄	—	292	2	no	—
345	C₂²⋊C₄	—	292	2	no	—
346	C₄²	—	146	2	no	—
347	C₄²	C₂³×D₇₃	1	1	yes	characteristic
348	C₂³	—	2	1	no	—
349	C₂³	—	2	1	no	—
350	C₂³	—	292	2	no	—
351	Q₈	—	584	2	no	—
352	Q₈	—	146	2	no	—
353	Q₈	—	146	2	no	—
354	D₄	—	2	1	no	—
355	D₄	—	2	1	no	—
356	D₄	—	2	1	no	—
357	D₄	—	292	2	no	—
358	D₄	—	292	2	no	—
359	D₄	—	292	2	no	—
360	D₄	—	292	2	no	—
361	D₄	—	584	2	no	—
362	D₄	—	584	4	no	—
363	D₄	—	1168	2	no	—
364	D₄	#14556613	1	2	yes	complemented
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	292	2	no	—
372	C₂×C₄	—	584	4	no	—
373	C₂×C₄	—	1168	2	no	—
374	C₂×C₄	—	146	2	no	—
375	C₂×C₄	—	146	2	no	—
376	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
377	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
378	C₂×C₄	#14556637	1	1	yes	characteristic
379	C₈	—	2	2	no	—
380	C₈	—	2	1	no	—
381	C₈	—	2	1	no	—
382	C₈	—	2	1	no	—
383	C₈	—	4	1	no	—
384	C₈	—	4	1	no	—
385	C₈	—	292	2	no	—
386	C₈	#14556615	1	2	yes	—
387	C₂²	—	2	2	no	—
388	C₂²	—	4	1	no	—
389	C₂²	—	4	1	no	—
390	C₂²	—	4	1	no	—
391	C₂²	—	4	1	no	—
392	C₂²	—	4	1	no	—
393	C₂²	—	584	2	no	—
394	C₂²	—	584	2	no	—
395	C₂²	—	584	2	no	—
396	C₂²	D₄×D₂₉₂	1	1	yes	characteristic
397	C₄	—	2	1	no	—
398	C₄	—	4	1	no	—
399	C₄	—	4	1	no	—
400	C₄	—	292	2	no	—
401	C₄	—	292	2	no	—
402	C₄	—	584	2	no	—
403	C₄	C₂×#14556613	1	1	yes	characteristic
404	C₄	C₂×#14556615	1	1	yes	characteristic
405	C₄	#17989394	1	1	yes	characteristic
406	C₄	#17989404	1	1	yes	characteristic
407	C₂	—	4	2	no	—
408	C₂	—	8	1	no	—
409	C₂	—	1168	2	no	—
410	C₂	#22621532	1	1	yes	characteristic
411	C₂	#22621674	1	1	yes	characteristic
412	C₂	#22621731	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₉₂
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₈×D₈×C₇₃	C₂
socle	C₂²×C₇₃	D₄²
hypercenter	C₈×D₈	D₇₃
hypocenter	C₇₃	#19937
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈

Complements

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

N	H ≅ G/N	type
1. #22622636	C₂	semidirect
2. #22622324	C₂	semidirect
3. #22621691	C₂	semidirect
4. #22609523	C₂	semidirect
5. #22609472	C₂	semidirect
6. #22609330	C₂	semidirect
7. #22609286	C₂	semidirect
8. #22609061	C₂	semidirect
9. #22609005	C₂	semidirect
10. #22608829	C₂	semidirect
11. C₈×D₈×C₇₃	C₂	semidirect
147. D₄×C₇₃	#135	semidirect
217. C₈×D₈	D₇₃	semidirect
218. C₇₃	#19937	semidirect
364. D₄	#14556613	semidirect

Maximal subgroups

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

subgroup	index
#22622636	2
#22622324	2
#22621691	2
#22609523	2
#22609472	2
#22609330	2
#22609286	2
#22609061	2
#22609005	2
#22608829	2
C₈×D₈×C₇₃	2
#19937	73