Subgroups — #33112557

Order 18688.

756 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620384	C₂	1	1	yes	maximal, characteristic
2	#22610421	C₂	1	2	yes	maximal, complemented
3	#22610415	C₂	1	2	yes	maximal, complemented
4	#22610411	C₂	1	2	yes	maximal, complemented
5	#22610409	C₂	1	2	yes	maximal, complemented
6	#22610328	C₂	1	2	yes	maximal, complemented
7	#22610315	C₂	1	1	yes	maximal, characteristic, complemented
8	#22610077	C₂	1	2	yes	maximal, complemented
9	C₇₃×#1884	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14556616	C₂²	1	2	yes	—
11	C₂×#14556616	C₂²	1	2	yes	—
12	#17989354	C₂²	1	1	yes	characteristic
13	C₂×#14556588	C₂²	1	1	yes	characteristic
14	#17989061	C₂²	1	2	yes	—
15	#17988977	C₄	1	2	yes	complemented
16	#17988977	C₄	1	2	yes	complemented
17	#17988963	C₄	1	2	yes	complemented
18	#17988961	C₄	1	2	yes	complemented
19	#17988604	C₂²	1	2	yes	—
20	#17988593	C₂²	1	2	yes	—
21	#17988591	C₂²	1	2	yes	—
22	C₂×#14556507	C₂²	1	2	yes	—
23	#17988561	C₂²	1	1	yes	characteristic
24	C₄×#11924012	C₂²	1	1	yes	characteristic
25	#17988359	C₂²	1	2	yes	—
26	#17988357	C₂²	1	2	yes	—
27	#17988347	C₂²	1	2	yes	—
28	#17988345	C₂²	1	2	yes	—
29	#17988343	C₂²	1	2	yes	—
30	#568×C₇₃	C₂²	1	1	yes	characteristic
31	#532×C₇₃	C₂²	1	1	yes	characteristic
32	#421×C₇₃	C₂²	1	2	yes	—
33	#410×C₇₃	C₂²	1	1	yes	characteristic
34	#339×C₇₃	C₂²	1	2	yes	—
35	C₂²×D₂₉₂	—	4	1	no	—
36	C₂²×D₂₉₂	C₂³	1	2	yes	—
37	#14556616	—	4	2	no	—
38	#14556616	—	4	2	no	—
39	#14556616	—	4	2	no	—
40	#14556616	—	4	2	no	—
41	C₂×#11924015	C₂×C₄	1	4	yes	complemented
42	C₂×#11924015	C₂×C₄	1	4	yes	complemented
43	C₂×#11924013	C₂×C₄	1	4	yes	complemented
44	C₂×#11924013	C₂×C₄	1	4	yes	complemented
45	#14556598	—	4	2	no	—
46	C₂×#11924012	—	4	1	no	—
47	C₂×#11924012	C₂³	1	2	yes	—
48	C₂×#11924012	C₂³	1	2	yes	—
49	#14556588	—	4	1	no	—
50	#14556588	C₂×C₄	1	2	yes	—
51	#14556588	C₂×C₄	1	2	yes	—
52	C₄×D₂₉₂	—	4	2	no	—
53	#14556553	—	4	2	no	—
54	#14556550	C₂³	1	2	yes	—
55	C₂²⋊C₄×D₇₃	—	4	2	no	—
56	#14556518	—	4	2	no	—
57	#14556513	—	4	2	no	—
58	#14556511	—	4	2	no	—
59	#14556507	—	4	2	no	—
60	#14556506	—	4	1	no	—
61	#14556502	C₂×C₄	1	2	yes	complemented
62	C₄×#10596831	C₂×C₄	1	2	yes	complemented
63	C₂×#41×C₇₃	C₂³	1	1	yes	characteristic
64	C₂×#41×C₇₃	C₂³	1	1	yes	characteristic
65	#131×C₇₃	—	4	2	no	—
66	C₂×M₁₆×C₇₃	—	4	1	no	—
67	C₂×M₁₆×C₇₃	C₂³	1	2	yes	—
68	C₄⋊Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
69	#127×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
70	#124×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
71	#124×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
72	#121×C₇₃	—	4	1	no	—
73	C₄×D₄×C₇₃	—	4	1	no	—
74	#117×C₇₃	C₂³	1	2	yes	—
75	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
76	#104×C₇₃	—	4	2	no	—
77	#99×C₇₃	—	4	1	no	—
78	C₂³×D₇₃	—	2	2	no	—
79	C₂×D₂₉₂	—	2	2	no	—
80	C₂×D₂₉₂	—	2	2	no	—
81	C₂×D₂₉₂	—	2	2	no	—
82	C₂×D₂₉₂	—	2	2	no	—
83	C₂×D₂₉₂	—	2	2	no	—
84	C₂×D₂₉₂	—	4	2	no	—
85	C₂×D₂₉₂	—	4	2	no	—
86	C₂×D₂₉₂	—	8	4	no	—
87	C₂×D₂₉₂	C₂²×C₄	1	4	yes	—
88	C₂×C₄×D₇₃	—	4	4	no	—
89	#11924015	—	4	4	no	—
90	#11924015	—	4	4	no	—
91	#11924013	—	4	4	no	—
92	#11924013	—	4	4	no	—
93	#11924012	—	2	2	no	—
94	#11924012	—	2	2	no	—
95	#11924012	—	2	2	no	—
96	#11924012	—	2	2	no	—
97	#11924012	—	4	2	no	—
98	C₂×#10596831	—	2	2	no	—
99	C₂×#10596831	—	4	2	no	—
100	C₂×#10596831	C₂²×C₄	1	4	yes	—
101	C₄⋊Dic₇₃	—	2	2	no	—
102	C₂²⋊Dic₇₃	—	2	2	no	—
103	#11924005	—	4	4	no	—
104	C₄×Dic₇₃	—	2	2	no	—
105	#41×C₇₃	—	2	2	no	—
106	#41×C₇₃	—	2	2	no	—
107	#41×C₇₃	—	2	2	no	—
108	#41×C₇₃	—	2	2	no	—
109	C₂×Q₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×Q₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×D₄×C₇₃	—	2	1	no	—
112	C₂×D₄×C₇₃	—	2	1	no	—
113	C₂×D₄×C₇₃	—	4	1	no	—
114	C₂×D₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×D₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₇₃	—	2	1	no	—
117	C₂²×C₄×C₇₃	—	2	1	no	—
118	C₂²×C₄×C₇₃	—	2	1	no	—
119	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
120	M₁₆×C₇₃	—	2	2	no	—
121	M₁₆×C₇₃	—	2	2	no	—
122	M₁₆×C₇₃	—	4	2	no	—
123	C₂×C₈×C₇₃	—	2	2	no	—
124	C₂×C₈×C₇₃	—	4	2	no	—
125	C₄⋊C₄×C₇₃	—	2	2	no	—
126	C₄⋊C₄×C₇₃	—	4	1	no	—
127	C₂²⋊C₄×C₇₃	—	2	2	no	—
128	C₂²⋊C₄×C₇₃	—	4	1	no	—
129	C₂²⋊C₄×C₇₃	—	4	1	no	—
130	C₄²×C₇₃	—	2	1	no	—
131	C₄²×C₇₃	—	2	2	no	—
132	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
134	C₂³×C₇₃	—	2	1	no	—
135	C₂³×C₇₃	—	2	1	no	—
136	C₂³×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂²×D₇₃	—	2	4	no	—
138	C₂²×D₇₃	—	4	2	no	—
139	C₂²×D₇₃	—	4	2	no	—
140	C₂²×D₇₃	—	8	4	no	—
141	Q₈×C₇₃	—	2	2	no	—
142	Q₈×C₇₃	—	2	2	no	—
143	D₄×C₇₃	—	2	2	no	—
144	D₄×C₇₃	—	2	2	no	—
145	D₄×C₇₃	—	4	2	no	—
146	D₄×C₇₃	—	4	2	no	—
147	D₄×C₇₃	—	8	2	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	—	2	1	no	—
152	C₂×C₄×C₇₃	—	4	2	no	—
153	C₂×C₄×C₇₃	—	4	1	no	—
154	C₂×C₄×C₇₃	—	4	1	no	—
155	C₂×C₄×C₇₃	—	4	1	no	—
156	C₂×C₄×C₇₃	—	4	1	no	—
157	C₂×C₄×C₇₃	—	4	2	no	—
158	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₇₃	H(5,𝔽₂)	1	2	yes	—
160	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂×C₄×C₇₃	H(5,𝔽₂)	1	2	yes	—
162	C₂×Dic₇₃	—	4	2	no	—
163	D₂₉₂	—	2	4	no	—
164	D₂₉₂	—	2	4	no	—
165	D₂₉₂	—	4	2	no	—
166	D₂₉₂	—	4	2	no	—
167	D₂₉₂	—	8	4	no	—
168	D₂₉₂	—	8	4	no	—
169	C₄×D₇₃	—	16	4	no	—
170	C₈×C₇₃	—	4	2	no	—
171	#10596831	—	2	4	no	—
172	#10596831	—	4	2	no	—
173	C₂²×C₇₃	—	4	1	no	—
174	C₂²×C₇₃	—	4	1	no	—
175	C₂²×C₇₃	—	4	1	no	—
176	C₂²×C₇₃	—	8	1	no	—
177	C₂²×C₇₃	—	8	1	no	—
178	C₂²×C₇₃	#518	1	1	yes	characteristic
179	C₂²×C₇₃	#560	1	2	yes	—
180	C₂×D₇₃	—	4	4	no	—
181	C₂×D₇₃	—	8	4	no	—
182	C₂×D₇₃	—	8	4	no	—
183	C₂×D₇₃	—	16	2	no	—
184	C₄×C₇₃	—	2	2	no	—
185	C₄×C₇₃	—	4	1	no	—
186	C₄×C₇₃	—	4	1	no	—
187	C₄×C₇₃	—	4	1	no	—
188	C₄×C₇₃	—	8	2	no	—
189	C₄×C₇₃	#518	1	2	yes	—
190	C₄×C₇₃	#560	1	1	yes	characteristic
191	C₄×C₇₃	#560	1	1	yes	characteristic
192	Dic₇₃	—	8	2	no	—
193	#23361	—	73	1	no	maximal, Sylow
194	C₂×C₇₃	—	2	2	no	—
195	C₂×C₇₃	—	2	1	no	—
196	C₂×C₇₃	—	8	1	no	—
197	C₂×C₇₃	—	8	1	no	—
198	C₂×C₇₃	#2380	1	1	yes	characteristic
199	D₇₃	—	8	4	no	—
200	#3151	—	73	1	no	—
201	#1904	—	73	2	no	—
202	#1904	—	73	2	no	—
203	#1903	—	73	2	no	—
204	#1903	—	73	2	no	—
205	#1887	—	73	2	no	—
206	#1884	—	73	1	no	—
207	#1884	D₇₃	1	1	yes	characteristic, complemented
208	#1841	—	73	2	no	—
209	C₇₃	#23361	1	1	yes	characteristic, complemented, Sylow
210	C₂×AGL(1,ℤ/8)	—	73	2	no	—
211	C₂×AGL(1,ℤ/8)	—	73	2	no	—
212	#568	—	73	1	no	—
213	#568	C₂×D₇₃	1	1	yes	characteristic
214	#532	C₂×D₇₃	1	1	yes	characteristic
215	C₂×#127	—	73	1	no	—
216	#518	—	73	2	no	—
217	#496	—	73	2	no	—
218	#496	—	73	2	no	—
219	#493	—	73	2	no	—
220	#492	—	73	2	no	—
221	#421	—	73	2	no	—
222	#421	C₂×D₇₃	1	2	yes	—
223	#418	—	73	2	no	—
224	#418	—	73	2	no	—
225	C₂×#100	—	73	2	no	—
226	#410	—	73	1	no	—
227	#410	C₂×D₇₃	1	1	yes	characteristic
228	C₄×M₁₆	—	73	1	no	—
229	#344	—	73	2	no	—
230	#344	—	73	2	no	—
231	#339	—	73	2	no	—
232	#339	C₂×D₇₃	1	2	yes	—
233	#338	—	73	2	no	—
234	H(3,ℤ/4)	—	73	2	no	—
235	C₂×#41	C₂²×D₇₃	1	1	yes	characteristic
236	C₂×#41	C₂²×D₇₃	1	1	yes	characteristic
237	C₂²×D₄	—	292	1	no	—
238	C₂²×D₄	—	73	2	no	—
239	AGL(1,ℤ/8)	—	292	2	no	—
240	AGL(1,ℤ/8)	—	292	2	no	—
241	AGL(1,ℤ/8)	—	292	2	no	—
242	AGL(1,ℤ/8)	—	292	2	no	—
243	C₂×SD₁₆	—	73	4	no	—
244	C₂×SD₁₆	—	73	4	no	—
245	C₂×D₈	—	73	4	no	—
246	C₂×D₈	—	73	4	no	—
247	#131	—	4	2	no	—
248	#131	—	292	2	no	—
249	C₂×M₁₆	—	4	1	no	—
250	C₂×M₁₆	—	292	1	no	—
251	C₂×M₁₆	—	73	2	no	—
252	C₂×M₁₆	—	73	2	no	—
253	C₂×M₁₆	C₂²×D₇₃	1	2	yes	—
254	C₄⋊Q₈	C₄×D₇₃	1	1	yes	characteristic, complemented
255	#127	—	292	1	no	—
256	#127	—	73	2	no	—
257	#127	—	73	2	no	—
258	#127	C₄×D₇₃	1	1	yes	characteristic, complemented
259	#124	C₄×D₇₃	1	1	yes	characteristic, complemented
260	#124	C₄×D₇₃	1	1	yes	characteristic, complemented
261	#121	—	4	1	no	—
262	C₄×D₄	—	4	1	no	—
263	C₄×D₄	—	292	2	no	—
264	C₄×D₄	—	292	2	no	—
265	#117	—	73	2	no	—
266	#117	C₂²×D₇₃	1	2	yes	—
267	C₂×C₂²⋊C₄	—	292	2	no	—
268	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
269	#104	—	4	2	no	—
270	#104	—	292	2	no	—
271	#102	—	292	2	no	—
272	#102	—	292	2	no	—
273	#100	—	292	2	no	—
274	#99	—	4	1	no	—
275	#99	—	292	1	no	—
276	#97	—	73	2	no	—
277	C₄×C₈	—	73	2	no	—
278	C₂⁴	—	146	2	no	—
279	#41	—	2	2	no	—
280	#41	—	2	2	no	—
281	#41	—	2	2	no	—
282	#41	—	2	2	no	—
283	C₂×Q₈	C₂×C₄×D₇₃	1	1	yes	characteristic
284	C₂×Q₈	C₂×C₄×D₇₃	1	1	yes	characteristic
285	C₂×D₄	—	2	1	no	—
286	C₂×D₄	—	2	1	no	—
287	C₂×D₄	—	4	1	no	—
288	C₂×D₄	—	292	2	no	—
289	C₂×D₄	—	292	2	no	—
290	C₂×D₄	—	584	4	no	—
291	C₂×D₄	—	73	4	no	—
292	C₂×D₄	—	146	2	no	—
293	C₂×D₄	—	146	2	no	—
294	C₂×D₄	—	146	2	no	—
295	C₂×D₄	—	146	2	no	—
296	C₂×D₄	—	146	2	no	—
297	C₂×D₄	C₂×C₄×D₇₃	1	1	yes	characteristic
298	C₂×D₄	C₂×C₄×D₇₃	1	1	yes	characteristic
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	2	1	no	—
302	C₂²×C₄	—	292	4	no	—
303	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
304	SD₁₆	—	292	4	no	—
305	SD₁₆	—	292	4	no	—
306	D₈	—	292	4	no	—
307	D₈	—	292	4	no	—
308	M₁₆	—	2	2	no	—
309	M₁₆	—	2	2	no	—
310	M₁₆	—	4	2	no	—
311	M₁₆	—	292	2	no	—
312	M₁₆	—	146	2	no	—
313	M₁₆	—	146	2	no	—
314	M₁₆	—	146	2	no	—
315	M₁₆	—	146	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	4	2	no	—
318	C₂×C₈	—	292	2	no	—
319	C₂×C₈	—	73	4	no	—
320	C₂×C₈	—	146	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	4	1	no	—
323	C₄⋊C₄	—	146	2	no	—
324	C₂²⋊C₄	—	2	2	no	—
325	C₂²⋊C₄	—	4	1	no	—
326	C₂²⋊C₄	—	4	1	no	—
327	C₂²⋊C₄	—	292	4	no	—
328	C₂²⋊C₄	—	146	2	no	—
329	C₄²	—	2	1	no	—
330	C₄²	—	2	2	no	—
331	C₄²	—	146	2	no	—
332	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
333	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
334	C₂³	—	2	1	no	—
335	C₂³	—	2	1	no	—
336	C₂³	—	292	2	no	—
337	C₂³	—	292	2	no	—
338	C₂³	—	584	4	no	—
339	C₂³	—	146	4	no	—
340	C₂³	#14556636	1	1	yes	characteristic
341	Q₈	—	2	2	no	—
342	Q₈	—	2	2	no	—
343	D₄	—	2	2	no	—
344	D₄	—	2	2	no	—
345	D₄	—	4	2	no	—
346	D₄	—	4	2	no	—
347	D₄	—	8	2	no	—
348	D₄	—	292	2	no	—
349	D₄	—	292	2	no	—
350	D₄	—	584	4	no	—
351	D₄	—	584	4	no	—
352	D₄	—	146	4	no	—
353	D₄	—	146	4	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	292	2	no	—
365	C₂×C₄	—	1168	4	no	—
366	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
367	C₂×C₄	#14556636	1	2	yes	—
368	C₂×C₄	#14556636	1	1	yes	characteristic
369	C₂×C₄	#14556637	1	2	yes	—
370	C₈	—	4	2	no	—
371	C₈	—	292	2	no	—
372	C₈	—	146	4	no	—
373	C₂²	—	4	1	no	—
374	C₂²	—	4	1	no	—
375	C₂²	—	4	1	no	—
376	C₂²	—	8	1	no	—
377	C₂²	—	8	1	no	—
378	C₂²	—	292	4	no	—
379	C₂²	—	584	4	no	—
380	C₂²	—	584	4	no	—
381	C₂²	—	1168	2	no	—
382	C₂²	#17989060	1	1	yes	characteristic
383	C₂²	#17989325	1	2	yes	—
384	C₄	—	2	2	no	—
385	C₄	—	4	1	no	—
386	C₄	—	4	1	no	—
387	C₄	—	4	1	no	—
388	C₄	—	8	2	no	—
389	C₄	—	584	2	no	—
390	C₄	#17989059	1	2	yes	—
391	C₄	#17989326	1	1	yes	characteristic
392	C₄	#17989327	1	1	yes	characteristic
393	C₂	—	2	1	no	—
394	C₂	—	2	2	no	—
395	C₂	—	8	1	no	—
396	C₂	—	8	1	no	—
397	C₂	—	584	4	no	—
398	C₂	#22613106	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22613106
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#1884	C₂
socle	C₂×C₇₃	#2380
prime radical	#22620384	C₂
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1884	D₇₃
hypocenter	C₇₃	#23361
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1884

Complements

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

N	H ≅ G/N	type
2. #22610421	C₂	semidirect
3. #22610415	C₂	semidirect
4. #22610411	C₂	semidirect
5. #22610409	C₂	semidirect
6. #22610328	C₂	semidirect
7. #22610315	C₂	semidirect
8. #22610077	C₂	semidirect
9. C₇₃×#1884	C₂	semidirect
15. #17988977	C₄	semidirect
16. #17988977	C₄	semidirect
17. #17988963	C₄	semidirect
18. #17988961	C₄	semidirect
41. C₂×#11924015	C₂×C₄	semidirect
42. C₂×#11924015	C₂×C₄	semidirect
43. C₂×#11924013	C₂×C₄	semidirect
44. C₂×#11924013	C₂×C₄	semidirect
61. #14556502	C₂×C₄	semidirect
62. C₄×#10596831	C₂×C₄	semidirect
68. C₄⋊Q₈×C₇₃	C₂×C₄	semidirect
69. #127×C₇₃	C₂×C₄	semidirect
70. #124×C₇₃	C₂×C₄	semidirect
71. #124×C₇₃	C₂×C₄	semidirect
207. #1884	D₇₃	semidirect
209. C₇₃	#23361	semidirect
254. C₄⋊Q₈	C₄×D₇₃	semidirect
258. #127	C₄×D₇₃	semidirect
259. #124	C₄×D₇₃	semidirect
260. #124	C₄×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22620384	2
#22610421	2
#22610415	2
#22610411	2
#22610409	2
#22610328	2
#22610315	2
#22610077	2
C₇₃×#1884	2
#23361	73