Subgroups — #34203321

Order 20224.

482 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23771822	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098348	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098346	C₂	1	1	yes	maximal, characteristic
4	M₃₂×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098338	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761489	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14579169	C₄	1	1	yes	characteristic
9	C₂×#14579167	C₂²	1	1	yes	characteristic
10	#19099031	C₄	1	1	yes	characteristic
11	C₂×#14579101	C₂²	1	1	yes	characteristic
12	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	#19098520	—	2	1	no	—
14	#19098519	C₄	1	1	yes	characteristic
15	#19098517	—	2	1	no	—
16	#19098515	—	2	1	no	—
17	M₁₆×Dic₇₉	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₇₉	—	2	1	no	—
20	#19098348	C₄	1	4	yes	complemented
21	#19098346	C₄	1	4	yes	complemented
22	#19098338	—	2	1	no	—
23	#19098338	—	2	1	no	—
24	#19098338	—	2	1	no	—
25	C₁₆×Dic₇₉	—	2	1	no	—
26	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
28	#405×C₇₉	C₂²	1	1	yes	characteristic
29	C₂×#11929624	C₂×C₄	1	1	yes	characteristic
30	#14579169	—	2	2	no	—
31	#14579169	C₈	1	4	yes	—
32	#14579167	—	2	1	no	—
33	#14579167	—	2	1	no	—
34	#14579167	C₂×C₄	1	2	yes	—
35	#14579167	C₂×C₄	1	2	yes	—
36	C₂×#11929608	C₂×C₄	1	1	yes	characteristic
37	C₂×#11929607	—	2	1	no	—
38	C₂×#11929607	—	2	1	no	—
39	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₇₉	—	2	1	no	—
41	C₂×C₈×D₇₉	—	2	1	no	—
42	#14579126	C₈	1	2	yes	—
43	#14579123	C₈	1	2	yes	—
44	(C₂²⋊D₇₉)⋊C₄	—	2	2	no	—
45	#14579118	—	2	1	no	—
46	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
47	C₄²×D₇₉	—	2	1	no	—
48	#14579101	—	2	1	no	—
49	#14579101	C₂×C₄	1	2	yes	complemented
50	C₂×C₇₉⋊C₁₆	—	2	1	no	—
51	C₂×C₇₉⋊C₁₆	—	2	1	no	—
52	C₂×C₇₉⋊C₁₆	—	2	1	no	—
53	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	complemented
54	#14579072	—	2	1	no	—
55	#14579071	—	2	1	no	—
56	C₄×C₇₉⋊C₈	—	2	1	no	—
57	C₈×Dic₇₉	—	2	1	no	—
58	C₈×Dic₇₉	D₄	1	1	yes	characteristic
59	C₈×Dic₇₉	D₄	1	1	yes	characteristic
60	C₈×Dic₇₉	D₄	1	1	yes	characteristic
61	C₈×Dic₇₉	D₄	1	1	yes	characteristic
62	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
64	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
65	M₃₂×C₇₉	—	2	1	no	—
66	M₃₂×C₇₉	C₂×C₄	1	2	yes	complemented
67	C₂×C₁₆×C₇₉	—	2	1	no	—
68	C₂×C₁₆×C₇₉	—	2	1	no	—
69	C₂×C₁₆×C₇₉	—	2	1	no	—
70	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	complemented
71	#97×C₇₉	—	2	1	no	—
72	C₄×C₈×C₇₉	—	2	1	no	—
73	#11929624	—	2	1	no	—
74	#11929624	—	2	1	no	—
75	#11929624	C₄²	1	2	yes	—
76	#11929624	C₂×C₈	1	2	yes	—
77	C₂×Dic₁₅₈	C₂×C₈	1	1	yes	characteristic
78	C₂×D₃₁₆	C₂×C₈	1	1	yes	characteristic
79	C₂×C₂²⋊D₇₉	—	2	1	no	—
80	C₂²×Dic₇₉	M₁₆	1	2	yes	—
81	C₂×C₄×D₇₉	—	2	2	no	—
82	C₂×C₄×D₇₉	—	2	1	no	—
83	#11929608	—	2	1	no	—
84	#11929608	C₄²	1	2	yes	—
85	#11929607	—	2	2	no	—
86	#11929607	—	2	2	no	—
87	#11929607	—	2	4	no	—
88	C₂×C₇₉⋊C₈	—	2	1	no	—
89	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	—
92	C₂×C₇₉⋊C₈	M₁₆	1	2	yes	—
93	C₈×D₇₉	—	2	2	no	—
94	C₈×D₇₉	—	2	2	no	—
95	C₈×D₇₉	—	2	4	no	—
96	Dic₇₉⋊C₄	—	2	2	no	—
97	(C₂×D₇₉)⋊C₄	—	2	2	no	—
98	C₄×Dic₇₉	—	2	2	no	—
99	C₄×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
100	C₄×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
101	C₄×Dic₇₉	C₂×C₈	1	2	yes	—
102	C₇₉⋊C₁₆	—	2	2	no	—
103	C₇₉⋊C₁₆	—	2	2	no	—
104	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
105	M₁₆×C₇₉	—	2	1	no	—
106	M₁₆×C₇₉	C₂×C₈	1	2	yes	—
107	C₂×C₈×C₇₉	—	2	1	no	—
108	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
113	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
114	C₄⋊C₄×C₇₉	C₂×C₈	1	2	yes	—
115	C₂²⋊C₄×C₇₉	—	2	1	no	—
116	C₄²×C₇₉	—	2	1	no	—
117	C₁₆×C₇₉	—	2	2	no	—
118	C₁₆×C₇₉	—	2	2	no	—
119	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
120	C₂²×D₇₉	—	2	1	no	—
121	C₂×C₄×C₇₉	—	2	2	no	—
122	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
125	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
128	C₂²⋊D₇₉	—	2	2	no	—
129	C₂²⋊D₇₉	—	2	2	no	—
130	C₂×Dic₇₉	—	2	1	no	—
131	C₂×Dic₇₉	—	4	2	no	—
132	C₂×Dic₇₉	C₂²⋊C₈	1	2	yes	—
133	C₂×Dic₇₉	C₂²⋊C₈	1	2	yes	—
134	D₃₁₆	—	2	1	no	—
135	D₃₁₆	C₄×C₈	1	2	yes	—
136	C₄×D₇₉	—	2	2	no	—
137	C₄×D₇₉	—	2	2	no	—
138	C₄×D₇₉	—	4	4	no	—
139	Dic₁₅₈	—	2	1	no	—
140	Dic₁₅₈	C₄×C₈	1	2	yes	—
141	C₈×C₇₉	—	2	2	no	—
142	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
143	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
144	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
145	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
146	C₇₉⋊C₈	—	2	2	no	—
147	C₇₉⋊C₈	C₂²⋊C₈	1	2	yes	—
148	C₇₉⋊C₈	C₂²⋊C₈	1	2	yes	—
149	C₂²×C₇₉	—	2	2	no	—
150	C₂²×C₇₉	C₄×M₁₆	1	1	yes	characteristic
151	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
152	C₂²×C₇₉	#431	1	1	yes	characteristic
153	C₂×D₇₉	—	2	2	no	—
154	C₂×D₇₉	—	4	2	no	—
155	C₄×C₇₉	—	4	2	no	—
156	C₄×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
158	C₄×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
159	C₄×C₇₉	#431	1	1	yes	characteristic
160	Dic₇₉	—	2	2	no	—
161	Dic₇₉	—	2	2	no	—
162	Dic₇₉	—	2	2	no	—
163	#12604	—	79	1	no	maximal, Sylow
164	C₂×C₇₉	—	2	1	no	—
165	C₂×C₇₉	—	2	1	no	—
166	C₂×C₇₉	C₄×C₂²⋊C₈	1	1	yes	characteristic
167	C₂×C₇₉	#2117	1	2	yes	—
168	D₇₉	—	4	2	no	—
169	C₄×#131	—	79	1	no	—
170	C₂×#350	—	79	1	no	—
171	C₂×#350	—	79	1	no	—
172	C₄×M₃₂	—	79	1	no	—
173	C₂×#346	—	79	1	no	—
174	#1379	—	79	1	no	—
175	#1379	D₇₉	1	1	yes	characteristic, complemented
176	C₇₉	#12604	1	1	yes	characteristic, complemented, Sylow
177	C₂×#131	—	79	1	no	—
178	C₂×#131	—	79	1	no	—
179	C₄×#41	—	79	1	no	—
180	C₂×M₃₂	—	79	1	no	—
181	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
182	C₂²×C₁₆	—	79	1	no	—
183	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
184	#405	—	79	1	no	—
185	#405	—	158	1	no	—
186	#405	—	158	1	no	—
187	#405	C₂×D₇₉	1	1	yes	characteristic
188	C₄×M₁₆	—	79	1	no	—
189	C₄×M₁₆	—	158	1	no	—
190	C₂×C₄×C₈	—	79	1	no	—
191	C₂×C₄×C₈	—	158	1	no	—
192	#350	—	79	4	no	—
193	#350	—	79	4	no	—
194	#346	—	158	1	no	—
195	#346	—	158	1	no	—
196	#346	—	158	1	no	—
197	C₄×C₁₆	—	158	1	no	—
198	C₂×#41	—	79	1	no	—
199	#131	—	79	4	no	—
200	#131	—	79	2	no	—
201	#131	—	79	2	no	—
202	#131	—	158	2	no	—
203	#131	—	158	1	no	—
204	#131	—	158	1	no	—
205	C₂×M₁₆	—	79	1	no	—
206	C₂×M₁₆	—	158	1	no	—
207	C₂×M₁₆	—	158	1	no	—
208	C₂×M₁₆	C₄×D₇₉	1	1	yes	characteristic
209	C₂²×C₈	—	79	1	no	—
210	C₂²×C₈	—	158	1	no	—
211	C₂²×C₈	—	158	1	no	—
212	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
213	C₄×Q₈	—	79	2	no	—
214	C₄×D₄	—	79	2	no	—
215	C₄×D₄	—	158	2	no	—
216	#117	—	158	1	no	—
217	#117	C₄×D₇₉	1	1	yes	characteristic
218	C₂×C₄²	—	79	1	no	—
219	C₂×C₄²	—	158	1	no	—
220	M₃₂	—	2	1	no	—
221	M₃₂	—	79	2	no	—
222	M₃₂	—	158	1	no	—
223	M₃₂	C₄×D₇₉	1	2	yes	complemented
224	C₂×C₁₆	—	2	1	no	—
225	C₂×C₁₆	—	2	1	no	—
226	C₂×C₁₆	—	2	1	no	—
227	C₂×C₁₆	—	79	2	no	—
228	C₂×C₁₆	—	158	1	no	—
229	C₂×C₁₆	—	158	1	no	—
230	C₂×C₁₆	—	158	1	no	—
231	C₂×C₁₆	C₄×D₇₉	1	2	yes	complemented
232	#97	—	2	1	no	—
233	#97	—	158	1	no	—
234	#97	—	158	1	no	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	—	79	1	no	—
237	C₄×C₈	—	79	1	no	—
238	C₄×C₈	—	79	1	no	—
239	C₄×C₈	—	79	1	no	—
240	C₄×C₈	—	158	1	no	—
241	C₄×C₈	—	158	1	no	—
242	#41	—	79	2	no	—
243	#41	—	79	2	no	—
244	#41	—	158	1	no	—
245	#41	—	158	1	no	—
246	C₂×Q₈	—	79	1	no	—
247	C₂×D₄	—	79	1	no	—
248	C₂×D₄	—	158	1	no	—
249	C₂²×C₄	—	79	2	no	—
250	C₂²×C₄	—	158	2	no	—
251	C₂²×C₄	—	158	1	no	—
252	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
253	M₁₆	—	2	1	no	—
254	M₁₆	—	79	2	no	—
255	M₁₆	—	158	4	no	—
256	M₁₆	—	158	1	no	—
257	M₁₆	—	158	2	no	—
258	M₁₆	—	158	2	no	—
259	M₁₆	C₈×D₇₉	1	2	yes	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	79	1	no	—
262	C₂×C₈	—	79	1	no	—
263	C₂×C₈	—	79	2	no	—
264	C₂×C₈	—	79	2	no	—
265	C₂×C₈	—	158	4	no	—
266	C₂×C₈	—	158	1	no	—
267	C₂×C₈	—	158	2	no	—
268	C₂×C₈	—	158	2	no	—
269	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
271	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
272	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
273	C₂×C₈	#11929623	1	1	yes	characteristic
274	C₂×C₈	#11929625	1	1	yes	characteristic
275	C₄⋊C₄	—	158	2	no	—
276	C₄⋊C₄	C₈×D₇₉	1	2	yes	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	158	2	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	79	1	no	—
281	C₄²	—	79	1	no	—
282	C₄²	—	79	2	no	—
283	C₄²	—	158	2	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	2	2	no	—
286	C₁₆	—	158	2	no	—
287	C₁₆	—	158	2	no	—
288	C₂³	—	158	1	no	—
289	C₂³	M₁₆×D₇₉	1	1	yes	characteristic
290	Q₈	—	79	2	no	—
291	Q₈	—	158	1	no	—
292	D₄	—	79	2	no	—
293	D₄	—	158	2	no	—
294	D₄	—	158	2	no	—
295	D₄	—	158	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	316	4	no	—
298	C₂×C₄	—	316	2	no	—
299	C₂×C₄	—	79	2	no	—
300	C₂×C₄	—	79	2	no	—
301	C₂×C₄	—	158	1	no	—
302	C₂×C₄	—	158	2	no	—
303	C₂×C₄	—	158	2	no	—
304	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
305	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
306	C₂×C₄	#14579119	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
309	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	79	2	no	—
312	C₈	—	79	2	no	—
313	C₈	—	158	2	no	—
314	C₈	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
315	C₈	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
316	C₈	#14579123	1	1	yes	characteristic
317	C₈	#14579123	1	1	yes	characteristic
318	C₂²	—	2	2	no	—
319	C₂²	—	316	2	no	—
320	C₂²	—	158	2	no	—
321	C₂²	#19098515	1	1	yes	characteristic
322	C₂²	C₂²⋊C₈×D₇₉	1	1	yes	characteristic
323	C₂²	#19098615	1	1	yes	characteristic
324	C₄	—	4	2	no	—
325	C₄	—	158	2	no	—
326	C₄	—	158	2	no	—
327	C₄	—	158	2	no	—
328	C₄	#19098410	1	1	yes	characteristic
329	C₄	C₄×C₈×D₇₉	1	1	yes	characteristic
330	C₄	C₂²⋊C₈×D₇₉	1	1	yes	characteristic
331	C₄	#19098615	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	316	2	no	—
335	C₂	#23763157	1	1	yes	characteristic
336	C₂	#23765026	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₇₉
commutator	C₄×C₇₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1379	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₈
prime radical	C₂×#11929624	C₂×C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1379	D₇₉
hypocenter	C₇₉	#12604
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
4. M₃₂×Dic₇₉	C₂	semidirect
5. C₂×#19098338	C₂	semidirect
6. #23761489	C₂	semidirect
7. C₇₉×#1379	C₂	semidirect
20. #19098348	C₄	semidirect
21. #19098346	C₄	semidirect
49. #14579101	C₂×C₄	semidirect
53. C₂×C₇₉⋊C₁₆	C₂×C₄	semidirect
66. M₃₂×C₇₉	C₂×C₄	semidirect
70. C₂×C₁₆×C₇₉	C₂×C₄	semidirect
175. #1379	D₇₉	semidirect
176. C₇₉	#12604	semidirect
223. M₃₂	C₄×D₇₉	semidirect
231. C₂×C₁₆	C₄×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23771822	2
C₂×#19098348	2
C₂×#19098346	2
M₃₂×Dic₇₉	2
C₂×#19098338	2
#23761489	2
C₇₉×#1379	2
#12604	79