Subgroups — #34178865

Order 20224.

654 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579126	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098295	C₂	1	1	yes	maximal, characteristic
5	C₂×#19098293	C₂	1	1	yes	maximal, characteristic
6	#23761241	C₂	1	1	yes	maximal, characteristic
7	C₇₉×#1279	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
9	C₂×#14579126	C₄	1	2	yes	—
10	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₇₉	—	2	2	no	—
13	C₂×C₈×Dic₇₉	—	2	1	no	—
14	C₄⋊C₄×Dic₇₉	C₄	1	2	yes	—
15	C₄×Dic₇₉⋊C₄	—	2	1	no	—
16	#19098414	—	2	2	no	—
17	C₄²×Dic₇₉	—	2	1	no	—
18	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
19	#19098317	—	2	1	no	—
20	#19098295	C₄	1	4	yes	complemented
21	#19098293	C₄	1	4	yes	complemented
22	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
25	C₂²×Dic₁₅₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇₉⋊C₈	—	2	1	no	—
27	C₂²×C₇₉⋊C₈	—	2	1	no	—
28	C₄×Dic₁₅₈	—	2	1	no	—
29	C₄×Dic₁₅₈	—	2	1	no	—
30	C₄×Dic₁₅₈	C₂×C₄	1	2	yes	—
31	C₄×Dic₁₅₈	C₂×C₄	1	2	yes	—
32	#14579126	—	2	4	no	—
33	#14579126	C₈	1	8	yes	complemented
34	C₂×C₄⋊Dic₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂×Dic₇₉⋊C₄	—	2	1	no	—
36	C₂×Dic₇₉⋊C₄	—	2	2	no	—
37	C₂×C₄×Dic₇₉	—	2	1	no	—
38	C₂×C₄×Dic₇₉	—	2	2	no	—
39	C₂×C₄×Dic₇₉	—	2	2	no	—
40	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
41	C₂×C₄×Dic₇₉	D₄	1	2	yes	—
42	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
43	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
44	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
45	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
46	C₄×C₇₉⋊C₈	—	2	1	no	—
47	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
48	C₈×Dic₇₉	—	2	8	no	—
49	C₈×Dic₇₉	—	4	2	no	—
50	C₇₉⋊#95	—	2	2	no	—
51	C₂²×C₈×C₇₉	—	2	1	no	—
52	C₂²×C₈×C₇₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₇₉	—	2	1	no	—
55	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₇₉	—	2	1	no	—
57	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₇₉	—	2	1	no	—
59	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
60	#95×C₇₉	—	2	1	no	—
61	C₂×Dic₁₅₈	—	2	1	no	—
62	C₂×Dic₁₅₈	—	2	2	no	—
63	C₂×Dic₁₅₈	C₄²	1	2	yes	—
64	C₂×Dic₁₅₈	C₂×C₈	1	4	yes	—
65	C₂²×Dic₇₉	—	2	1	no	—
66	C₂²×Dic₇₉	—	2	1	no	—
67	C₂²×Dic₇₉	C₂²⋊C₄	1	2	yes	—
68	C₂²×Dic₇₉	C₂²⋊C₄	1	2	yes	—
69	C₂×C₇₉⋊C₈	—	2	4	no	—
70	C₂×C₇₉⋊C₈	—	2	2	no	—
71	C₂×C₇₉⋊C₈	—	2	2	no	—
72	C₂×C₇₉⋊C₈	—	4	1	no	—
73	C₂×C₇₉⋊C₈	—	4	1	no	—
74	C₄⋊Dic₇₉	—	2	1	no	—
75	C₄⋊Dic₇₉	C₄²	1	2	yes	—
76	Dic₇₉⋊C₄	—	2	8	no	—
77	Dic₇₉⋊C₄	—	2	2	no	—
78	Dic₇₉⋊C₄	—	2	2	no	—
79	C₄×Dic₇₉	—	2	8	no	—
80	C₄×Dic₇₉	—	2	4	no	—
81	C₄×Dic₇₉	—	2	2	no	—
82	C₄×Dic₇₉	—	2	2	no	—
83	C₄×Dic₇₉	—	2	1	no	—
84	C₄×Dic₇₉	—	2	1	no	—
85	C₄×Dic₇₉	—	4	4	no	—
86	C₄×Dic₇₉	C₂×C₈	1	4	yes	—
87	C₄×Dic₇₉	M₁₆	1	4	yes	—
88	C₄×Dic₇₉	SD₁₆	1	2	yes	—
89	C₄×Dic₇₉	Q₁₆	1	2	yes	—
90	C₂²×C₄×C₇₉	—	2	2	no	—
91	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₇₉	—	2	4	no	—
95	C₂×C₈×C₇₉	—	2	2	no	—
96	C₂×C₈×C₇₉	—	2	2	no	—
97	C₂×C₈×C₇₉	—	4	1	no	—
98	C₂×C₈×C₇₉	—	4	1	no	—
99	C₄⋊C₄×C₇₉	—	2	2	no	—
100	C₄⋊C₄×C₇₉	C₂×C₈	1	4	yes	—
101	C₄²×C₇₉	—	4	1	no	—
102	C₄²×C₇₉	—	4	1	no	—
103	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₇₉	#41	1	1	yes	characteristic
106	C₄²×C₇₉	#41	1	1	yes	characteristic
107	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₇₉	—	2	2	no	—
109	C₂×C₄×C₇₉	—	2	4	no	—
110	C₂×C₄×C₇₉	—	4	2	no	—
111	C₂×C₄×C₇₉	—	4	2	no	—
112	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
120	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
122	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
123	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
124	C₂×Dic₇₉	—	2	2	no	—
125	C₂×Dic₇₉	—	2	4	no	—
126	C₂×Dic₇₉	—	2	2	no	—
127	C₂×Dic₇₉	—	2	2	no	—
128	C₂×Dic₇₉	—	2	2	no	—
129	C₂×Dic₇₉	—	4	2	no	—
130	C₂×Dic₇₉	C₂²⋊C₈	1	4	yes	—
131	C₂×Dic₇₉	C₂²⋊C₈	1	4	yes	—
132	C₂×Dic₇₉	#103	1	4	yes	—
133	C₂×Dic₇₉	#104	1	4	yes	—
134	Dic₁₅₈	—	2	2	no	—
135	Dic₁₅₈	—	2	4	no	—
136	Dic₁₅₈	C₄×C₈	1	4	yes	complemented
137	C₈×C₇₉	—	2	4	no	—
138	C₈×C₇₉	—	4	2	no	—
139	C₇₉⋊C₈	—	2	4	no	—
140	C₇₉⋊C₈	—	4	2	no	—
141	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₇₉	C₂×C₂²⋊C₈	1	2	yes	—
143	C₂²×C₇₉	C₂×#103	1	1	yes	characteristic
144	C₂²×C₇₉	#416	1	1	yes	characteristic
145	C₂²×C₇₉	C₂×#104	1	2	yes	—
146	C₄×C₇₉	—	2	1	no	—
147	C₄×C₇₉	—	2	1	no	—
148	C₄×C₇₉	—	4	4	no	—
149	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
150	C₄×C₇₉	C₄×M₁₆	1	1	yes	characteristic
151	C₄×C₇₉	#431	1	2	yes	—
152	C₄×C₇₉	C₄×SD₁₆	1	1	yes	characteristic
153	C₄×C₇₉	C₄×SD₁₆	1	1	yes	characteristic
154	C₄×C₇₉	C₄×Q₁₆	1	1	yes	characteristic
155	C₄×C₇₉	C₄×Q₁₆	1	1	yes	characteristic
156	Dic₇₉	—	2	4	no	—
157	Dic₇₉	—	2	4	no	—
158	Dic₇₉	—	4	2	no	—
159	Dic₇₉	Q₈⋊C₈	1	8	yes	complemented
160	C₄×Q₈⋊C₈	—	79	1	no	maximal, Sylow
161	C₂×C₇₉	C₂×Q₈⋊C₈	1	2	yes	—
162	C₂×C₇₉	#1476	1	2	yes	—
163	C₂×C₇₉	C₄×C₂²⋊C₈	1	1	yes	characteristic
164	C₂×C₇₉	C₄×#104	1	1	yes	characteristic
165	C₂×C₇₉	C₄×#103	1	1	yes	characteristic
166	C₄²×Q₈	—	79	1	no	—
167	C₄×C₄⋊C₈	—	79	1	no	—
168	C₄²×C₈	—	79	1	no	—
169	C₂×Q₈⋊C₈	—	79	1	no	—
170	C₂×Q₈⋊C₈	—	79	1	no	—
171	#1279	—	79	1	no	—
172	#1279	D₇₉	1	1	yes	characteristic
173	C₇₉	C₄×Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
174	C₂×C₄×Q₈	—	79	2	no	—
175	C₂×C₄×Q₈	—	79	1	no	—
176	C₂×C₄⋊C₈	—	79	1	no	—
177	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
178	C₂×C₄×C₈	—	79	1	no	—
179	C₂×C₄×C₈	—	158	1	no	—
180	C₂×C₄×C₈	—	158	2	no	—
181	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
182	C₄×C₄⋊C₄	—	79	2	no	—
183	C₄×C₄⋊C₄	—	158	2	no	—
184	C₄×C₄⋊C₄	—	158	1	no	—
185	C₄×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
186	C₄³	—	79	1	no	—
187	C₄³	—	158	1	no	—
188	#336	—	158	1	no	—
189	Q₈⋊C₈	—	79	4	no	—
190	Q₈⋊C₈	—	79	4	no	—
191	C₂²×Q₈	—	79	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	1	no	—
194	C₂²×C₈	—	158	1	no	—
195	C₂²×C₈	—	158	1	no	—
196	C₄×Q₈	—	79	8	no	—
197	C₄×Q₈	—	79	2	no	—
198	C₄×Q₈	—	79	2	no	—
199	C₄×Q₈	—	158	4	no	—
200	C₄×Q₈	—	158	1	no	—
201	C₄×Q₈	—	158	1	no	—
202	C₂×C₄⋊C₄	—	79	1	no	—
203	C₂×C₄⋊C₄	—	158	2	no	—
204	C₂×C₄⋊C₄	—	158	1	no	—
205	C₂×C₄⋊C₄	C₄×D₇₉	1	2	yes	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	—	79	2	no	—
208	C₂×C₄²	—	79	1	no	—
209	C₂×C₄²	—	79	2	no	—
210	C₂×C₄²	—	79	1	no	—
211	C₂×C₄²	—	158	2	no	—
212	C₂×C₄²	—	158	2	no	—
213	C₂×C₄²	—	158	1	no	—
214	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	79	2	no	—
217	C₄⋊C₈	—	158	1	no	—
218	C₄⋊C₈	C₄×D₇₉	1	2	yes	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	316	2	no	—
221	C₄×C₈	—	79	2	no	—
222	C₄×C₈	—	158	8	no	—
223	C₄×C₈	—	158	1	no	—
224	C₄×C₈	C₄×D₇₉	1	2	yes	—
225	#95	—	2	1	no	—
226	#95	—	158	2	no	—
227	C₂×Q₈	—	79	2	no	—
228	C₂×Q₈	—	79	4	no	—
229	C₂×Q₈	—	158	1	no	—
230	C₂×Q₈	—	158	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	79	2	no	—
233	C₂²×C₄	—	79	2	no	—
234	C₂²×C₄	—	158	1	no	—
235	C₂²×C₄	—	158	1	no	—
236	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
237	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
238	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	316	1	no	—
245	C₂×C₈	—	316	1	no	—
246	C₂×C₈	—	158	2	no	—
247	C₂×C₈	—	158	4	no	—
248	C₂×C₈	—	158	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	79	2	no	—
251	C₄⋊C₄	—	158	8	no	—
252	C₄⋊C₄	—	158	1	no	—
253	C₄⋊C₄	—	158	2	no	—
254	C₄⋊C₄	—	158	2	no	—
255	C₄⋊C₄	C₈×D₇₉	1	4	yes	—
256	C₄²	—	4	1	no	—
257	C₄²	—	4	1	no	—
258	C₄²	—	316	4	no	—
259	C₄²	—	79	4	no	—
260	C₄²	—	79	4	no	—
261	C₄²	—	79	2	no	—
262	C₄²	—	79	2	no	—
263	C₄²	—	158	4	no	—
264	C₄²	—	158	8	no	—
265	C₄²	—	158	2	no	—
266	C₄²	—	158	2	no	—
267	C₄²	—	158	1	no	—
268	C₄²	—	158	1	no	—
269	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
270	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
271	C₄²	#11929623	1	1	yes	characteristic
272	C₄²	#11929625	1	1	yes	characteristic
273	C₂³	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
274	Q₈	—	79	4	no	—
275	Q₈	—	158	2	no	—
276	Q₈	—	158	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	316	2	no	—
282	C₂×C₄	—	79	4	no	—
283	C₂×C₄	—	79	4	no	—
284	C₂×C₄	—	79	4	no	—
285	C₂×C₄	—	79	4	no	—
286	C₂×C₄	—	158	2	no	—
287	C₂×C₄	—	158	2	no	—
288	C₂×C₄	—	158	4	no	—
289	C₂×C₄	—	158	2	no	—
290	C₂×C₄	—	158	2	no	—
291	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
292	C₂×C₄	#14579119	1	1	yes	characteristic
293	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
294	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
295	C₂×C₄	#14579123	1	1	yes	characteristic
296	C₂×C₄	#14579123	1	1	yes	characteristic
297	C₂×C₄	C₂×C₈×D₇₉	1	2	yes	—
298	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
299	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
300	C₂×C₄	Q₁₆×D₇₉	1	1	yes	characteristic
301	C₂×C₄	#14579180	1	1	yes	characteristic
302	C₂×C₄	#14579182	1	1	yes	characteristic
303	C₈	—	2	4	no	—
304	C₈	—	4	2	no	—
305	C₈	—	316	2	no	—
306	C₈	—	158	4	no	—
307	C₂²	#19098410	1	1	yes	characteristic
308	C₂²	C₂²⋊C₈×D₇₉	1	2	yes	—
309	C₂²	#103×D₇₉	1	1	yes	characteristic
310	C₂²	#19098556	1	1	yes	characteristic
311	C₂²	#104×D₇₉	1	2	yes	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	4	4	no	—
315	C₄	—	316	2	no	—
316	C₄	—	79	8	no	—
317	C₄	—	158	4	no	—
318	C₄	—	158	4	no	—
319	C₄	C₄×C₈×D₇₉	1	1	yes	characteristic
320	C₄	#19098515	1	1	yes	characteristic
321	C₄	#19098615	1	2	yes	—
322	C₄	#19098650	1	1	yes	characteristic
323	C₄	#19098654	1	1	yes	characteristic
324	C₄	#19098655	1	1	yes	characteristic
325	C₄	#19098657	1	1	yes	characteristic
326	C₂	Q₈⋊C₈×D₇₉	1	2	yes	—
327	C₂	#23761779	1	2	yes	—
328	C₂	#23763157	1	1	yes	characteristic
329	C₂	#23763204	1	1	yes	characteristic
330	C₂	#23763219	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₉
commutator	C₄×C₇₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1279	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1279	D₇₉
hypocenter	C₇₉	C₄×Q₈⋊C₈
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1279

Complements

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

N	H ≅ G/N	type
20. #19098295	C₄	semidirect
21. #19098293	C₄	semidirect
33. #14579126	C₈	semidirect
136. Dic₁₅₈	C₄×C₈	semidirect
159. Dic₇₉	Q₈⋊C₈	semidirect
173. C₇₉	C₄×Q₈⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579126	2
C₄⋊C₈×Dic₇₉	2
C₄×C₈×Dic₇₉	2
C₂×#19098295	2
C₂×#19098293	2
#23761241	2
C₇₉×#1279	2
C₄×Q₈⋊C₈	79