Subgroups — #34175232

Order 20224.

462 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098289	C₂	1	2	yes	maximal
2	C₇₉×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
4	#19098289	—	2	2	no	—
5	#19098289	C₄	1	4	yes	complemented
6	C₂²×C₄⋊C₄×C₇₉	C₄	1	1	yes	characteristic
7	C₂²×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₇₉	C₄	1	1	yes	characteristic
9	C₂×#95×C₇₉	—	2	2	no	—
10	C₂²×C₇₉⋊C₈	—	4	2	no	—
11	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
12	C₂²⋊(C₇₉⋊C₈)	—	4	2	no	—
13	C₂³×C₄×C₇₉	—	2	1	no	—
14	C₂³×C₄×C₇₉	—	2	1	no	—
15	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₇₉	D₄	1	2	yes	—
17	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
18	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
19	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
20	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
21	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
22	C₂×C₄⋊C₄×C₇₉	C₈	1	2	yes	complemented
23	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₇₉	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₇₉	—	2	1	no	—
27	C₂×C₄²×C₇₉	—	4	1	no	—
28	C₂×C₄²×C₇₉	—	4	1	no	—
29	C₂×C₄²×C₇₉	C₈	1	2	yes	complemented
30	#95×C₇₉	—	4	4	no	—
31	C₂×C₇₉⋊C₈	—	4	4	no	—
32	C₂×C₇₉⋊C₈	—	8	2	no	—
33	C₂×C₇₉⋊C₈	—	8	2	no	—
34	C₂⁴×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	—	2	1	no	—
36	C₂²×C₄×C₇₉	—	2	1	no	—
37	C₂²×C₄×C₇₉	—	2	1	no	—
38	C₂²×C₄×C₇₉	—	2	1	no	—
39	C₂²×C₄×C₇₉	—	2	2	no	—
40	C₂²×C₄×C₇₉	—	2	2	no	—
41	C₂²×C₄×C₇₉	—	2	2	no	—
42	C₂²×C₄×C₇₉	—	2	2	no	—
43	C₂²×C₄×C₇₉	—	2	2	no	—
44	C₂²×C₄×C₇₉	—	2	2	no	—
45	C₂²×C₄×C₇₉	—	4	1	no	—
46	C₂²×C₄×C₇₉	—	4	1	no	—
47	C₂²×C₄×C₇₉	—	4	1	no	—
48	C₂²×C₄×C₇₉	—	4	1	no	—
49	C₂²×C₄×C₇₉	—	4	2	no	—
50	C₂²×C₄×C₇₉	—	4	2	no	—
51	C₂²×C₄×C₇₉	—	4	2	no	—
52	C₂²×C₄×C₇₉	—	4	2	no	—
53	C₂²×C₄×C₇₉	—	4	1	no	—
54	C₂²×C₄×C₇₉	—	4	1	no	—
55	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
56	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
63	C₄⋊C₄×C₇₉	—	4	2	no	—
64	C₄⋊C₄×C₇₉	—	4	1	no	—
65	C₄⋊C₄×C₇₉	—	4	1	no	—
66	C₄⋊C₄×C₇₉	—	4	2	no	—
67	C₄⋊C₄×C₇₉	—	4	2	no	—
68	C₄²×C₇₉	—	4	2	no	—
69	C₄²×C₇₉	—	4	2	no	—
70	C₂³×C₇₉	—	2	1	no	—
71	C₂³×C₇₉	—	2	1	no	—
72	C₂³×C₇₉	—	2	1	no	—
73	C₂³×C₇₉	—	2	1	no	—
74	C₂³×C₇₉	#95	1	1	yes	characteristic
75	C₂³×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
76	C₂³×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
77	C₂³×C₇₉	#99	1	1	yes	characteristic
78	C₂³×C₇₉	#99	1	1	yes	characteristic
79	C₂³×C₇₉	#100	1	1	yes	characteristic
80	C₂³×C₇₉	#101	1	1	yes	characteristic
81	C₂×C₄×C₇₉	—	2	1	no	—
82	C₂×C₄×C₇₉	—	2	1	no	—
83	C₂×C₄×C₇₉	—	2	1	no	—
84	C₂×C₄×C₇₉	—	2	1	no	—
85	C₂×C₄×C₇₉	—	2	1	no	—
86	C₂×C₄×C₇₉	—	2	1	no	—
87	C₂×C₄×C₇₉	—	2	1	no	—
88	C₂×C₄×C₇₉	—	2	1	no	—
89	C₂×C₄×C₇₉	—	2	1	no	—
90	C₂×C₄×C₇₉	—	2	1	no	—
91	C₂×C₄×C₇₉	—	2	1	no	—
92	C₂×C₄×C₇₉	—	2	1	no	—
93	C₂×C₄×C₇₉	—	2	4	no	—
94	C₂×C₄×C₇₉	—	2	2	no	—
95	C₂×C₄×C₇₉	—	2	2	no	—
96	C₂×C₄×C₇₉	—	4	2	no	—
97	C₂×C₄×C₇₉	—	4	2	no	—
98	C₂×C₄×C₇₉	—	4	2	no	—
99	C₂×C₄×C₇₉	—	4	2	no	—
100	C₂×C₄×C₇₉	—	4	2	no	—
101	C₂×C₄×C₇₉	—	4	2	no	—
102	C₂×C₄×C₇₉	—	4	2	no	—
103	C₂×C₄×C₇₉	—	4	2	no	—
104	C₂×C₄×C₇₉	—	4	2	no	—
105	C₂×C₄×C₇₉	—	4	2	no	—
106	C₂×C₄×C₇₉	—	4	4	no	—
107	C₂×C₄×C₇₉	—	4	2	no	—
108	C₂×C₄×C₇₉	—	8	2	no	—
109	C₂×C₄×C₇₉	—	8	2	no	—
110	C₂×C₄×C₇₉	—	8	1	no	—
111	C₂×C₄×C₇₉	—	8	1	no	—
112	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
113	C₂×C₄×C₇₉	#97	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
116	C₇₉⋊C₈	—	8	4	no	—
117	C₂²×C₇₉	—	2	1	no	—
118	C₂²×C₇₉	—	2	1	no	—
119	C₂²×C₇₉	—	2	1	no	—
120	C₂²×C₇₉	—	2	1	no	—
121	C₂²×C₇₉	—	2	1	no	—
122	C₂²×C₇₉	—	2	1	no	—
123	C₂²×C₇₉	—	2	1	no	—
124	C₂²×C₇₉	—	2	1	no	—
125	C₂²×C₇₉	—	2	1	no	—
126	C₂²×C₇₉	—	2	1	no	—
127	C₂²×C₇₉	—	2	1	no	—
128	C₂²×C₇₉	—	2	1	no	—
129	C₂²×C₇₉	#323	1	1	yes	characteristic
130	C₂²×C₇₉	#323	1	1	yes	characteristic
131	C₂²×C₇₉	#324	1	1	yes	characteristic
132	C₂²×C₇₉	#324	1	1	yes	characteristic
133	C₂²×C₇₉	#336	1	1	yes	characteristic
134	C₂²×C₇₉	#342	1	1	yes	characteristic
135	C₂²×C₇₉	#343	1	1	yes	characteristic
136	C₂²×C₇₉	#353	1	1	yes	characteristic
137	C₂²×C₇₉	#354	1	1	yes	characteristic
138	C₂²×C₇₉	#355	1	1	yes	characteristic
139	C₂²×C₇₉	#356	1	1	yes	characteristic
140	C₄×C₇₉	—	2	1	no	—
141	C₄×C₇₉	—	2	1	no	—
142	C₄×C₇₉	—	2	1	no	—
143	C₄×C₇₉	—	2	1	no	—
144	C₄×C₇₉	—	4	4	no	—
145	C₄×C₇₉	—	4	2	no	—
146	C₄×C₇₉	—	4	2	no	—
147	C₄×C₇₉	—	8	2	no	—
148	#7189	—	79	1	no	maximal, Sylow
149	C₂×C₇₉	—	2	1	no	—
150	C₂×C₇₉	—	2	1	no	—
151	C₂×C₇₉	—	2	1	no	—
152	C₂×C₇₉	—	2	1	no	—
153	C₂×C₇₉	#1280	1	1	yes	characteristic
154	C₂×C₇₉	#1318	1	1	yes	characteristic
155	C₂×C₇₉	#1319	1	1	yes	characteristic
156	C₂×C₇₉	#1324	1	1	yes	characteristic
157	C₂×C₇₉	#1325	1	1	yes	characteristic
158	C₂×C₇₉	#1393	1	1	yes	characteristic
159	C₂×C₇₉	#1394	1	1	yes	characteristic
160	C₂×#324	—	79	2	no	—
161	C₄⋊#95	D₇₉	1	1	yes	characteristic
162	C₇₉	#7189	1	1	yes	characteristic, complemented, Sylow
163	C₂²×C₄⋊C₄	Dic₇₉	1	1	yes	characteristic
164	C₂²×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
165	C₂²×C₄²	Dic₇₉	1	1	yes	characteristic
166	C₂×C₂²⋊C₈	—	158	2	no	—
167	C₂×#95	—	2	2	no	—
168	#324	—	79	4	no	—
169	#324	—	158	2	no	—
170	C₂³×C₄	—	2	1	no	—
171	C₂³×C₄	—	2	1	no	—
172	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
173	C₂³×C₄	C₂²⋊D₇₉	1	2	yes	—
174	C₂²×C₈	—	316	2	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	—	4	1	no	—
177	C₂×C₄⋊C₄	—	4	1	no	—
178	C₂×C₄⋊C₄	—	4	1	no	—
179	C₂×C₄⋊C₄	—	4	1	no	—
180	C₂×C₄⋊C₄	C₇₉⋊C₈	1	2	yes	complemented
181	C₂×C₄⋊C₄	Dic₁₅₈	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₄×D₇₉	1	2	yes	—
183	C₂×C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
184	C₂×C₄²	—	2	1	no	—
185	C₂×C₄²	—	4	1	no	—
186	C₂×C₄²	—	4	1	no	—
187	C₂×C₄²	C₇₉⋊C₈	1	2	yes	complemented
188	C₂²⋊C₈	—	316	2	no	—
189	C₂²⋊C₈	—	158	4	no	—
190	#95	—	4	4	no	—
191	C₂⁴	C₂²⋊Dic₇₉	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	—	4	1	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
213	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
214	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
215	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
218	C₂²×C₄	#11929608	1	1	yes	characteristic
219	C₂²×C₄	#11929608	1	1	yes	characteristic
220	C₂×C₈	—	316	4	no	—
221	C₂×C₈	—	632	2	no	—
222	C₂×C₈	—	632	2	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄²	—	4	2	no	—
229	C₄²	—	4	2	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	1	no	—
234	C₂³	C₇₉⋊#95	1	1	yes	characteristic
235	C₂³	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
236	C₂³	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
237	C₂³	#14579076	1	1	yes	characteristic
238	C₂³	#14579076	1	1	yes	characteristic
239	C₂³	#14579078	1	1	yes	characteristic
240	C₂³	#14579080	1	1	yes	characteristic
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	8	2	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	C₄×C₇₉⋊C₈	1	1	yes	characteristic
273	C₂×C₄	#14579072	1	1	yes	characteristic
274	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
275	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
276	C₈	—	632	4	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	#19098287	1	1	yes	characteristic
290	C₂²	#19098287	1	1	yes	characteristic
291	C₂²	#19098289	1	1	yes	characteristic
292	C₂²	#19098289	1	1	yes	characteristic
293	C₂²	#19098318	1	1	yes	characteristic
294	C₂²	#19098330	1	1	yes	characteristic
295	C₂²	#19098332	1	1	yes	characteristic
296	C₂²	#19098356	1	1	yes	characteristic
297	C₂²	#19098358	1	1	yes	characteristic
298	C₂²	#19098360	1	1	yes	characteristic
299	C₂²	#19098362	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	4	no	—
305	C₄	—	4	2	no	—
306	C₄	—	4	2	no	—
307	C₄	—	8	2	no	—
308	C₂	—	2	1	no	—
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	#23761243	1	1	yes	characteristic
313	C₂	#23761332	1	1	yes	characteristic
314	C₂	#23761334	1	1	yes	characteristic
315	C₂	#23761344	1	1	yes	characteristic
316	C₂	#23761346	1	1	yes	characteristic
317	C₂	#23761525	1	1	yes	characteristic
318	C₂	#23761527	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579076
commutator	C₂×C₄×C₇₉	C₄×C₈
Frattini	C₂²×C₄⋊C₄	C₂×D₇₉
Fitting	C₇₉×C₄⋊#95	C₂
socle	C₂³×C₇₉	#99
prime radical	C₂⁴×C₇₉	C₂²⋊C₄
cosocle	C₂²×C₄⋊C₄×C₇₉	C₂²
hypercenter	C₄⋊#95	D₇₉
hypocenter	C₇₉	#7189
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ C₄⋊#95

Complements

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

N	H ≅ G/N	type
5. #19098289	C₄	semidirect
22. C₂×C₄⋊C₄×C₇₉	C₈	semidirect
29. C₂×C₄²×C₇₉	C₈	semidirect
162. C₇₉	#7189	semidirect
180. C₂×C₄⋊C₄	C₇₉⋊C₈	semidirect
187. C₂×C₄²	C₇₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098289	2
C₇₉×C₄⋊#95	2
#7189	79