Subgroups — #32984862

Order 18688.

538 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11924017	C₂	1	1	yes	maximal, characteristic
2	#22610867	C₂	1	1	yes	maximal, characteristic, complemented
3	#22610725	C₂	1	1	yes	maximal, characteristic
4	C₂×#17988340	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17988319	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988316	C₂	1	1	yes	maximal, characteristic
7	C₂×#322×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11924017	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
11	C₄×#11924017	—	2	2	no	—
12	C₄×#11924017	—	2	2	no	—
13	C₂×#14556523	C₄	1	1	yes	characteristic
14	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
16	C₂×#14556516	C₄	1	1	yes	characteristic
17	C₂×#14556513	C₄	1	1	yes	characteristic
18	C₂×#14556503	—	2	2	no	—
19	#17988341	—	2	2	no	—
20	#17988340	—	2	2	no	—
21	#17988319	—	2	2	no	—
22	#17988316	—	2	2	no	—
23	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
26	#322×C₇₃	—	2	2	no	—
27	C₂²×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₂₉₂	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₇₃	—	2	1	no	—
30	C₂×#11924017	—	2	2	no	—
31	C₂×#11924017	—	2	2	no	—
32	C₂×#11924017	C₈	1	4	yes	complemented
33	C₂²×#10596831	—	2	1	no	—
34	C₂²×#10596831	—	2	1	no	—
35	C₄×Dic₁₄₆	—	2	1	no	—
36	C₄×Dic₁₄₆	—	2	1	no	—
37	C₄×Dic₁₄₆	—	2	1	no	—
38	C₄×Dic₁₄₆	—	2	1	no	—
39	C₄×D₂₉₂	—	2	1	no	—
40	C₄×D₂₉₂	—	2	1	no	—
41	C₄×D₂₉₂	—	2	1	no	—
42	C₄×D₂₉₂	—	2	1	no	—
43	C₂×C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂×#11924007	—	2	1	no	—
46	C₂×#11924005	—	2	1	no	—
47	C₂×C₄×Dic₇₃	—	2	1	no	—
48	#14556523	—	2	2	no	—
49	C₄⋊#10596831	—	2	1	no	—
50	C₄⋊#10596831	—	2	1	no	—
51	C₄⋊#10596831	—	2	1	no	—
52	C₄⋊#10596831	—	2	1	no	—
53	#14556516	—	2	2	no	—
54	#14556513	—	2	2	no	—
55	#14556503	—	4	4	no	—
56	C₂²×C₈×C₇₃	—	2	1	no	—
57	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₇₃	D₄	1	2	yes	—
60	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₇₃	—	2	1	no	—
62	C₄×C₈×C₇₃	—	2	1	no	—
63	C₄×C₈×C₇₃	—	2	1	no	—
64	C₄×C₈×C₇₃	D₄	1	2	yes	—
65	C₄×C₈×C₇₃	D₄	1	2	yes	—
66	C₄×C₈×C₇₃	D₄	1	2	yes	—
67	C₂³×D₇₃	—	2	1	no	—
68	C₂×Dic₁₄₆	—	2	1	no	—
69	C₂×Dic₁₄₆	—	2	1	no	—
70	C₂×Dic₁₄₆	—	2	1	no	—
71	C₂×Dic₁₄₆	—	2	1	no	—
72	C₂×Dic₁₄₆	—	2	1	no	—
73	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
74	C₂×D₂₉₂	—	2	1	no	—
75	C₂×D₂₉₂	—	2	1	no	—
76	C₂×D₂₉₂	—	2	1	no	—
77	C₂×D₂₉₂	—	2	1	no	—
78	C₂×D₂₉₂	—	2	1	no	—
79	C₂×D₂₉₂	C₂×C₈	1	2	yes	—
80	C₂²×Dic₇₃	—	2	1	no	—
81	C₂²×Dic₇₃	—	2	1	no	—
82	C₂²×Dic₇₃	—	2	1	no	—
83	C₂×C₄×D₇₃	—	4	1	no	—
84	C₂×C₄×D₇₃	—	4	1	no	—
85	C₂×C₄×D₇₃	—	4	2	no	—
86	C₂×C₄×D₇₃	—	4	2	no	—
87	#11924017	—	2	8	no	—
88	C₂×#10596831	—	4	1	no	—
89	C₂×#10596831	—	4	1	no	—
90	C₂×#10596831	—	4	2	no	—
91	C₂×#10596831	—	4	2	no	—
92	C₄⋊Dic₇₃	—	2	1	no	—
93	C₄⋊Dic₇₃	—	2	1	no	—
94	C₄⋊Dic₇₃	—	2	1	no	—
95	C₄⋊Dic₇₃	—	2	1	no	—
96	#11924007	—	4	1	no	—
97	#11924007	—	4	1	no	—
98	#11924005	—	4	1	no	—
99	#11924005	—	4	1	no	—
100	C₄×Dic₇₃	—	4	1	no	—
101	C₄×Dic₇₃	—	4	1	no	—
102	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
105	C₂×C₈×C₇₃	—	2	1	no	—
106	C₂×C₈×C₇₃	—	2	1	no	—
107	C₂×C₈×C₇₃	—	2	2	no	—
108	C₂×C₈×C₇₃	—	2	2	no	—
109	C₂×C₈×C₇₃	—	2	2	no	—
110	C₂×C₈×C₇₃	—	2	2	no	—
111	C₂×C₈×C₇₃	—	4	2	no	—
112	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
113	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
114	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
115	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
116	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₇₃	#41	1	1	yes	characteristic
120	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
121	C₂²×D₇₃	—	2	2	no	—
122	C₂²×D₇₃	—	4	1	no	—
123	C₂²×D₇₃	—	4	1	no	—
124	C₂²×D₇₃	—	4	2	no	—
125	C₂²×D₇₃	—	4	2	no	—
126	C₂×C₄×C₇₃	—	2	2	no	—
127	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
132	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
135	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
136	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
137	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₇₃	#137	1	2	yes	—
139	C₂×Dic₇₃	—	2	2	no	—
140	C₂×Dic₇₃	—	4	1	no	—
141	C₂×Dic₇₃	—	4	1	no	—
142	C₂×Dic₇₃	—	4	1	no	—
143	C₂×Dic₇₃	—	4	1	no	—
144	C₂×Dic₇₃	—	4	2	no	—
145	C₂×Dic₇₃	—	4	2	no	—
146	D₂₉₂	—	2	2	no	—
147	D₂₉₂	—	2	2	no	—
148	D₂₉₂	—	4	1	no	—
149	D₂₉₂	—	4	1	no	—
150	C₄×D₇₃	—	8	2	no	—
151	C₄×D₇₃	—	8	1	no	—
152	C₄×D₇₃	—	8	1	no	—
153	Dic₁₄₆	—	2	2	no	—
154	Dic₁₄₆	—	2	2	no	—
155	Dic₁₄₆	—	4	1	no	—
156	Dic₁₄₆	—	4	1	no	—
157	C₈×C₇₃	—	2	2	no	—
158	C₈×C₇₃	—	4	2	no	—
159	C₈×C₇₃	—	4	2	no	—
160	C₈×C₇₃	C₂²⋊C₈	1	4	yes	complemented
161	#10596831	—	8	1	no	—
162	#10596831	—	8	1	no	—
163	C₂²×C₇₃	#390	1	1	yes	characteristic
164	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
165	C₂²×C₇₃	#436	1	1	yes	characteristic
166	C₂²×C₇₃	#440	1	2	yes	—
167	C₂²×C₇₃	#444	1	2	yes	—
168	C₂×D₇₃	—	4	4	no	—
169	C₂×D₇₃	—	4	2	no	—
170	C₂×D₇₃	—	8	2	no	—
171	C₂×D₇₃	—	8	1	no	—
172	C₂×D₇₃	—	8	1	no	—
173	C₄×C₇₃	—	2	1	no	—
174	C₄×C₇₃	—	2	1	no	—
175	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
176	C₄×C₇₃	#408	1	1	yes	characteristic
177	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
178	C₄×C₇₃	#436	1	1	yes	characteristic
179	C₄×C₇₃	#468	1	2	yes	—
180	C₄×C₇₃	#496	1	1	yes	characteristic
181	C₄×C₇₃	#497	1	1	yes	characteristic
182	Dic₇₃	—	4	2	no	—
183	Dic₇₃	—	8	1	no	—
184	Dic₇₃	—	8	1	no	—
185	#9660	—	73	1	no	maximal, Sylow
186	C₂×C₇₃	SD₁₆⋊C₈	1	2	yes	—
187	C₂×C₇₃	#1592	1	2	yes	—
188	C₂×C₇₃	#1954	1	1	yes	characteristic
189	C₂×C₇₃	#1973	1	1	yes	characteristic
190	C₂×C₇₃	#1975	1	1	yes	characteristic
191	D₇₃	—	8	2	no	—
192	C₂×C₄×SD₁₆	—	73	1	no	—
193	#1984	—	73	1	no	—
194	#1954	—	73	1	no	—
195	C₂×#335	—	73	1	no	—
196	C₂×Q₈⋊C₈	—	73	1	no	—
197	C₂×D₄⋊C₈	—	73	1	no	—
198	C₂×#322	D₇₃	1	1	yes	characteristic, complemented
199	C₇₃	#9660	1	1	yes	characteristic, complemented, Sylow
200	C₂²×SD₁₆	—	73	1	no	—
201	C₂×C₄×Q₈	—	73	1	no	—
202	C₂×C₄×D₄	—	73	1	no	—
203	C₄×SD₁₆	—	146	2	no	—
204	C₄×SD₁₆	—	146	2	no	—
205	C₂×#106	—	73	1	no	—
206	C₂×C₄⋊C₈	—	73	1	no	—
207	C₂×C₄⋊C₈	—	73	1	no	—
208	C₂×#103	—	73	1	no	—
209	C₂×#102	—	73	1	no	—
210	C₂×C₂²⋊C₈	—	146	2	no	—
211	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
212	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
213	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
214	#336	—	146	2	no	—
215	#335	—	146	2	no	—
216	Q₈⋊C₈	—	146	2	no	—
217	D₄⋊C₈	—	146	2	no	—
218	#322	—	2	2	no	—
219	C₂²×Q₈	—	73	1	no	—
220	C₂²×D₄	—	73	1	no	—
221	C₂³×C₄	—	146	1	no	—
222	C₂×SD₁₆	—	73	4	no	—
223	C₂×SD₁₆	—	146	2	no	—
224	C₂×SD₁₆	—	146	2	no	—
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	146	1	no	—
227	C₂²×C₈	—	146	1	no	—
228	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
229	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
230	C₂²×C₈	D₂₉₂	1	2	yes	—
231	C₄×Q₈	—	146	1	no	—
232	C₄×Q₈	—	146	1	no	—
233	C₄×Q₈	—	146	1	no	—
234	C₄×Q₈	—	146	1	no	—
235	C₄×D₄	—	146	1	no	—
236	C₄×D₄	—	146	1	no	—
237	C₄×D₄	—	146	1	no	—
238	C₄×D₄	—	146	1	no	—
239	C₂×C₄⋊C₄	—	73	1	no	—
240	C₂×C₄⋊C₄	—	73	1	no	—
241	C₂×C₄⋊C₄	—	146	1	no	—
242	C₂×C₂²⋊C₄	—	146	1	no	—
243	C₂×C₄²	—	146	1	no	—
244	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
245	#106	—	146	2	no	—
246	C₄⋊C₈	—	146	1	no	—
247	C₄⋊C₈	—	146	1	no	—
248	C₄⋊C₈	—	146	1	no	—
249	C₄⋊C₈	—	146	1	no	—
250	#103	—	146	2	no	—
251	#102	—	146	2	no	—
252	C₂²⋊C₈	—	292	4	no	—
253	C₄×C₈	—	2	1	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	—	2	1	no	—
256	C₄×C₈	D₂₉₂	1	2	yes	—
257	C₄×C₈	D₂₉₂	1	2	yes	—
258	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
259	C₂⁴	—	146	1	no	—
260	C₂×Q₈	—	73	2	no	—
261	C₂×Q₈	—	146	1	no	—
262	C₂×Q₈	—	146	1	no	—
263	C₂×Q₈	—	146	1	no	—
264	C₂×Q₈	—	146	1	no	—
265	C₂×Q₈	—	146	1	no	—
266	C₂×D₄	—	73	2	no	—
267	C₂×D₄	—	146	1	no	—
268	C₂×D₄	—	146	1	no	—
269	C₂×D₄	—	146	1	no	—
270	C₂×D₄	—	146	1	no	—
271	C₂×D₄	—	146	1	no	—
272	C₂²×C₄	—	292	1	no	—
273	C₂²×C₄	—	292	1	no	—
274	C₂²×C₄	—	292	2	no	—
275	C₂²×C₄	—	292	2	no	—
276	C₂²×C₄	—	146	1	no	—
277	C₂²×C₄	—	146	1	no	—
278	C₂²×C₄	—	146	1	no	—
279	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
281	C₂²×C₄	#11924028	1	1	yes	characteristic
282	SD₁₆	—	146	8	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	4	2	no	—
290	C₂×C₈	—	292	1	no	—
291	C₂×C₈	—	292	1	no	—
292	C₂×C₈	—	292	2	no	—
293	C₂×C₈	—	292	2	no	—
294	C₂×C₈	#11924005	1	2	yes	—
295	C₂×C₈	#11924005	1	2	yes	—
296	C₂×C₈	C₈×D₇₃	1	2	yes	—
297	C₂×C₈	#11924011	1	2	yes	—
298	C₄⋊C₄	—	292	1	no	—
299	C₄⋊C₄	—	292	1	no	—
300	C₄⋊C₄	—	146	1	no	—
301	C₄⋊C₄	—	146	1	no	—
302	C₄⋊C₄	—	146	1	no	—
303	C₄⋊C₄	—	146	1	no	—
304	C₂²⋊C₄	—	292	1	no	—
305	C₂²⋊C₄	—	292	1	no	—
306	C₄²	—	292	1	no	—
307	C₄²	—	292	1	no	—
308	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
309	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
310	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
311	C₄²	#11924028	1	1	yes	characteristic
312	C₂³	—	292	1	no	—
313	C₂³	—	292	2	no	—
314	C₂³	—	292	2	no	—
315	C₂³	—	292	1	no	—
316	C₂³	—	146	2	no	—
317	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
318	Q₈	—	292	1	no	—
319	Q₈	—	292	1	no	—
320	Q₈	—	146	2	no	—
321	Q₈	—	146	2	no	—
322	D₄	—	292	1	no	—
323	D₄	—	292	1	no	—
324	D₄	—	146	2	no	—
325	D₄	—	146	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	292	1	no	—
328	C₂×C₄	—	292	1	no	—
329	C₂×C₄	—	292	2	no	—
330	C₂×C₄	—	292	2	no	—
331	C₂×C₄	—	292	1	no	—
332	C₂×C₄	—	292	1	no	—
333	C₂×C₄	—	584	1	no	—
334	C₂×C₄	—	584	1	no	—
335	C₂×C₄	—	584	2	no	—
336	C₂×C₄	—	146	2	no	—
337	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
338	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
339	C₂×C₄	#14556565	1	1	yes	characteristic
340	C₂×C₄	#14556565	1	1	yes	characteristic
341	C₂×C₄	#14556579	1	1	yes	characteristic
342	C₂×C₄	#14556588	1	1	yes	characteristic
343	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
344	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
345	C₂×C₄	#14556597	1	1	yes	characteristic
346	C₂×C₄	#14556597	1	1	yes	characteristic
347	C₂×C₄	#14556619	1	2	yes	—
348	C₂×C₄	#14556624	1	2	yes	—
349	C₈	—	2	2	no	—
350	C₈	—	4	2	no	—
351	C₈	—	4	2	no	—
352	C₈	—	584	1	no	—
353	C₈	—	584	1	no	—
354	C₈	#14556503	1	4	yes	complemented
355	C₂²	—	292	2	no	—
356	C₂²	—	292	4	no	—
357	C₂²	—	584	1	no	—
358	C₂²	—	584	1	no	—
359	C₂²	—	584	2	no	—
360	C₂²	#17988489	1	1	yes	characteristic
361	C₂²	C₈×D₂₉₂	1	1	yes	characteristic
362	C₂²	#17988666	1	1	yes	characteristic
363	C₂²	#17988689	1	2	yes	—
364	C₂²	#17988710	1	2	yes	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	292	2	no	—
368	C₄	—	584	1	no	—
369	C₄	—	584	1	no	—
370	C₄	C₂×#14556503	1	1	yes	characteristic
371	C₄	#17988553	1	1	yes	characteristic
372	C₄	C₈×D₂₉₂	1	1	yes	characteristic
373	C₄	#17988666	1	1	yes	characteristic
374	C₄	#17988833	1	2	yes	—
375	C₄	#17988976	1	1	yes	characteristic
376	C₄	#17988981	1	1	yes	characteristic
377	C₂	—	584	2	no	—
378	C₂	#22608745	1	2	yes	—
379	C₂	#22608835	1	2	yes	—
380	C₂	#22610725	1	1	yes	characteristic
381	C₂	#22610812	1	1	yes	characteristic
382	C₂	#22610824	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#322×C₇₃	C₂
socle	C₂³×C₇₃	C₄×D₄
prime radical	C₂²×D₂₉₂	C₂×C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#322	D₇₃
hypocenter	C₇₃	#9660
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #22610867	C₂	semidirect
4. C₂×#17988340	C₂	semidirect
5. C₂×#17988319	C₂	semidirect
7. C₂×#322×C₇₃	C₂	semidirect
32. C₂×#11924017	C₈	semidirect
160. C₈×C₇₃	C₂²⋊C₈	semidirect
198. C₂×#322	D₇₃	semidirect
199. C₇₃	#9660	semidirect
354. C₈	#14556503	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11924017	2
#22610867	2
#22610725	2
C₂×#17988340	2
C₂×#17988319	2
C₂×#17988316	2
C₂×#322×C₇₃	2
#9660	73