Subgroups — #34258131

Order 20224.

646 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772513	C₂	1	2	yes	maximal, complemented
2	C₄×#14579183	C₂	1	1	yes	maximal, characteristic
3	#23762315	C₂	1	2	yes	maximal, complemented
4	#23762310	C₂	1	2	yes	maximal, complemented
5	#23762292	C₂	1	4	yes	maximal, complemented
6	C₂×#19098314	C₂	1	1	yes	maximal, characteristic, complemented
7	#23761788	C₂	1	2	yes	maximal, complemented
8	C₇₉×#1450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14579183	C₄	1	1	yes	characteristic
10	C₄×#11929624	C₂²	1	2	yes	—
11	#19098693	C₂²	1	4	yes	—
12	C₄×Dic₃₁₆	C₄	1	1	yes	characteristic
13	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
14	C₄×#11929613	C₄	1	2	yes	—
15	C₄×#11929613	C₂²	1	2	yes	—
16	C₄×D₆₃₂	C₄	1	1	yes	characteristic
17	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
18	#19098642	C₂²	1	4	yes	—
19	#19098631	—	2	4	no	—
20	#19098619	—	2	2	no	—
21	#19098601	C₄	1	1	yes	characteristic
22	#19098585	—	2	2	no	—
23	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
24	#19098560	C₄	1	2	yes	—
25	#19098314	C₂²	1	2	yes	—
26	#19098314	C₂²	1	2	yes	—
27	#19098295	C₂²	1	4	yes	—
28	#19098292	C₂²	1	4	yes	—
29	#431×C₇₉	C₂²	1	1	yes	characteristic
30	#431×C₇₉	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
32	#322×C₇₉	C₂²	1	2	yes	—
33	C₈²×C₇₉	C₂²	1	2	yes	—
34	C₂×#11929624	C₂×C₄	1	2	yes	—
35	#14579183	—	2	4	no	—
36	C₂×Dic₃₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×#11929613	C₂×C₄	1	2	yes	—
38	C₂×D₆₃₂	C₂×C₄	1	1	yes	characteristic
39	C₂×#11929607	—	2	4	no	—
40	C₂²×C₇₉⋊C₈	—	2	2	no	—
41	C₄×Dic₁₅₈	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₅₈	C₂³	1	2	yes	—
43	C₄×D₃₁₆	C₂×C₄	1	2	yes	—
44	C₄×D₃₁₆	C₂³	1	2	yes	—
45	C₄×C₂²⋊D₇₉	—	2	2	no	—
46	C₄×C₂²⋊D₇₉	—	2	2	no	—
47	#14579120	C₂×C₄	1	2	yes	—
48	#14579118	—	2	2	no	—
49	C₄²×D₇₉	—	2	2	no	—
50	#14579095	C₂×C₄	1	1	yes	characteristic
51	#14579095	C₂×C₄	1	1	yes	characteristic
52	#14579093	C₂×C₄	1	2	yes	—
53	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
54	C₄⋊(C₇₉⋊C₈)	C₂³	1	4	yes	—
55	Dic₇₉⋊C₈	—	2	4	no	—
56	#14579086	C₂×C₄	1	2	yes	—
57	#14579086	C₂×C₄	1	2	yes	—
58	#14579083	C₂×C₄	1	2	yes	—
59	#14579083	C₂×C₄	1	2	yes	—
60	(C₂×D₇₉)⋊C₈	—	2	4	no	—
61	#14579071	—	2	4	no	—
62	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₇₉	—	2	1	no	—
66	C₄⋊C₈×C₇₉	D₄	1	2	yes	—
67	C₂²⋊C₈×C₇₉	—	2	1	no	—
68	C₂²⋊C₈×C₇₉	D₄	1	2	yes	—
69	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₇₉	C₂³	1	2	yes	—
72	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₇₉	C₂³	1	2	yes	—
75	#11929624	—	2	4	no	—
76	#11929624	—	2	4	no	—
77	C₂×Dic₁₅₈	C₂²×C₄	1	2	yes	—
78	C₂×D₃₁₆	C₂²×C₄	1	2	yes	—
79	C₂×C₂²⋊D₇₉	—	2	2	no	—
80	C₂×C₄×D₇₉	—	2	2	no	—
81	C₂×C₄×D₇₉	—	2	2	no	—
82	C₂×C₄×D₇₉	—	2	2	no	—
83	Dic₃₁₆	—	2	2	no	—
84	#11929613	—	2	4	no	—
85	D₆₃₂	—	2	2	no	—
86	#11929607	—	8	4	no	—
87	C₂×C₇₉⋊C₈	—	2	4	no	—
88	C₂×C₇₉⋊C₈	—	8	2	no	—
89	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
90	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
91	Dic₇₉⋊C₄	—	2	2	no	—
92	Dic₇₉⋊C₄	—	2	2	no	—
93	C₂²⋊Dic₇₉	—	2	2	no	—
94	(C₂×D₇₉)⋊C₄	—	2	2	no	—
95	(C₂×D₇₉)⋊C₄	—	2	2	no	—
96	C₄×Dic₇₉	—	2	2	no	—
97	C₄×Dic₇₉	—	2	2	no	—
98	C₄×Dic₇₉	—	2	2	no	—
99	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
102	C₂×C₈×C₇₉	—	2	2	no	—
103	C₂×C₈×C₇₉	—	2	2	no	—
104	C₂×C₈×C₇₉	M₁₆	1	4	yes	complemented
105	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
110	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₇₉	#41	1	1	yes	characteristic
113	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
114	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
115	C₂²×D₇₉	—	2	2	no	—
116	C₂×C₄×C₇₉	—	2	2	no	—
117	C₂×C₄×C₇₉	—	4	1	no	—
118	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
122	C₂×C₄×C₇₉	#131	1	2	yes	—
123	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
126	C₂²⋊D₇₉	—	4	4	no	—
127	C₂×Dic₇₉	—	2	2	no	—
128	C₂×Dic₇₉	—	2	2	no	—
129	C₂×Dic₇₉	—	2	2	no	—
130	D₃₁₆	—	2	2	no	—
131	D₃₁₆	—	2	2	no	—
132	C₄×D₇₉	—	4	2	no	—
133	C₄×D₇₉	—	4	2	no	—
134	C₄×D₇₉	—	4	4	no	—
135	C₄×D₇₉	—	8	2	no	—
136	Dic₁₅₈	—	2	2	no	—
137	Dic₁₅₈	—	2	2	no	—
138	C₈×C₇₉	—	2	1	no	—
139	C₈×C₇₉	—	2	1	no	—
140	C₈×C₇₉	—	2	4	no	—
141	C₈×C₇₉	—	4	2	no	—
142	C₈×C₇₉	C₂×M₁₆	1	2	yes	—
143	C₈×C₇₉	C₂×M₁₆	1	2	yes	—
144	C₇₉⋊C₈	—	4	4	no	—
145	C₂²×C₇₉	—	2	1	no	—
146	C₂²×C₇₉	—	2	1	no	—
147	C₂²×C₇₉	#436	1	2	yes	—
148	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×D₇₉	—	4	2	no	—
150	C₂×D₇₉	—	4	2	no	—
151	C₂×D₇₉	—	4	2	no	—
152	C₄×C₇₉	—	2	1	no	—
153	C₄×C₇₉	—	2	1	no	—
154	C₄×C₇₉	—	2	1	no	—
155	C₄×C₇₉	#436	1	2	yes	—
156	C₄×C₇₉	C₂²×M₁₆	1	1	yes	characteristic
157	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
158	C₄×C₇₉	#577	1	2	yes	—
159	Dic₇₉	—	4	2	no	—
160	Dic₇₉	—	4	2	no	—
161	Dic₇₉	—	4	2	no	—
162	#17899	—	79	1	no	maximal, Sylow
163	C₂×C₇₉	—	2	2	no	—
164	C₂×C₇₉	C₂×#436	1	1	yes	characteristic
165	C₂×C₇₉	#2952	1	1	yes	characteristic
166	C₂×C₇₉	#2957	1	1	yes	characteristic
167	D₇₉	—	8	2	no	—
168	#2989	—	79	2	no	—
169	C₄×#135	—	79	1	no	—
170	#1594	—	79	2	no	—
171	#1593	—	79	2	no	—
172	#1590	—	79	4	no	—
173	C₂×#334	—	79	1	no	—
174	#1477	—	79	2	no	—
175	#1450	D₇₉	1	1	yes	characteristic, complemented
176	C₇₉	#17899	1	1	yes	characteristic, complemented, Sylow
177	C₂×#135	—	79	1	no	—
178	C₄×#41	—	79	2	no	—
179	#446	—	79	4	no	—
180	C₄×Q₁₆	—	79	1	no	—
181	C₄×Q₁₆	—	79	1	no	—
182	C₄×SD₁₆	—	79	2	no	—
183	C₄×SD₁₆	—	79	2	no	—
184	C₄×D₈	—	79	1	no	—
185	C₄×D₈	—	79	1	no	—
186	#436	—	79	4	no	—
187	#435	—	158	4	no	—
188	#432	—	158	2	no	—
189	#431	C₂×D₇₉	1	1	yes	characteristic
190	#431	C₂×D₇₉	1	1	yes	characteristic
191	#427	—	79	1	no	—
192	#423	—	158	2	no	—
193	C₂×C₄⋊C₈	—	79	2	no	—
194	#416	—	79	2	no	—
195	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
196	#334	—	79	2	no	—
197	#334	—	79	2	no	—
198	Q₈⋊C₈	—	79	4	no	—
199	D₄⋊C₈	—	79	4	no	—
200	#322	C₂×D₇₉	1	2	yes	—
201	C₈²	C₂×D₇₉	1	2	yes	—
202	C₂×#41	—	79	2	no	—
203	#135	—	158	4	no	—
204	C₂×Q₁₆	—	79	1	no	—
205	C₂×SD₁₆	—	79	2	no	—
206	C₂×D₈	—	79	1	no	—
207	C₂×M₁₆	—	158	4	no	—
208	C₂²×C₈	—	158	2	no	—
209	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
210	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
211	C₄×Q₈	—	79	2	no	—
212	C₄×Q₈	—	79	2	no	—
213	C₄×D₄	—	79	2	no	—
214	C₄×D₄	—	79	2	no	—
215	C₄×D₄	—	158	2	no	—
216	C₄×D₄	—	158	2	no	—
217	#117	—	79	2	no	—
218	#117	—	158	2	no	—
219	C₂×C₄²	—	158	2	no	—
220	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
221	#107	—	79	1	no	—
222	#107	—	79	1	no	—
223	#106	—	79	2	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	79	4	no	—
226	C₄⋊C₈	—	158	4	no	—
227	C₄⋊C₈	—	158	2	no	—
228	C₄⋊C₈	D₃₁₆	1	2	yes	—
229	#103	—	79	2	no	—
230	#103	—	79	2	no	—
231	#102	—	79	2	no	—
232	#102	—	79	2	no	—
233	C₂²⋊C₈	—	2	1	no	—
234	C₂²⋊C₈	—	158	4	no	—
235	C₂²⋊C₈	D₃₁₆	1	2	yes	—
236	#97	—	158	4	no	—
237	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
238	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₇₉	1	2	yes	—
240	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₇₉	1	2	yes	—
243	#41	—	158	4	no	—
244	#41	—	158	4	no	—
245	C₂×Q₈	—	79	2	no	—
246	C₂×D₄	—	79	2	no	—
247	C₂×D₄	—	158	2	no	—
248	C₂²×C₄	—	158	2	no	—
249	C₂²×C₄	—	158	2	no	—
250	C₂²×C₄	—	158	2	no	—
251	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
253	C₂²×C₄	#11929624	1	1	yes	characteristic
254	Q₁₆	—	158	2	no	—
255	SD₁₆	—	158	4	no	—
256	D₈	—	158	2	no	—
257	M₁₆	—	632	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	632	2	no	—
261	C₂×C₈	—	158	4	no	—
262	C₂×C₈	#11929607	1	4	yes	complemented
263	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
264	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
267	C₂×C₈	C₂×D₃₁₆	1	4	yes	—
268	C₄⋊C₄	—	79	2	no	—
269	C₄⋊C₄	—	79	2	no	—
270	C₄⋊C₄	—	158	2	no	—
271	C₄⋊C₄	—	158	2	no	—
272	C₂²⋊C₄	—	158	2	no	—
273	C₂²⋊C₄	—	158	2	no	—
274	C₂²⋊C₄	—	158	2	no	—
275	C₄²	—	158	2	no	—
276	C₄²	—	158	2	no	—
277	C₄²	—	158	2	no	—
278	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
279	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
280	C₄²	#11929624	1	1	yes	characteristic
281	C₄²	C₂³×D₇₉	1	1	yes	characteristic
282	C₂³	—	158	2	no	—
283	C₂³	C₄×D₃₁₆	1	1	yes	characteristic
284	Q₈	—	158	2	no	—
285	Q₈	—	158	2	no	—
286	D₄	—	316	4	no	—
287	D₄	—	158	2	no	—
288	D₄	—	158	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	316	4	no	—
292	C₂×C₄	—	316	2	no	—
293	C₂×C₄	—	316	2	no	—
294	C₂×C₄	—	632	2	no	—
295	C₂×C₄	—	158	2	no	—
296	C₂×C₄	—	158	2	no	—
297	C₂×C₄	—	158	2	no	—
298	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
299	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
300	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
301	C₂×C₄	C₂×#11929607	1	2	yes	—
302	C₂×C₄	#14579167	1	2	yes	—
303	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
305	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	4	no	—
309	C₈	—	4	2	no	—
310	C₈	—	316	4	no	—
311	C₈	C₂×#11929607	1	2	yes	—
312	C₈	C₂×#11929607	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	316	2	no	—
316	C₂²	—	316	2	no	—
317	C₂²	—	316	2	no	—
318	C₂²	#19098642	1	2	yes	—
319	C₂²	C₂×C₄×D₃₁₆	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	316	2	no	—
324	C₄	—	316	2	no	—
325	C₄	—	316	2	no	—
326	C₄	#19098642	1	2	yes	—
327	C₄	C₂²×#11929607	1	1	yes	characteristic
328	C₄	C₂×#14579167	1	1	yes	characteristic
329	C₄	#19099375	1	2	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	632	2	no	—
332	C₂	C₂×#19098642	1	1	yes	characteristic
333	C₂	#23772285	1	1	yes	characteristic
334	C₂	#23772318	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1450	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	C₂×#14579183	C₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	#1450	D₇₉
hypocenter	C₇₉	#17899
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
1. #23772513	C₂	semidirect
3. #23762315	C₂	semidirect
4. #23762310	C₂	semidirect
5. #23762292	C₂	semidirect
6. C₂×#19098314	C₂	semidirect
7. #23761788	C₂	semidirect
8. C₇₉×#1450	C₂	semidirect
104. C₂×C₈×C₇₉	M₁₆	semidirect
175. #1450	D₇₉	semidirect
176. C₇₉	#17899	semidirect
262. C₂×C₈	#11929607	semidirect

Maximal subgroups

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

subgroup	index
#23772513	2
C₄×#14579183	2
#23762315	2
#23762310	2
#23762292	2
C₂×#19098314	2
#23761788	2
C₇₉×#1450	2
#17899	79