Subgroups — #34030116

Order 18688.

610 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626299	C₂	1	2	yes	maximal, complemented
2	#22626298	C₂	1	2	yes	maximal, complemented
3	#22626297	C₂	1	2	yes	maximal, complemented
4	#22626296	C₂	1	2	yes	maximal, complemented
5	#22626261	C₂	1	2	yes	maximal, complemented
6	#22626156	C₂	1	2	yes	maximal
7	#22626132	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#14556664	C₂	1	1	yes	maximal, characteristic, complemented
9	#405×D₇₃	C₂	1	1	yes	maximal, characteristic
10	C₂×#14556664	C₂²	1	2	yes	—
11	C₂×#14556664	C₂²	1	2	yes	—
12	#17989508	C₂²	1	4	yes	complemented
13	C₄⋊#11924031	C₂²	1	4	yes	complemented
14	#17989482	C₂²	1	4	yes	—
15	C₂²⋊#11924031	C₂²	1	4	yes	—
16	#17989476	C₂²	1	2	yes	complemented
17	#17989475	C₂²	1	2	yes	complemented
18	#17989473	C₂²	1	2	yes	complemented
19	C₄×#11924031	C₂²	1	2	yes	complemented
20	C₂×M₁₆×D₇₃	C₄	1	1	yes	characteristic
21	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
22	#117×D₇₃	C₄	1	1	yes	characteristic
23	#17988545	C₂²	1	1	yes	characteristic
24	#17988544	C₂²	1	1	yes	characteristic
25	#17988544	C₂²	1	1	yes	characteristic
26	#17988543	C₂²	1	1	yes	characteristic
27	#17988542	C₄	1	1	yes	characteristic
28	#17988541	C₄	1	1	yes	characteristic
29	#17988541	C₄	1	1	yes	characteristic
30	#97×D₇₃	C₄	1	1	yes	characteristic
31	#97×D₇₃	C₄	1	1	yes	characteristic
32	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
34	#405×C₇₃	C₄	1	1	yes	characteristic
35	#14556664	—	2	4	no	—
36	#14556664	—	2	4	no	—
37	C₂×#11924031	C₂³	1	4	yes	—
38	C₂×#11924031	C₂³	1	4	yes	—
39	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
41	C₂×#11924011	C₂×C₄	1	1	yes	characteristic
42	C₂×#11924011	C₂×C₄	1	1	yes	characteristic
43	M₁₆×D₇₃	—	2	4	no	—
44	C₂²×#10596831	C₂³	1	1	yes	characteristic
45	C₂×C₈×D₇₃	—	2	4	no	—
46	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
52	#14556550	C₂×C₄	1	1	yes	characteristic
53	#14556549	C₈	1	2	yes	—
54	#14556548	C₂×C₄	1	1	yes	characteristic
55	#14556548	C₂×C₄	1	1	yes	characteristic
56	#14556547	C₈	1	2	yes	—
57	C₄⋊C₄×D₇₃	C₈	1	2	yes	—
58	C₂²⋊C₄×D₇₃	C₈	1	2	yes	—
59	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
60	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
61	#14556502	C₂×C₄	1	1	yes	characteristic
62	#14556502	C₂×C₄	1	1	yes	characteristic
63	#14556501	C₂×C₄	1	1	yes	characteristic
64	#14556501	C₂×C₄	1	1	yes	characteristic
65	#14556501	C₂×C₄	1	1	yes	characteristic
66	#14556501	C₂×C₄	1	1	yes	characteristic
67	C₄×#10596831	C₂³	1	1	yes	characteristic
68	C₄×#10596831	C₂³	1	1	yes	characteristic
69	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
72	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
75	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
76	#97×C₇₃	C₂×C₄	1	1	yes	characteristic
77	#97×C₇₃	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
79	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
80	#11924031	—	2	4	no	—
81	#11924031	—	2	4	no	—
82	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
83	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₇₃	—	2	2	no	—
85	C₂×C₄×D₇₃	—	2	2	no	—
86	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
87	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
88	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
89	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
90	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
92	#11924012	—	2	2	no	—
93	#11924011	—	2	2	no	—
94	#11924011	—	2	2	no	—
95	C₂×#10596831	—	2	2	no	—
96	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
97	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
98	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
99	C₂×#10596831	C₂⁴	1	1	yes	characteristic
100	C₈×D₇₃	—	2	2	no	—
101	C₈×D₇₃	—	2	2	no	—
102	C₈×D₇₃	—	2	2	no	—
103	C₈×D₇₃	—	2	2	no	—
104	C₈×D₇₃	M₁₆	1	4	yes	—
105	C₈×D₇₃	#41	1	4	yes	—
106	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
107	#11924007	C₂×C₈	1	2	yes	—
108	#11924007	C₂×C₈	1	2	yes	—
109	C₂²⋊Dic₇₃	C₂×C₈	1	2	yes	—
110	#11924005	C₂×C₈	1	2	yes	—
111	#11924005	C₂×C₈	1	2	yes	—
112	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
113	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
114	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
115	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
117	M₁₆×C₇₃	—	2	2	no	—
118	C₂×C₈×C₇₃	—	2	2	no	—
119	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
123	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
124	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
125	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
127	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
128	C₂²×D₇₃	—	2	2	no	—
129	C₂²×D₇₃	—	2	4	no	—
130	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
131	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₇₃	—	2	1	no	—
133	C₂×C₄×C₇₃	—	2	1	no	—
134	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×Dic₇₃	—	2	2	no	—
139	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
140	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
141	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
142	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
143	C₄×D₇₃	—	2	2	no	—
144	C₄×D₇₃	—	2	4	no	—
145	C₄×D₇₃	—	2	4	no	—
146	C₄×D₇₃	—	2	1	no	—
147	C₄×D₇₃	—	2	1	no	—
148	C₄×D₇₃	—	2	2	no	—
149	C₄×D₇₃	#117	1	2	yes	—
150	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
151	C₈×C₇₃	—	2	1	no	—
152	C₈×C₇₃	—	2	1	no	—
153	C₈×C₇₃	#117	1	2	yes	—
154	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
155	#10596831	—	2	1	no	—
156	#10596831	—	2	1	no	—
157	#10596831	C₂×M₁₆	1	2	yes	—
158	#10596831	C₂×#41	1	2	yes	—
159	C₂²×C₇₃	—	2	2	no	—
160	C₂²×C₇₃	—	2	1	no	—
161	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
162	C₂×D₇₃	—	2	2	no	—
163	C₂×D₇₃	—	2	1	no	—
164	C₂×D₇₃	—	4	4	no	—
165	C₂×D₇₃	#431	1	2	yes	—
166	C₂×D₇₃	#504	1	4	yes	—
167	C₄×C₇₃	—	2	2	no	—
168	C₄×C₇₃	—	2	2	no	—
169	C₄×C₇₃	—	2	1	no	—
170	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
171	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
172	Dic₇₃	—	2	2	no	—
173	Dic₇₃	—	2	2	no	—
174	Dic₇₃	—	2	1	no	—
175	Dic₇₃	#431	1	2	yes	—
176	C₂×#2164	—	73	1	no	maximal, Sylow
177	C₂×C₇₃	—	4	1	no	—
178	C₂×C₇₃	C₂×#431	1	1	yes	characteristic
179	C₂×C₇₃	C₂×#504	1	2	yes	—
180	D₇₃	—	4	1	no	—
181	D₇₃	#2164	1	4	yes	complemented
182	C₂×#405	—	73	1	no	—
183	#2164	—	73	2	no	—
184	#2164	—	73	2	no	—
185	#2164	—	73	2	no	—
186	#2164	—	73	2	no	—
187	C₂×C₄⋊C₁₆	—	73	2	no	—
188	C₂×C₂²⋊C₁₆	—	73	2	no	—
189	C₂×#346	—	73	1	no	—
190	C₂×C₄×C₁₆	—	73	1	no	—
191	C₇₃	C₂×#2164	1	1	yes	characteristic, complemented, Sylow
192	C₂²×M₁₆	—	73	1	no	—
193	C₂³×C₈	—	73	1	no	—
194	C₂×#117	—	73	1	no	—
195	C₂²×C₁₆	—	73	2	no	—
196	C₂²×C₁₆	—	73	2	no	—
197	#405	—	73	1	no	—
198	#405	—	73	1	no	—
199	#405	—	73	1	no	—
200	#405	—	73	1	no	—
201	#405	—	73	1	no	—
202	#405	—	73	1	no	—
203	#405	—	73	1	no	—
204	#405	#64408	1	1	yes	characteristic
205	C₂×#97	—	73	1	no	—
206	C₂×#97	—	73	1	no	—
207	C₂×C₄×C₈	—	73	1	no	—
208	C₂×C₄×C₈	—	73	1	no	—
209	C₄⋊C₁₆	—	73	4	no	—
210	C₄⋊C₁₆	—	73	4	no	—
211	C₂²⋊C₁₆	—	73	4	no	—
212	C₂²⋊C₁₆	—	73	4	no	—
213	#346	—	73	2	no	—
214	#346	—	73	2	no	—
215	C₄×C₁₆	—	73	2	no	—
216	C₄×C₁₆	—	73	2	no	—
217	C₂³×C₄	—	73	1	no	—
218	C₂×M₁₆	—	73	1	no	—
219	C₂×M₁₆	—	73	1	no	—
220	C₂×M₁₆	—	73	1	no	—
221	C₂×M₁₆	—	146	4	no	—
222	C₂×M₁₆	C₂×#64408	1	1	yes	characteristic
223	C₂²×C₈	—	73	1	no	—
224	C₂²×C₈	—	73	1	no	—
225	C₂²×C₈	—	73	1	no	—
226	C₂²×C₈	—	73	1	no	—
227	C₂²×C₈	—	73	1	no	—
228	C₂²×C₈	—	73	1	no	—
229	C₂²×C₈	—	73	1	no	—
230	C₂²×C₈	—	146	4	no	—
231	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
232	#117	—	73	1	no	—
233	#117	—	73	2	no	—
234	#117	—	73	2	no	—
235	#117	—	73	1	no	—
236	#117	—	73	1	no	—
237	#117	C₂×#64408	1	1	yes	characteristic
238	C₂×C₄⋊C₄	—	73	2	no	—
239	C₂×C₂²⋊C₄	—	73	2	no	—
240	C₂×C₄²	—	73	1	no	—
241	C₂×C₄²	—	73	1	no	—
242	C₂×C₁₆	—	73	4	no	—
243	C₂×C₁₆	—	73	4	no	—
244	C₂×C₁₆	—	146	4	no	—
245	C₂×C₁₆	—	146	4	no	—
246	#97	—	73	1	no	—
247	#97	—	73	1	no	—
248	#97	—	73	1	no	—
249	#97	—	73	1	no	—
250	#97	—	73	1	no	—
251	#97	—	73	1	no	—
252	#97	C₂×#64408	1	1	yes	characteristic
253	#97	C₂×#64408	1	1	yes	characteristic
254	C₄×C₈	—	73	1	no	—
255	C₄×C₈	—	73	1	no	—
256	C₄×C₈	—	73	1	no	—
257	C₄×C₈	—	73	1	no	—
258	C₄×C₈	—	73	1	no	—
259	C₄×C₈	—	73	1	no	—
260	C₄×C₈	C₂×#64408	1	1	yes	characteristic
261	C₄×C₈	C₂×#64408	1	1	yes	characteristic
262	C₂⁴	—	73	1	no	—
263	C₂²×C₄	—	73	1	no	—
264	C₂²×C₄	—	73	1	no	—
265	C₂²×C₄	—	73	2	no	—
266	C₂²×C₄	—	73	2	no	—
267	C₂²×C₄	—	73	1	no	—
268	C₂²×C₄	—	73	1	no	—
269	C₂²×C₄	—	73	1	no	—
270	C₂²×C₄	—	146	2	no	—
271	C₂²×C₄	—	146	2	no	—
272	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
273	M₁₆	—	2	2	no	—
274	M₁₆	—	146	2	no	—
275	M₁₆	—	146	2	no	—
276	M₁₆	—	146	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	73	1	no	—
279	C₂×C₈	—	73	1	no	—
280	C₂×C₈	—	73	4	no	—
281	C₂×C₈	—	73	4	no	—
282	C₂×C₈	—	73	1	no	—
283	C₂×C₈	—	73	1	no	—
284	C₂×C₈	—	146	2	no	—
285	C₂×C₈	—	146	2	no	—
286	C₂×C₈	—	146	2	no	—
287	C₂×C₈	—	146	2	no	—
288	C₂×C₈	—	146	2	no	—
289	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
290	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
291	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
292	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
293	C₄⋊C₄	—	73	2	no	—
294	C₄⋊C₄	—	73	2	no	—
295	C₄⋊C₄	—	73	2	no	—
296	C₄⋊C₄	#11924036	1	2	yes	—
297	C₂²⋊C₄	—	73	2	no	—
298	C₂²⋊C₄	—	73	2	no	—
299	C₂²⋊C₄	—	73	2	no	—
300	C₂²⋊C₄	#11924036	1	2	yes	—
301	C₄²	—	73	2	no	—
302	C₄²	—	73	2	no	—
303	C₄²	—	73	1	no	—
304	C₄²	—	73	1	no	—
305	C₄²	C₂²×#64408	1	1	yes	characteristic
306	C₄²	C₂²×#64408	1	1	yes	characteristic
307	C₁₆	—	146	4	no	—
308	C₁₆	—	146	4	no	—
309	C₂³	—	73	1	no	—
310	C₂³	—	73	1	no	—
311	C₂³	—	146	2	no	—
312	C₂³	—	146	4	no	—
313	C₂³	C₂×#11924036	1	1	yes	characteristic
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	73	1	no	—
317	C₂×C₄	—	73	2	no	—
318	C₂×C₄	—	73	2	no	—
319	C₂×C₄	—	73	2	no	—
320	C₂×C₄	—	73	2	no	—
321	C₂×C₄	—	73	1	no	—
322	C₂×C₄	—	146	2	no	—
323	C₂×C₄	—	146	2	no	—
324	C₂×C₄	—	146	4	no	—
325	C₂×C₄	—	146	4	no	—
326	C₂×C₄	—	146	2	no	—
327	C₂×C₄	—	146	1	no	—
328	C₂×C₄	—	146	1	no	—
329	C₂×C₄	C₂×#11924036	1	2	yes	—
330	C₂×C₄	C₂×#11924036	1	2	yes	—
331	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
332	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	73	2	no	—
336	C₈	—	73	2	no	—
337	C₈	—	146	1	no	—
338	C₈	—	146	1	no	—
339	C₈	#14556670	1	2	yes	—
340	C₈	D₇₃⋊M₁₆	1	2	yes	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	292	4	no	—
344	C₂²	—	73	2	no	—
345	C₂²	—	73	4	no	—
346	C₂²	—	146	1	no	—
347	C₂²	—	146	2	no	—
348	C₂²	C₂²×#11924036	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	2	no	—
352	C₄	—	73	2	no	—
353	C₄	—	146	1	no	—
354	C₄	—	146	2	no	—
355	C₄	—	146	2	no	—
356	C₄	C₂×#14556670	1	1	yes	characteristic
357	C₄	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
358	C₂	—	4	1	no	—
359	C₂	—	292	1	no	—
360	C₂	—	73	4	no	—
361	C₂	#22626595	1	1	yes	characteristic
362	C₂	#22626896	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#64408
Fitting	#405×C₇₃	C₄
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	#405	#64408
hypocenter	C₇₃	C₂×#2164
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
1. #22626299	C₂	semidirect
2. #22626298	C₂	semidirect
3. #22626297	C₂	semidirect
4. #22626296	C₂	semidirect
5. #22626261	C₂	semidirect
7. #22626132	C₂	semidirect
8. C₄×#14556664	C₂	semidirect
12. #17989508	C₂²	semidirect
13. C₄⋊#11924031	C₂²	semidirect
16. #17989476	C₂²	semidirect
17. #17989475	C₂²	semidirect
18. #17989473	C₂²	semidirect
19. C₄×#11924031	C₂²	semidirect
181. D₇₃	#2164	semidirect
191. C₇₃	C₂×#2164	semidirect

Maximal subgroups

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

subgroup	index
#22626299	2
#22626298	2
#22626297	2
#22626296	2
#22626261	2
#22626156	2
#22626132	2
C₄×#14556664	2
#405×D₇₃	2
C₂×#2164	73