Subgroups — #34443724

Order 20224.

794 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099005	C₂	1	2	yes	maximal, complemented
2	C₂×#19098996	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₃₂×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#23773888	C₂	1	2	yes	maximal, complemented
5	#23765669	C₂	1	2	yes	maximal, complemented
6	#23765668	C₂	1	2	yes	maximal, complemented
7	#23765662	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765661	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765651	C₂	1	1	yes	maximal, characteristic, complemented
10	#23764995	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#2247	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579193	—	2	2	no	—
13	C₂×#14579184	C₂²	1	2	yes	—
14	C₂×Q₁₆×D₇₉	C₂²	1	2	yes	—
15	#19099138	—	2	2	no	—
16	#19099005	—	2	8	no	—
17	#19098996	—	2	4	no	—
18	C₂×Dic₆₃₂	C₂²	1	1	yes	characteristic
19	C₂×#14579107	C₂²	1	2	yes	—
20	C₂×#14579107	C₂²	1	2	yes	—
21	Q₃₂×D₇₉	—	2	4	no	—
22	C₂×#14579106	C₂²	1	2	yes	—
23	C₂×#14579104	C₂²	1	2	yes	—
24	C₂×#14579104	C₂²	1	2	yes	—
25	C₂×D₁₂₆₄	C₂²	1	1	yes	characteristic
26	C₂×#14579100	C₂²	1	1	yes	characteristic
27	C₂×C₁₆×D₇₉	C₂²	1	1	yes	characteristic
28	#19098956	—	2	2	no	—
29	#19098955	—	2	2	no	—
30	#19098948	C₂²	1	2	yes	—
31	#19098947	C₂²	1	2	yes	—
32	#19098941	C₂²	1	2	yes	—
33	#19098517	C₂²	1	1	yes	characteristic
34	#19098339	C₂²	1	1	yes	characteristic
35	#19098338	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
37	C₂×Q₃₂×C₇₉	C₂²	1	1	yes	characteristic
38	C₂×Q₃₂×C₇₉	C₂²	1	2	yes	—
39	C₂×Q₃₂×C₇₉	C₂²	1	1	yes	characteristic
40	#494×C₇₉	C₂²	1	2	yes	—
41	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
42	C₂×#11929625	—	2	2	no	—
43	C₂×Q₈×D₇₉	—	2	2	no	—
44	#14579193	—	4	8	no	—
45	#14579184	—	2	4	no	—
46	#14579184	—	2	4	no	—
47	C₂×Dic₃₁₆	C₂³	1	2	yes	—
48	C₂×#11929614	—	2	2	no	—
49	C₂×#11929614	—	2	2	no	—
50	Q₁₆×D₇₉	—	2	4	no	—
51	Q₁₆×D₇₉	—	2	4	no	—
52	C₂×#11929613	—	2	2	no	—
53	C₂×#11929611	—	2	2	no	—
54	C₂×#11929611	—	2	2	no	—
55	C₂×D₆₃₂	C₂³	1	2	yes	—
56	C₂×#11929607	D₄	1	1	yes	characteristic
57	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
58	#14579159	—	2	2	no	—
59	#14579150	—	2	2	no	—
60	#14579149	—	2	2	no	—
61	#14579146	—	4	4	no	—
62	#14579139	—	4	4	no	—
63	#14579118	D₄	1	1	yes	characteristic
64	Dic₆₃₂	—	2	2	no	—
65	#14579107	—	2	4	no	—
66	#14579107	—	2	4	no	—
67	#14579106	—	2	4	no	—
68	#14579104	—	2	4	no	—
69	#14579104	—	2	4	no	—
70	D₁₂₆₄	—	2	2	no	—
71	#14579100	D₄	1	4	yes	—
72	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
74	C₁₆×D₇₉	D₄	1	4	yes	—
75	#14579072	D₄	1	1	yes	characteristic
76	#14579071	D₄	1	1	yes	characteristic
77	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
78	C₂×Q₁₆×C₇₉	—	2	2	no	—
79	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
80	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
81	C₄⋊Q₈×C₇₉	—	2	2	no	—
82	Q₃₂×C₇₉	—	2	2	no	—
83	Q₃₂×C₇₉	—	2	2	no	—
84	Q₃₂×C₇₉	—	2	4	no	—
85	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
86	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
88	#11929625	—	4	4	no	—
89	#11929625	—	4	4	no	—
90	C₂×Dic₁₅₈	—	2	2	no	—
91	C₂×Dic₁₅₈	—	4	2	no	—
92	Q₈×D₇₉	—	4	4	no	—
93	Q₈×D₇₉	—	4	4	no	—
94	C₂×D₃₁₆	—	2	2	no	—
95	C₂×D₃₁₆	—	4	2	no	—
96	C₂×C₄×D₇₉	—	4	2	no	—
97	C₂×C₄×D₇₉	—	4	2	no	—
98	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
99	Dic₃₁₆	—	2	2	no	—
100	Dic₃₁₆	—	2	2	no	—
101	#11929614	—	4	4	no	—
102	#11929614	—	4	4	no	—
103	#11929613	—	4	4	no	—
104	#11929611	—	4	4	no	—
105	#11929611	—	4	4	no	—
106	D₆₃₂	—	2	2	no	—
107	D₆₃₂	—	2	2	no	—
108	#11929607	D₈	1	4	yes	—
109	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
111	C₈×D₇₉	C₂×D₄	1	2	yes	—
112	C₈×D₇₉	C₂×D₄	1	2	yes	—
113	Dic₇₉⋊C₄	—	4	4	no	—
114	Dic₇₉⋊C₄	D₈	1	2	yes	—
115	(C₂×D₇₉)⋊C₄	—	4	4	no	—
116	(C₂×D₇₉)⋊C₄	—	4	4	no	—
117	(C₂×D₇₉)⋊C₄	D₈	1	2	yes	—
118	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
119	C₇₉⋊C₁₆	C₂×D₄	1	2	yes	—
120	C₇₉⋊C₁₆	C₂×D₄	1	2	yes	—
121	C₂×Q₈×C₇₉	—	2	2	no	—
122	C₂×Q₈×C₇₉	—	2	2	no	—
123	Q₁₆×C₇₉	—	2	2	no	—
124	Q₁₆×C₇₉	—	2	2	no	—
125	Q₁₆×C₇₉	—	2	2	no	—
126	Q₁₆×C₇₉	—	2	2	no	—
127	Q₁₆×C₇₉	—	4	4	no	—
128	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
130	C₄⋊C₄×C₇₉	—	4	4	no	—
131	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
132	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
133	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
134	C₂²×D₇₉	—	4	2	no	—
135	C₂²×D₇₉	C₂×D₈	1	1	yes	characteristic
136	Q₈×C₇₉	—	4	2	no	—
137	Q₈×C₇₉	—	4	2	no	—
138	Q₈×C₇₉	—	4	2	no	—
139	Q₈×C₇₉	—	4	2	no	—
140	C₂×C₄×C₇₉	—	4	2	no	—
141	C₂×C₄×C₇₉	—	4	2	no	—
142	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×Dic₇₉	—	4	2	no	—
145	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
146	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
147	D₃₁₆	—	4	2	no	—
148	D₃₁₆	—	4	2	no	—
149	D₃₁₆	—	8	4	no	—
150	C₄×D₇₉	—	8	4	no	—
151	C₄×D₇₉	—	8	4	no	—
152	C₄×D₇₉	#127	1	2	yes	—
153	C₄×D₇₉	C₂×D₈	1	2	yes	—
154	Dic₁₅₈	—	4	2	no	—
155	Dic₁₅₈	—	4	2	no	—
156	Dic₁₅₈	—	8	4	no	—
157	C₈×C₇₉	C₂×D₈	1	2	yes	—
158	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
159	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
160	C₇₉⋊C₈	#127	1	2	yes	—
161	C₇₉⋊C₈	C₂×D₈	1	2	yes	—
162	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
163	C₂×D₇₉	—	2	2	no	—
164	C₂×D₇₉	—	8	2	no	—
165	C₂×D₇₉	—	8	4	no	—
166	C₂×D₇₉	#493	1	2	yes	—
167	C₄×C₇₉	—	2	2	no	—
168	C₄×C₇₉	—	8	2	no	—
169	C₄×C₇₉	—	8	2	no	—
170	C₄×C₇₉	C₂×#127	1	1	yes	characteristic
171	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
172	Dic₇₉	—	2	2	no	—
173	Dic₇₉	—	8	2	no	—
174	Dic₇₉	#493	1	2	yes	—
175	#33721	—	79	1	no	maximal, Sylow
176	C₂×C₇₉	C₂×#493	1	1	yes	characteristic
177	C₂×C₇₉	#3417	1	2	yes	—
178	D₇₉	—	2	2	no	—
179	D₇₉	—	16	2	no	—
180	C₂×#510	—	79	2	no	—
181	C₂×#508	—	79	1	no	—
182	C₂²×Q₃₂	—	79	1	no	—
183	#3156	—	79	2	no	—
184	#2251	—	79	2	no	—
185	#2251	—	79	2	no	—
186	#2249	—	79	1	no	—
187	#2249	—	79	1	no	—
188	#2247	—	79	1	no	—
189	#2247	D₇₉	1	1	yes	characteristic, complemented
190	#2108	—	79	1	no	—
191	C₇₉	#33721	1	1	yes	characteristic, complemented, Sylow
192	C₂×#137	—	158	2	no	—
193	C₂×#135	—	79	2	no	—
194	C₂²×Q₁₆	—	79	2	no	—
195	#536	—	158	2	no	—
196	#510	—	158	8	no	—
197	#508	—	158	4	no	—
198	C₂×Q₃₂	—	79	2	no	—
199	C₂×Q₃₂	—	79	2	no	—
200	C₂×Q₃₂	—	79	1	no	—
201	C₂×Q₃₂	—	158	4	no	—
202	C₂×Q₃₂	C₂×D₇₉	1	1	yes	characteristic
203	C₂×Q₃₂	C₂×D₇₉	1	1	yes	characteristic
204	C₂×Q₃₂	C₂×D₇₉	1	2	yes	—
205	C₂×SD₃₂	—	79	2	no	—
206	C₂×SD₃₂	—	79	2	no	—
207	C₂×SD₃₂	—	79	2	no	—
208	C₂×D₁₆	—	79	1	no	—
209	C₂×M₃₂	—	79	1	no	—
210	C₂²×C₁₆	—	79	1	no	—
211	#497	—	158	2	no	—
212	#497	—	158	2	no	—
213	#495	—	79	2	no	—
214	#495	—	79	2	no	—
215	#494	—	79	2	no	—
216	#494	C₂×D₇₉	1	2	yes	—
217	#405	—	79	1	no	—
218	#346	—	79	1	no	—
219	#346	—	79	1	no	—
220	C₄×C₁₆	—	79	1	no	—
221	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
222	C₂×#41	—	158	2	no	—
223	C₂²×Q₈	—	158	2	no	—
224	#137	—	316	8	no	—
225	#135	—	158	4	no	—
226	#135	—	158	4	no	—
227	C₂×Q₁₆	—	2	2	no	—
228	C₂×Q₁₆	—	79	2	no	—
229	C₂×Q₁₆	—	158	4	no	—
230	C₂×Q₁₆	—	158	4	no	—
231	C₂×Q₁₆	—	158	2	no	—
232	C₂×Q₁₆	—	158	2	no	—
233	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
234	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
235	C₂×SD₁₆	—	158	2	no	—
236	C₂×SD₁₆	—	158	2	no	—
237	C₂×SD₁₆	—	158	2	no	—
238	C₂×D₈	—	79	2	no	—
239	C₂×M₁₆	—	79	1	no	—
240	C₂²×C₈	—	79	1	no	—
241	C₄⋊Q₈	—	2	2	no	—
242	C₄⋊Q₈	—	158	2	no	—
243	#124	—	158	2	no	—
244	#124	—	158	2	no	—
245	#123	—	316	4	no	—
246	C₂²⋊Q₈	—	316	4	no	—
247	#117	—	79	1	no	—
248	Q₃₂	—	2	2	no	—
249	Q₃₂	—	2	2	no	—
250	Q₃₂	—	2	4	no	—
251	Q₃₂	—	158	4	no	—
252	Q₃₂	—	158	4	no	—
253	Q₃₂	—	158	2	no	—
254	SD₃₂	—	158	4	no	—
255	SD₃₂	—	158	4	no	—
256	SD₃₂	—	158	4	no	—
257	D₁₆	—	158	2	no	—
258	M₃₂	—	79	4	no	—
259	C₂×C₁₆	—	79	1	no	—
260	C₂×C₁₆	—	79	1	no	—
261	C₂×C₁₆	—	79	4	no	—
262	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
263	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
264	#97	—	79	1	no	—
265	#97	—	79	1	no	—
266	C₄×C₈	—	79	1	no	—
267	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
268	#41	—	316	4	no	—
269	#41	—	316	4	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	316	2	no	—
273	C₂×Q₈	—	316	4	no	—
274	C₂×Q₈	—	316	4	no	—
275	C₂×Q₈	—	158	2	no	—
276	C₂×D₄	—	316	2	no	—
277	C₂×D₄	—	158	2	no	—
278	C₂²×C₄	—	316	2	no	—
279	C₂²×C₄	—	316	2	no	—
280	C₂²×C₄	—	79	1	no	—
281	Q₁₆	—	2	2	no	—
282	Q₁₆	—	2	2	no	—
283	Q₁₆	—	2	2	no	—
284	Q₁₆	—	2	2	no	—
285	Q₁₆	—	4	4	no	—
286	Q₁₆	—	316	4	no	—
287	Q₁₆	—	316	4	no	—
288	Q₁₆	—	158	2	no	—
289	Q₁₆	—	158	2	no	—
290	SD₁₆	—	316	4	no	—
291	SD₁₆	—	316	4	no	—
292	SD₁₆	—	316	4	no	—
293	D₈	—	158	2	no	—
294	D₈	—	158	2	no	—
295	M₁₆	—	79	4	no	—
296	C₂×C₈	—	79	1	no	—
297	C₂×C₈	—	79	1	no	—
298	C₂×C₈	—	79	2	no	—
299	C₂×C₈	—	79	2	no	—
300	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
301	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
302	C₄⋊C₄	—	4	4	no	—
303	C₄⋊C₄	—	316	4	no	—
304	C₄⋊C₄	—	79	2	no	—
305	C₂²⋊C₄	—	316	4	no	—
306	C₂²⋊C₄	—	316	4	no	—
307	C₂²⋊C₄	—	79	2	no	—
308	C₄²	—	79	1	no	—
309	C₄²	D₄×D₇₉	1	1	yes	characteristic
310	C₁₆	—	79	2	no	—
311	C₁₆	—	79	2	no	—
312	C₁₆	D₄×D₇₉	1	2	yes	—
313	C₁₆	D₄×D₇₉	1	2	yes	—
314	C₂³	—	316	2	no	—
315	C₂³	—	79	1	no	—
316	Q₈	—	4	2	no	—
317	Q₈	—	4	2	no	—
318	Q₈	—	4	2	no	—
319	Q₈	—	4	2	no	—
320	Q₈	—	316	2	no	—
321	Q₈	—	316	2	no	—
322	Q₈	—	632	4	no	—
323	D₄	—	316	2	no	—
324	D₄	—	316	2	no	—
325	D₄	—	632	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	316	2	no	—
329	C₂×C₄	—	79	1	no	—
330	C₂×C₄	—	79	2	no	—
331	C₂×C₄	—	79	2	no	—
332	C₂×C₄	—	79	2	no	—
333	C₂×C₄	—	632	4	no	—
334	C₂×C₄	—	632	4	no	—
335	C₂×C₄	D₈×D₇₉	1	2	yes	—
336	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
337	C₈	—	79	2	no	—
338	C₈	—	79	2	no	—
339	C₈	D₈×D₇₉	1	2	yes	—
340	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
341	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
342	C₂²	—	79	2	no	—
343	C₂²	—	632	4	no	—
344	C₂²	—	632	2	no	—
345	C₂²	—	158	2	no	—
346	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
347	C₄	—	2	2	no	—
348	C₄	—	8	2	no	—
349	C₄	—	8	2	no	—
350	C₄	—	79	2	no	—
351	C₄	—	632	2	no	—
352	C₄	—	158	2	no	—
353	C₄	#127×D₇₉	1	1	yes	characteristic
354	C₄	C₂×D₈×D₇₉	1	1	yes	characteristic
355	C₂	—	158	2	no	—
356	C₂	—	1264	2	no	—
357	C₂	#493×D₇₉	1	1	yes	characteristic
358	C₂	#23776549	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2247	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×#19098996	C₂
hypercenter	#2247	D₇₉
hypocenter	C₇₉	#33721
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

Complements

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

N	H ≅ G/N	type
1. C₂×#19099005	C₂	semidirect
3. C₂×Q₃₂×D₇₉	C₂	semidirect
4. #23773888	C₂	semidirect
5. #23765669	C₂	semidirect
6. #23765668	C₂	semidirect
7. #23765662	C₂	semidirect
8. #23765661	C₂	semidirect
9. #23765651	C₂	semidirect
10. #23764995	C₂	semidirect
11. C₇₉×#2247	C₂	semidirect
189. #2247	D₇₉	semidirect
191. C₇₉	#33721	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099005	2
C₂×#19098996	2
C₂×Q₃₂×D₇₉	2
#23773888	2
#23765669	2
#23765668	2
#23765662	2
#23765661	2
#23765651	2
#23764995	2
C₇₉×#2247	2
#33721	79