Subgroups — #34206638

Order 20224.

510 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579102	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098388	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098364	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19098363	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#23764266	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1970	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₆₃₂	C₂²	1	1	yes	characteristic
9	C₂×#14579102	—	2	2	no	—
10	C₂×#14579102	—	2	2	no	—
11	C₂×#14579102	C₄	1	4	yes	complemented
12	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×#14579095	C₂²	1	1	yes	characteristic
15	C₂×#14579083	—	2	1	no	—
16	#19098465	—	2	1	no	—
17	#19098388	—	2	2	no	—
18	#19098364	—	2	2	no	—
19	#19098363	—	2	2	no	—
20	C₄×C₇₉⋊C₁₆	—	2	2	no	—
21	C₂²×D₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×#102×C₇₉	—	2	1	no	—
24	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
25	#390×C₇₉	—	2	1	no	—
26	C₂²×D₃₁₆	—	2	1	no	—
27	C₂×D₆₃₂	—	2	1	no	—
28	C₂×D₆₃₂	—	2	1	no	—
29	C₂×D₆₃₂	—	2	1	no	—
30	C₂×D₆₃₂	—	2	1	no	—
31	C₂×D₆₃₂	—	2	1	no	—
32	C₂×D₆₃₂	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₇₉	—	2	1	no	—
34	C₂×(C₂×D₇₉)⋊C₄	—	4	2	no	—
35	#14579102	—	2	8	no	—
36	C₂×C₇₉⋊C₁₆	—	2	1	no	—
37	C₂×C₇₉⋊C₁₆	—	2	1	no	—
38	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
40	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
41	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
42	#14579095	—	2	1	no	—
43	#14579095	—	2	1	no	—
44	#14579083	—	4	2	no	—
45	C₂²×D₄×C₇₉	—	2	1	no	—
46	C₂×D₈×C₇₉	—	2	1	no	—
47	C₂×D₈×C₇₉	—	2	1	no	—
48	C₂×D₈×C₇₉	—	2	1	no	—
49	C₂×D₈×C₇₉	—	2	1	no	—
50	C₂×D₈×C₇₉	—	2	1	no	—
51	C₂×D₈×C₇₉	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
55	C₂×C₂²⋊C₄×C₇₉	—	4	2	no	—
56	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
57	#107×C₇₉	—	2	1	no	—
58	#107×C₇₉	—	2	1	no	—
59	#102×C₇₉	—	4	2	no	—
60	C₄×C₈×C₇₉	—	2	1	no	—
61	C₄×C₈×C₇₉	—	2	1	no	—
62	C₂³×D₇₉	—	4	1	no	—
63	C₂×D₃₁₆	—	2	2	no	—
64	C₂×D₃₁₆	—	4	1	no	—
65	C₂×D₃₁₆	—	4	1	no	—
66	C₂×D₃₁₆	—	4	1	no	—
67	C₂×D₃₁₆	—	4	1	no	—
68	C₂×D₃₁₆	—	4	1	no	—
69	C₂²×Dic₇₉	—	4	1	no	—
70	D₆₃₂	—	2	2	no	—
71	D₆₃₂	—	2	2	no	—
72	D₆₃₂	—	4	1	no	—
73	D₆₃₂	—	4	1	no	—
74	C₄⋊Dic₇₉	—	4	1	no	—
75	C₄⋊Dic₇₉	—	4	1	no	—
76	(C₂×D₇₉)⋊C₄	—	8	4	no	—
77	C₇₉⋊C₁₆	—	2	2	no	—
78	C₇₉⋊C₁₆	C₂²⋊C₄	1	4	yes	complemented
79	C₂⁴×C₇₉	—	4	1	no	—
80	C₂×D₄×C₇₉	—	2	2	no	—
81	C₂×D₄×C₇₉	—	4	1	no	—
82	C₂×D₄×C₇₉	—	4	1	no	—
83	C₂×D₄×C₇₉	—	4	1	no	—
84	C₂×D₄×C₇₉	—	4	1	no	—
85	C₂×D₄×C₇₉	—	4	1	no	—
86	C₂²×C₄×C₇₉	—	4	1	no	—
87	C₂²×C₄×C₇₉	D₈	1	2	yes	—
88	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
89	D₈×C₇₉	—	2	2	no	—
90	D₈×C₇₉	—	2	2	no	—
91	D₈×C₇₉	—	4	1	no	—
92	D₈×C₇₉	—	4	1	no	—
93	C₂×C₈×C₇₉	—	2	1	no	—
94	C₂×C₈×C₇₉	—	2	1	no	—
95	C₂×C₈×C₇₉	D₈	1	2	yes	—
96	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
101	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
102	C₄⋊C₄×C₇₉	—	4	1	no	—
103	C₄⋊C₄×C₇₉	—	4	1	no	—
104	C₂²⋊C₄×C₇₉	—	8	4	no	—
105	C₄²×C₇₉	—	2	1	no	—
106	C₄²×C₇₉	—	2	1	no	—
107	C₂³×C₇₉	—	4	2	no	—
108	C₂³×C₇₉	—	8	1	no	—
109	C₂³×C₇₉	—	8	1	no	—
110	C₂³×C₇₉	—	8	2	no	—
111	C₂³×C₇₉	—	8	2	no	—
112	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
113	C₂²×D₇₉	—	4	2	no	—
114	C₂²×D₇₉	—	8	1	no	—
115	C₂²×D₇₉	—	8	1	no	—
116	C₂²×D₇₉	—	8	2	no	—
117	C₂²×D₇₉	—	8	2	no	—
118	D₄×C₇₉	—	4	2	no	—
119	D₄×C₇₉	—	4	2	no	—
120	D₄×C₇₉	—	8	1	no	—
121	D₄×C₇₉	—	8	1	no	—
122	C₂×C₄×C₇₉	—	2	2	no	—
123	C₂×C₄×C₇₉	—	2	2	no	—
124	C₂×C₄×C₇₉	—	8	1	no	—
125	C₂×C₄×C₇₉	—	8	2	no	—
126	C₂×C₄×C₇₉	D₁₆	1	4	yes	complemented
127	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
129	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
130	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
132	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
133	C₂×Dic₇₉	—	8	1	no	—
134	C₂×Dic₇₉	—	8	2	no	—
135	D₃₁₆	—	4	2	no	—
136	D₃₁₆	—	4	2	no	—
137	D₃₁₆	—	8	1	no	—
138	D₃₁₆	—	8	1	no	—
139	C₈×C₇₉	—	2	2	no	—
140	C₈×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
142	C₈×C₇₉	#121	1	1	yes	characteristic
143	C₈×C₇₉	#124	1	1	yes	characteristic
144	C₂²×C₇₉	—	8	4	no	—
145	C₂²×C₇₉	—	8	2	no	—
146	C₂²×C₇₉	—	16	2	no	—
147	C₂²×C₇₉	—	16	1	no	—
148	C₂²×C₇₉	—	16	1	no	—
149	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
150	C₂²×C₇₉	#466	1	1	yes	characteristic
151	C₂²×C₇₉	#493	1	1	yes	characteristic
152	C₂²×C₇₉	C₂×D₁₆	1	2	yes	—
153	C₂²×C₇₉	#508	1	2	yes	—
154	C₂×D₇₉	—	8	4	no	—
155	C₂×D₇₉	—	8	2	no	—
156	C₂×D₇₉	—	16	2	no	—
157	C₂×D₇₉	—	16	1	no	—
158	C₂×D₇₉	—	16	1	no	—
159	C₄×C₇₉	—	2	4	no	—
160	C₄×C₇₉	—	16	1	no	—
161	C₄×C₇₉	#390	1	1	yes	characteristic
162	C₄×C₇₉	C₄×D₈	1	1	yes	characteristic
163	C₄×C₇₉	#466	1	1	yes	characteristic
164	C₄×C₇₉	#495	1	1	yes	characteristic
165	Dic₇₉	—	16	1	no	—
166	#13235	—	79	1	no	maximal, Sylow
167	C₂×C₇₉	—	16	2	no	—
168	C₂×C₇₉	#1970	1	1	yes	characteristic
169	C₂×C₇₉	C₄×D₁₆	1	2	yes	—
170	C₂×C₇₉	#2215	1	2	yes	—
171	C₂×C₇₉	#2246	1	1	yes	characteristic
172	C₂×C₇₉	#2249	1	1	yes	characteristic
173	D₇₉	—	16	2	no	—
174	C₂²×D₁₆	—	79	1	no	—
175	C₂×#366	—	79	1	no	—
176	C₂×#357	—	79	1	no	—
177	C₂×#357	—	79	1	no	—
178	C₂×C₄×C₁₆	—	79	1	no	—
179	#1970	—	79	1	no	—
180	#1970	D₇₉	1	1	yes	characteristic, complemented
181	C₇₉	#13235	1	1	yes	characteristic, complemented, Sylow
182	C₂²×D₈	—	79	1	no	—
183	C₂²×D₈	C₂×D₇₉	1	1	yes	characteristic
184	C₂×D₁₆	—	79	4	no	—
185	C₂×D₁₆	—	158	2	no	—
186	C₂×D₁₆	—	158	2	no	—
187	C₂²×C₁₆	—	79	1	no	—
188	C₂²×C₁₆	—	79	1	no	—
189	C₂×#107	—	79	1	no	—
190	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
191	C₂×#102	—	2	1	no	—
192	C₂×#102	—	158	1	no	—
193	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
194	#390	—	2	1	no	—
195	#390	—	158	1	no	—
196	#366	—	158	2	no	—
197	#357	—	158	2	no	—
198	#357	—	158	2	no	—
199	C₄×C₁₆	—	158	2	no	—
200	C₂²×D₄	—	2	1	no	—
201	C₂²×D₄	—	158	1	no	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	—	2	1	no	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	—	2	1	no	—
207	C₂×D₈	—	79	2	no	—
208	C₂×D₈	—	158	1	no	—
209	C₂×D₈	—	158	1	no	—
210	C₂×D₈	—	158	1	no	—
211	C₂×D₈	—	158	1	no	—
212	C₂×D₈	—	158	1	no	—
213	C₂×D₈	C₄×D₇₉	1	2	yes	—
214	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	158	1	no	—
218	C₂×C₂²⋊C₄	—	4	2	no	—
219	C₂×C₂²⋊C₄	—	316	2	no	—
220	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
221	D₁₆	—	158	8	no	—
222	C₂×C₁₆	—	79	2	no	—
223	C₂×C₁₆	—	79	2	no	—
224	C₂×C₁₆	—	79	2	no	—
225	C₂×C₁₆	—	79	2	no	—
226	C₂×C₁₆	—	158	1	no	—
227	C₂×C₁₆	—	158	1	no	—
228	#107	—	2	1	no	—
229	#107	—	2	1	no	—
230	#107	—	158	1	no	—
231	#107	—	158	1	no	—
232	#102	—	4	2	no	—
233	#102	—	316	2	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	—	2	1	no	—
236	C₂⁴	—	4	1	no	—
237	C₂⁴	—	316	1	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	316	1	no	—
245	C₂×D₄	—	316	1	no	—
246	C₂×D₄	—	316	1	no	—
247	C₂×D₄	—	316	1	no	—
248	C₂×D₄	—	316	1	no	—
249	C₂×D₄	—	158	2	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	316	1	no	—
252	C₂²×C₄	#11929609	1	2	yes	—
253	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
254	D₈	—	2	2	no	—
255	D₈	—	2	2	no	—
256	D₈	—	4	1	no	—
257	D₈	—	4	1	no	—
258	D₈	—	316	1	no	—
259	D₈	—	316	1	no	—
260	D₈	—	158	2	no	—
261	D₈	—	158	2	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	#11929609	1	2	yes	—
265	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
266	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
267	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
268	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
269	C₂×C₈	#11929623	1	1	yes	characteristic
270	C₂×C₈	#11929624	1	1	yes	characteristic
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	316	1	no	—
274	C₄⋊C₄	—	316	1	no	—
275	C₂²⋊C₄	—	8	4	no	—
276	C₂²⋊C₄	—	632	4	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₁₆	—	79	4	no	—
280	C₁₆	—	158	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	8	1	no	—
283	C₂³	—	8	2	no	—
284	C₂³	—	8	2	no	—
285	C₂³	—	8	1	no	—
286	C₂³	—	316	2	no	—
287	C₂³	—	632	1	no	—
288	C₂³	—	632	2	no	—
289	C₂³	—	632	2	no	—
290	C₂³	—	632	1	no	—
291	C₂³	C₂×#11929609	1	1	yes	characteristic
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	8	1	no	—
295	D₄	—	8	1	no	—
296	D₄	—	316	2	no	—
297	D₄	—	316	2	no	—
298	D₄	—	632	1	no	—
299	D₄	—	632	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	—	632	1	no	—
305	C₂×C₄	—	632	2	no	—
306	C₂×C₄	#14579102	1	4	yes	complemented
307	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
308	C₂×C₄	#14579133	1	1	yes	characteristic
309	C₂×C₄	#14579157	1	1	yes	characteristic
310	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
311	C₂×C₄	#14579181	1	1	yes	characteristic
312	C₂×C₄	#14579181	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
315	C₈	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
316	C₈	#14579134	1	1	yes	characteristic
317	C₈	#14579147	1	1	yes	characteristic
318	C₂²	—	8	2	no	—
319	C₂²	—	8	4	no	—
320	C₂²	—	16	1	no	—
321	C₂²	—	16	1	no	—
322	C₂²	—	16	2	no	—
323	C₂²	—	632	2	no	—
324	C₂²	—	632	4	no	—
325	C₂²	—	1264	1	no	—
326	C₂²	—	1264	1	no	—
327	C₂²	—	1264	2	no	—
328	C₂²	C₄×#11929609	1	1	yes	characteristic
329	C₂²	#19098797	1	1	yes	characteristic
330	C₂²	#19098939	1	1	yes	characteristic
331	C₂²	C₂×#14579102	1	2	yes	—
332	C₂²	#19098993	1	2	yes	—
333	C₄	—	2	4	no	—
334	C₄	—	16	1	no	—
335	C₄	—	1264	1	no	—
336	C₄	#19098463	1	1	yes	characteristic
337	C₄	C₄×#11929609	1	1	yes	characteristic
338	C₄	#19098797	1	1	yes	characteristic
339	C₄	#19098945	1	1	yes	characteristic
340	C₂	—	16	2	no	—
341	C₂	—	1264	2	no	—
342	C₂	#23764265	1	1	yes	characteristic
343	C₂	C₄×#14579102	1	2	yes	—
344	C₂	#23765484	1	2	yes	—
345	C₂	#23765649	1	1	yes	characteristic
346	C₂	#23765659	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1970	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂²×#14579102	C₂
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1970	D₇₉
hypocenter	C₇₉	#13235
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1970

Complements

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

N	H ≅ G/N	type
2. C₂×#19098388	C₂	semidirect
3. C₂×#19098364	C₂	semidirect
4. C₂×#19098363	C₂	semidirect
5. C₂×C₄×C₇₉⋊C₁₆	C₂	semidirect
6. #23764266	C₂	semidirect
7. C₇₉×#1970	C₂	semidirect
11. C₂×#14579102	C₄	semidirect
78. C₇₉⋊C₁₆	C₂²⋊C₄	semidirect
126. C₂×C₄×C₇₉	D₁₆	semidirect
180. #1970	D₇₉	semidirect
181. C₇₉	#13235	semidirect
306. C₂×C₄	#14579102	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14579102	2
C₂×#19098388	2
C₂×#19098364	2
C₂×#19098363	2
C₂×C₄×C₇₉⋊C₁₆	2
#23764266	2
C₇₉×#1970	2
#13235	79