Subgroups — #34178952

Order 20224.

498 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579169	C₂	1	1	yes	maximal, characteristic
2	#23761783	C₂	1	1	yes	maximal, characteristic
3	#23761780	C₂	1	1	yes	maximal, characteristic
4	#23761693	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098285	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761238	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14579169	C₄	1	2	yes	—
9	C₄×#11929624	C₂²	1	1	yes	characteristic
10	#19098616	C₂²	1	1	yes	characteristic
11	#19098521	C₄	1	2	yes	—
12	#19098520	—	2	2	no	—
13	C₄×#11929607	—	2	1	no	—
14	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₇₉	—	2	1	no	—
17	#19098295	C₄	1	2	yes	—
18	#19098293	C₄	1	2	yes	—
19	#19098292	C₄	1	2	yes	—
20	#19098290	C₄	1	2	yes	—
21	#19098285	—	2	2	no	—
22	#19098285	—	2	1	no	—
23	#19098284	—	2	1	no	—
24	#431×C₇₉	C₂²	1	1	yes	characteristic
25	C₄×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×#11929624	C₂×C₄	1	1	yes	characteristic
28	#14579169	—	2	4	no	—
29	#14579169	—	2	4	no	—
30	C₂×#11929607	—	2	2	no	—
31	C₂²×C₇₉⋊C₈	—	2	1	no	—
32	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₇₉	—	2	2	no	—
34	C₄×Dic₁₅₈	C₂×C₄	1	1	yes	characteristic
35	C₄×Dic₁₅₈	C₂×C₄	1	1	yes	characteristic
36	C₄×D₃₁₆	C₂×C₄	1	1	yes	characteristic
37	C₄×D₃₁₆	C₂×C₄	1	1	yes	characteristic
38	C₄×C₂²⋊D₇₉	—	2	1	no	—
39	C₄×C₂²⋊D₇₉	—	2	1	no	—
40	#14579120	C₂×C₄	1	1	yes	characteristic
41	#14579118	—	2	1	no	—
42	C₄²×D₇₉	—	2	1	no	—
43	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
44	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
45	#14579071	—	2	2	no	—
46	C₄×C₇₉⋊C₈	—	2	1	no	—
47	C₄×C₇₉⋊C₈	—	2	1	no	—
48	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
49	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
50	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
51	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
52	C₈×Dic₇₉	—	2	2	no	—
53	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
54	C₂²×C₈×C₇₉	—	2	1	no	—
55	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
57	C₂²⋊C₈×C₇₉	—	2	1	no	—
58	#97×C₇₉	—	2	1	no	—
59	C₄×C₈×C₇₉	—	2	1	no	—
60	C₄×C₈×C₇₉	—	2	1	no	—
61	C₄×C₈×C₇₉	—	2	1	no	—
62	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
63	#11929624	—	2	2	no	—
64	#11929624	M₁₆	1	4	yes	—
65	C₂×Dic₁₅₈	C₄²	1	1	yes	characteristic
66	C₂×D₃₁₆	C₄²	1	1	yes	characteristic
67	C₂×C₂²⋊D₇₉	—	2	1	no	—
68	C₂×C₄×D₇₉	—	2	1	no	—
69	C₂×C₄×D₇₉	—	2	1	no	—
70	C₂×C₄×D₇₉	—	2	1	no	—
71	#11929607	—	2	8	no	—
72	C₂×C₇₉⋊C₈	—	2	4	no	—
73	C₂×C₇₉⋊C₈	—	2	2	no	—
74	C₂×C₇₉⋊C₈	—	2	2	no	—
75	C₂×C₇₉⋊C₈	—	2	2	no	—
76	C₂×C₇₉⋊C₈	—	2	2	no	—
77	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
79	C₈×D₇₉	—	2	8	no	—
80	C₄⋊Dic₇₉	C₄²	1	1	yes	characteristic
81	C₄⋊Dic₇₉	C₄²	1	1	yes	characteristic
82	Dic₇₉⋊C₄	—	2	1	no	—
83	Dic₇₉⋊C₄	—	2	1	no	—
84	C₂²⋊Dic₇₉	—	2	1	no	—
85	(C₂×D₇₉)⋊C₄	—	2	1	no	—
86	(C₂×D₇₉)⋊C₄	—	2	1	no	—
87	C₄×Dic₇₉	—	2	1	no	—
88	C₄×Dic₇₉	—	2	1	no	—
89	C₄×Dic₇₉	—	2	1	no	—
90	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
92	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
93	M₁₆×C₇₉	—	2	2	no	—
94	M₁₆×C₇₉	M₁₆	1	4	yes	—
95	C₂×C₈×C₇₉	—	2	2	no	—
96	C₂×C₈×C₇₉	—	2	2	no	—
97	C₂×C₈×C₇₉	—	2	2	no	—
98	C₂×C₈×C₇₉	—	4	2	no	—
99	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
103	C₂³×C₇₉	#117	1	1	yes	characteristic
104	C₂²×D₇₉	—	2	1	no	—
105	C₂×C₄×C₇₉	—	2	2	no	—
106	C₂×C₄×C₇₉	—	2	2	no	—
107	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
115	C₂²⋊D₇₉	—	2	4	no	—
116	C₂×Dic₇₉	—	2	1	no	—
117	C₂×Dic₇₉	—	2	1	no	—
118	C₂×Dic₇₉	—	2	1	no	—
119	D₃₁₆	—	2	1	no	—
120	D₃₁₆	#97	1	2	yes	—
121	C₄×D₇₉	—	2	4	no	—
122	C₄×D₇₉	—	4	2	no	—
123	C₄×D₇₉	—	4	2	no	—
124	Dic₁₅₈	—	2	1	no	—
125	Dic₁₅₈	#97	1	2	yes	—
126	C₈×C₇₉	—	2	4	no	—
127	C₈×C₇₉	—	4	2	no	—
128	C₈×C₇₉	—	4	2	no	—
129	C₇₉⋊C₈	—	2	4	no	—
130	C₇₉⋊C₈	—	2	2	no	—
131	C₇₉⋊C₈	—	2	2	no	—
132	C₇₉⋊C₈	—	4	2	no	—
133	C₂²×C₇₉	—	2	1	no	—
134	C₂²×C₇₉	—	2	1	no	—
135	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₂²×C₇₉	#432	1	2	yes	—
137	C₂×D₇₉	—	2	2	no	—
138	C₂×D₇₉	—	4	1	no	—
139	C₂×D₇₉	—	4	1	no	—
140	C₄×C₇₉	—	2	1	no	—
141	C₄×C₇₉	—	2	1	no	—
142	C₄×C₇₉	C₂×#97	1	1	yes	characteristic
143	C₄×C₇₉	C₄×M₁₆	1	1	yes	characteristic
144	C₄×C₇₉	#407	1	2	yes	—
145	C₄×C₇₉	#443	1	2	yes	—
146	C₄×C₇₉	#444	1	2	yes	—
147	Dic₇₉	—	2	2	no	—
148	Dic₇₉	—	4	1	no	—
149	Dic₇₉	—	4	1	no	—
150	#8186	—	79	1	no	maximal, Sylow
151	C₂×C₇₉	—	2	2	no	—
152	C₂×C₇₉	#1749	1	1	yes	characteristic
153	C₂×C₇₉	#1764	1	1	yes	characteristic
154	C₂×C₇₉	#1765	1	1	yes	characteristic
155	D₇₉	—	4	2	no	—
156	C₄×#131	—	79	1	no	—
157	#1476	—	79	1	no	—
158	#1476	—	79	1	no	—
159	#1452	—	79	1	no	—
160	C₂×#322	—	79	1	no	—
161	#1278	—	79	1	no	—
162	#1278	D₇₉	1	1	yes	characteristic, complemented
163	C₇₉	#8186	1	1	yes	characteristic, complemented, Sylow
164	C₂×#131	—	79	2	no	—
165	C₄×#41	—	79	1	no	—
166	#431	—	79	1	no	—
167	#431	C₂×D₇₉	1	1	yes	characteristic
168	#405	—	79	2	no	—
169	#405	—	158	2	no	—
170	C₄×M₁₆	—	158	1	no	—
171	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
172	C₂×C₄×C₈	—	79	1	no	—
173	C₂×C₄×C₈	—	79	1	no	—
174	C₂×C₄×C₈	—	158	1	no	—
175	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
176	Q₈⋊C₈	—	79	2	no	—
177	Q₈⋊C₈	—	79	2	no	—
178	D₄⋊C₈	—	79	2	no	—
179	D₄⋊C₈	—	79	2	no	—
180	#322	—	158	2	no	—
181	#322	—	158	1	no	—
182	#322	—	158	1	no	—
183	C₂×#41	—	79	1	no	—
184	#131	—	158	4	no	—
185	#131	—	158	4	no	—
186	C₂×M₁₆	—	158	2	no	—
187	C₂×M₁₆	C₄×D₇₉	1	2	yes	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	79	2	no	—
190	C₂²×C₈	—	158	2	no	—
191	C₂²×C₈	—	158	1	no	—
192	C₄×Q₈	—	79	1	no	—
193	C₄×Q₈	—	79	1	no	—
194	C₄×D₄	—	79	1	no	—
195	C₄×D₄	—	79	1	no	—
196	C₄×D₄	—	158	1	no	—
197	C₄×D₄	—	158	1	no	—
198	#117	—	79	1	no	—
199	#117	—	158	1	no	—
200	C₂×C₄²	—	158	1	no	—
201	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
202	C₄⋊C₈	—	79	2	no	—
203	C₄⋊C₈	C₄×D₇₉	1	2	yes	—
204	C₂²⋊C₈	—	2	1	no	—
205	C₂²⋊C₈	—	158	1	no	—
206	#97	—	2	1	no	—
207	#97	—	158	2	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	79	2	no	—
212	C₄×C₈	—	79	2	no	—
213	C₄×C₈	—	79	1	no	—
214	C₄×C₈	—	79	1	no	—
215	C₄×C₈	—	158	2	no	—
216	C₄×C₈	—	158	1	no	—
217	C₄×C₈	—	158	1	no	—
218	C₄×C₈	C₄×D₇₉	1	2	yes	—
219	#41	—	79	4	no	—
220	#41	—	158	2	no	—
221	C₂×Q₈	—	79	1	no	—
222	C₂×D₄	—	79	1	no	—
223	C₂×D₄	—	158	1	no	—
224	C₂²×C₄	—	158	1	no	—
225	C₂²×C₄	—	158	1	no	—
226	C₂²×C₄	—	158	1	no	—
227	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
228	C₂²×C₄	#11929623	1	1	yes	characteristic
229	C₂²×C₄	#11929625	1	1	yes	characteristic
230	M₁₆	—	2	2	no	—
231	M₁₆	—	158	8	no	—
232	M₁₆	#11929607	1	4	yes	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	79	2	no	—
238	C₂×C₈	—	79	2	no	—
239	C₂×C₈	—	158	8	no	—
240	C₂×C₈	—	158	2	no	—
241	C₂×C₈	—	158	4	no	—
242	C₂×C₈	—	158	2	no	—
243	C₂×C₈	—	158	2	no	—
244	C₂×C₈	—	158	2	no	—
245	C₄⋊C₄	—	79	1	no	—
246	C₄⋊C₄	—	79	1	no	—
247	C₄⋊C₄	—	158	1	no	—
248	C₄⋊C₄	—	158	1	no	—
249	C₂²⋊C₄	—	158	1	no	—
250	C₂²⋊C₄	—	158	1	no	—
251	C₂²⋊C₄	—	158	1	no	—
252	C₄²	—	158	1	no	—
253	C₄²	—	158	1	no	—
254	C₄²	—	158	1	no	—
255	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
256	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
257	C₄²	D₄×D₇₉	1	1	yes	characteristic
258	C₄²	D₄×D₇₉	1	1	yes	characteristic
259	C₂³	—	158	1	no	—
260	C₂³	#14579119	1	1	yes	characteristic
261	Q₈	—	79	2	no	—
262	Q₈	—	158	1	no	—
263	D₄	—	79	2	no	—
264	D₄	—	158	4	no	—
265	D₄	—	158	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	316	2	no	—
269	C₂×C₄	—	316	2	no	—
270	C₂×C₄	—	158	1	no	—
271	C₂×C₄	—	158	1	no	—
272	C₂×C₄	—	158	1	no	—
273	C₂×C₄	—	158	4	no	—
274	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
275	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
276	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
277	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
278	C₂×C₄	#14579123	1	1	yes	characteristic
279	C₂×C₄	#14579123	1	1	yes	characteristic
280	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
281	C₂×C₄	C₂×#11929607	1	2	yes	—
282	C₈	—	2	4	no	—
283	C₈	—	4	2	no	—
284	C₈	—	4	2	no	—
285	C₈	—	316	2	no	—
286	C₈	—	158	4	no	—
287	C₈	—	158	2	no	—
288	C₈	—	158	2	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	316	1	no	—
292	C₂²	—	316	1	no	—
293	C₂²	—	158	2	no	—
294	C₂²	#19098410	1	1	yes	characteristic
295	C₂²	#19098619	1	2	yes	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	316	1	no	—
299	C₄	—	316	1	no	—
300	C₄	—	158	2	no	—
301	C₄	#97×D₇₉	1	1	yes	characteristic
302	C₄	C₄×#11929607	1	1	yes	characteristic
303	C₄	#19098525	1	2	yes	—
304	C₄	#19098679	1	2	yes	—
305	C₄	#19098682	1	2	yes	—
306	C₂	—	2	2	no	—
307	C₂	—	316	2	no	—
308	C₂	#23763161	1	1	yes	characteristic
309	C₂	#23763236	1	1	yes	characteristic
310	C₂	#23763241	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1278	C₂
socle	C₂²×C₇₉	C₄×C₂²⋊C₄
prime radical	C₂×#11929624	C₂×C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1278	D₇₉
hypocenter	C₇₉	#8186
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
4. #23761693	C₂	semidirect
5. C₂×#19098285	C₂	semidirect
6. #23761238	C₂	semidirect
7. C₇₉×#1278	C₂	semidirect
162. #1278	D₇₉	semidirect
163. C₇₉	#8186	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579169	2
#23761783	2
#23761780	2
#23761693	2
C₂×#19098285	2
#23761238	2
C₇₉×#1278	2
#8186	79