Subgroups — #34314900

Order 20224.

632 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579153	C₂	1	1	yes	maximal, characteristic
2	#23764117	C₂	1	2	yes	maximal
3	#23764115	C₂	1	2	yes	maximal
4	#23763960	C₂	1	2	yes	maximal
5	#23763958	C₂	1	2	yes	maximal
6	#23763956	C₂	1	2	yes	maximal
7	#23763954	C₂	1	2	yes	maximal
8	#23763517	C₂	1	1	yes	maximal, characteristic
9	C₇₉×#1831	C₂	1	1	yes	maximal, characteristic
10	C₂×#14579153	C₄	1	1	yes	characteristic
11	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₇₉⋊C₈	C₂²	1	2	yes	—
14	C₂×Dic₇₉⋊C₈	C₂²	1	2	yes	—
15	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
16	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
17	#19098461	C₄	1	1	yes	characteristic
18	#19098461	C₄	1	1	yes	characteristic
19	#19098447	C₄	1	2	yes	—
20	#19098447	C₄	1	2	yes	—
21	#19098429	C₄	1	1	yes	characteristic
22	C₄×Dic₇₉⋊C₄	C₂²	1	2	yes	—
23	C₄×Dic₇₉⋊C₄	C₂²	1	2	yes	—
24	C₄×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
25	C₄×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
26	#19098318	C₂²	1	2	yes	—
27	#19098318	C₂²	1	2	yes	—
28	#19098317	C₂²	1	2	yes	—
29	#19098317	C₂²	1	2	yes	—
30	#19098317	C₂²	1	2	yes	—
31	#19098317	C₂²	1	2	yes	—
32	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
34	C₄³×C₇₉	C₂²	1	1	yes	characteristic
35	#336×C₇₉	C₂²	1	2	yes	—
36	#336×C₇₉	C₂²	1	2	yes	—
37	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
38	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
39	#14579153	C₈	1	8	yes	complemented
40	C₂×C₄⋊Dic₇₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₇₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊Dic₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₇₉	C₂×C₄	1	1	yes	characteristic
44	C₂×Dic₇₉⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×Dic₇₉⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×Dic₇₉⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×Dic₇₉⋊C₄	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
49	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
50	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
51	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
52	Dic₇₉⋊C₈	—	4	2	no	—
53	Dic₇₉⋊C₈	—	4	2	no	—
54	C₈×Dic₇₉	—	4	2	no	—
55	C₈×Dic₇₉	—	4	2	no	—
56	C₇₉⋊#95	C₂×C₄	1	2	yes	—
57	C₇₉⋊#95	C₂×C₄	1	2	yes	—
58	C₇₉⋊#95	C₂×C₄	1	2	yes	—
59	C₇₉⋊#95	C₂×C₄	1	2	yes	—
60	C₇₉⋊#95	C₂×C₄	1	2	yes	—
61	C₇₉⋊#95	C₂×C₄	1	2	yes	—
62	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
63	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
64	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₇₉	—	4	1	no	—
72	C₄⋊C₈×C₇₉	—	4	1	no	—
73	C₂²×Dic₇₉	C₂²×C₄	1	2	yes	—
74	C₂²×Dic₇₉	C₂²×C₄	1	2	yes	—
75	C₂²×Dic₇₉	C₂²×C₄	1	2	yes	—
76	C₂²×Dic₇₉	C₂²×C₄	1	2	yes	—
77	C₂×C₇₉⋊C₈	—	2	2	no	—
78	C₂×C₇₉⋊C₈	—	2	2	no	—
79	C₂×C₇₉⋊C₈	—	2	2	no	—
80	C₂×C₇₉⋊C₈	—	2	2	no	—
81	C₂×C₇₉⋊C₈	—	2	2	no	—
82	C₂×C₇₉⋊C₈	—	2	2	no	—
83	C₄⋊Dic₇₉	—	4	1	no	—
84	C₄⋊Dic₇₉	—	4	1	no	—
85	C₄⋊Dic₇₉	C₂×C₈	1	4	yes	—
86	C₄⋊Dic₇₉	C₂×C₈	1	4	yes	—
87	Dic₇₉⋊C₄	—	4	2	no	—
88	Dic₇₉⋊C₄	—	4	2	no	—
89	Dic₇₉⋊C₄	C₂×C₈	1	8	yes	—
90	Dic₇₉⋊C₄	C₂×C₈	1	8	yes	—
91	C₄×Dic₇₉	—	2	2	no	—
92	C₄×Dic₇₉	—	2	2	no	—
93	C₄×Dic₇₉	—	2	2	no	—
94	C₄×Dic₇₉	—	2	2	no	—
95	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
102	C₂×C₈×C₇₉	—	2	2	no	—
103	C₂×C₈×C₇₉	—	2	2	no	—
104	C₂×C₈×C₇₉	—	2	2	no	—
105	C₂×C₈×C₇₉	—	2	2	no	—
106	C₂×C₈×C₇₉	—	2	2	no	—
107	C₂×C₈×C₇₉	—	2	2	no	—
108	C₄²×C₇₉	—	2	1	no	—
109	C₄²×C₇₉	—	2	1	no	—
110	C₄²×C₇₉	—	2	1	no	—
111	C₄²×C₇₉	—	2	1	no	—
112	C₄²×C₇₉	—	2	1	no	—
113	C₄²×C₇₉	—	2	1	no	—
114	C₄²×C₇₉	—	4	1	no	—
115	C₄²×C₇₉	—	4	1	no	—
116	C₄²×C₇₉	—	4	1	no	—
117	C₄²×C₇₉	C₂×C₈	1	4	yes	—
118	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₇₉	—	2	1	no	—
120	C₂×C₄×C₇₉	—	2	1	no	—
121	C₂×C₄×C₇₉	—	2	1	no	—
122	C₂×C₄×C₇₉	—	2	1	no	—
123	C₂×C₄×C₇₉	—	2	1	no	—
124	C₂×C₄×C₇₉	—	2	1	no	—
125	C₂×C₄×C₇₉	—	2	1	no	—
126	C₂×C₄×C₇₉	—	2	1	no	—
127	C₂×C₄×C₇₉	—	2	1	no	—
128	C₂×C₄×C₇₉	—	2	1	no	—
129	C₂×C₄×C₇₉	—	2	1	no	—
130	C₂×C₄×C₇₉	—	2	1	no	—
131	C₂×C₄×C₇₉	—	2	1	no	—
132	C₂×C₄×C₇₉	—	2	1	no	—
133	C₂×C₄×C₇₉	—	2	1	no	—
134	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
141	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
142	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂×Dic₇₉	—	2	2	no	—
144	C₂×Dic₇₉	—	2	2	no	—
145	C₂×Dic₇₉	—	2	2	no	—
146	C₂×Dic₇₉	—	2	2	no	—
147	C₂×Dic₇₉	—	2	2	no	—
148	C₂×Dic₇₉	—	2	2	no	—
149	C₂×Dic₇₉	—	2	2	no	—
150	C₂×Dic₇₉	—	2	2	no	—
151	C₂×Dic₇₉	—	2	2	no	—
152	C₂×Dic₇₉	—	2	2	no	—
153	C₂×Dic₇₉	C₂²×C₈	1	4	yes	—
154	C₂×Dic₇₉	C₂²×C₈	1	4	yes	—
155	C₈×C₇₉	—	4	2	no	—
156	C₈×C₇₉	—	4	2	no	—
157	C₇₉⋊C₈	—	4	2	no	—
158	C₇₉⋊C₈	—	4	2	no	—
159	C₂²×C₇₉	#519	1	1	yes	characteristic
160	C₂²×C₇₉	#520	1	1	yes	characteristic
161	C₂²×C₇₉	#520	1	1	yes	characteristic
162	C₂²×C₇₉	C₂³×C₈	1	1	yes	characteristic
163	C₂²×C₇₉	#568	1	1	yes	characteristic
164	C₂²×C₇₉	#568	1	1	yes	characteristic
165	C₂²×C₇₉	#568	1	1	yes	characteristic
166	C₄×C₇₉	—	2	2	no	—
167	C₄×C₇₉	—	2	2	no	—
168	C₄×C₇₉	—	2	2	no	—
169	C₄×C₇₉	—	4	1	no	—
170	C₄×C₇₉	—	4	1	no	—
171	C₄×C₇₉	—	4	1	no	—
172	C₄×C₇₉	#562	1	1	yes	characteristic
173	C₄×C₇₉	#563	1	1	yes	characteristic
174	C₄×C₇₉	#563	1	1	yes	characteristic
175	C₄×C₇₉	#564	1	1	yes	characteristic
176	Dic₇₉	—	2	4	no	—
177	Dic₇₉	—	2	4	no	—
178	Dic₇₉	—	4	2	no	—
179	Dic₇₉	—	4	2	no	—
180	#23369	—	79	1	no	maximal, Sylow
181	C₂×C₇₉	#2382	1	1	yes	characteristic
182	C₂×C₇₉	#2982	1	1	yes	characteristic
183	C₂×C₇₉	#2987	1	1	yes	characteristic
184	C₂×C₇₉	#2988	1	1	yes	characteristic
185	C₂×C₇₉	#2988	1	1	yes	characteristic
186	C₂×C₇₉	#2995	1	1	yes	characteristic
187	C₂×C₇₉	#2995	1	1	yes	characteristic
188	C₄×#125	—	79	1	no	—
189	#1939	—	79	2	no	—
190	#1939	—	79	2	no	—
191	#1916	—	79	2	no	—
192	#1916	—	79	2	no	—
193	#1916	—	79	2	no	—
194	#1916	—	79	2	no	—
195	#1831	—	79	1	no	—
196	#1831	D₇₉	1	1	yes	characteristic
197	C₇₉	#23369	1	1	yes	characteristic, complemented, Sylow
198	C₂×#125	—	79	1	no	—
199	C₂×C₄⋊C₈	—	79	2	no	—
200	C₂×C₄⋊C₈	—	79	2	no	—
201	C₂×C₄⋊C₈	—	79	1	no	—
202	C₂×C₄⋊C₈	—	79	1	no	—
203	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
204	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
205	C₂×C₄×C₈	—	79	2	no	—
206	C₂×C₄×C₈	—	79	2	no	—
207	#389	—	79	1	no	—
208	#389	—	79	1	no	—
209	#387	—	79	2	no	—
210	#387	—	79	2	no	—
211	#382	—	79	1	no	—
212	C₄×C₄⋊C₄	—	79	1	no	—
213	C₄×C₄⋊C₄	—	79	1	no	—
214	C₄×C₄⋊C₄	—	79	2	no	—
215	C₄×C₄⋊C₄	—	79	2	no	—
216	C₄³	C₂×D₇₉	1	1	yes	characteristic
217	#336	—	79	2	no	—
218	#336	—	79	2	no	—
219	#336	—	79	2	no	—
220	#336	—	79	2	no	—
221	#336	—	79	2	no	—
222	#336	—	79	2	no	—
223	#336	C₂×D₇₉	1	2	yes	—
224	#336	C₂×D₇₉	1	2	yes	—
225	C₂²×C₈	—	79	2	no	—
226	C₂²×C₈	—	79	2	no	—
227	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
228	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
229	#125	—	79	8	no	—
230	C₂×C₄⋊C₄	—	79	1	no	—
231	C₂×C₄⋊C₄	—	79	1	no	—
232	C₂×C₄⋊C₄	—	79	2	no	—
233	C₂×C₄⋊C₄	—	79	2	no	—
234	C₂×C₄⋊C₄	—	79	2	no	—
235	C₂×C₄⋊C₄	—	79	2	no	—
236	C₂×C₄⋊C₄	—	79	1	no	—
237	C₂×C₄⋊C₄	—	79	1	no	—
238	C₂×C₄²	—	79	2	no	—
239	C₂×C₄²	—	79	2	no	—
240	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
241	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
242	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
243	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
244	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
245	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
246	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
247	C₄⋊C₈	—	4	1	no	—
248	C₄⋊C₈	—	4	1	no	—
249	C₄⋊C₈	—	316	1	no	—
250	C₄⋊C₈	—	316	1	no	—
251	C₄⋊C₈	—	316	2	no	—
252	C₄⋊C₈	—	316	2	no	—
253	C₄×C₈	—	316	2	no	—
254	C₄×C₈	—	316	2	no	—
255	#95	—	79	2	no	—
256	#95	—	79	2	no	—
257	#95	—	79	2	no	—
258	#95	—	79	2	no	—
259	#95	—	79	2	no	—
260	#95	—	79	2	no	—
261	C₂²×C₄	—	79	2	no	—
262	C₂²×C₄	—	79	2	no	—
263	C₂²×C₄	—	79	2	no	—
264	C₂²×C₄	—	79	2	no	—
265	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
271	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	158	2	no	—
279	C₂×C₈	—	158	2	no	—
280	C₂×C₈	—	158	2	no	—
281	C₂×C₈	—	158	2	no	—
282	C₂×C₈	—	158	2	no	—
283	C₂×C₈	—	158	2	no	—
284	C₄⋊C₄	—	316	2	no	—
285	C₄⋊C₄	—	316	2	no	—
286	C₄⋊C₄	—	316	1	no	—
287	C₄⋊C₄	—	316	1	no	—
288	C₄⋊C₄	—	79	8	no	—
289	C₄⋊C₄	—	79	8	no	—
290	C₄⋊C₄	—	79	4	no	—
291	C₄⋊C₄	—	79	4	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	4	1	no	—
299	C₄²	—	4	1	no	—
300	C₄²	—	4	1	no	—
301	C₄²	—	158	2	no	—
302	C₄²	—	158	2	no	—
303	C₄²	—	158	2	no	—
304	C₄²	—	158	2	no	—
305	C₄²	C₈×D₇₉	1	4	yes	—
306	C₂³	C₂²×C₄×D₇₉	1	1	yes	characteristic
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	79	4	no	—
323	C₂×C₄	—	79	4	no	—
324	C₂×C₄	—	158	2	no	—
325	C₂×C₄	—	158	2	no	—
326	C₂×C₄	—	158	2	no	—
327	C₂×C₄	—	158	2	no	—
328	C₂×C₄	—	158	2	no	—
329	C₂×C₄	—	158	2	no	—
330	C₂×C₄	—	158	2	no	—
331	C₂×C₄	—	158	2	no	—
332	C₂×C₄	—	158	2	no	—
333	C₂×C₄	—	158	2	no	—
334	C₂×C₄	C₂×C₈×D₇₉	1	2	yes	—
335	C₂×C₄	C₂×C₈×D₇₉	1	2	yes	—
336	C₂×C₄	C₂×C₈×D₇₉	1	2	yes	—
337	C₂×C₄	#14579206	1	1	yes	characteristic
338	C₂×C₄	#14579207	1	1	yes	characteristic
339	C₂×C₄	#14579208	1	1	yes	characteristic
340	C₂×C₄	#14579208	1	1	yes	characteristic
341	C₂×C₄	#14579208	1	1	yes	characteristic
342	C₂×C₄	#14579209	1	1	yes	characteristic
343	C₈	—	4	2	no	—
344	C₈	—	4	2	no	—
345	C₈	—	316	2	no	—
346	C₈	—	316	2	no	—
347	C₂²	#19099040	1	1	yes	characteristic
348	C₂²	#19099043	1	1	yes	characteristic
349	C₂²	#19099044	1	1	yes	characteristic
350	C₂²	C₂²×C₈×D₇₉	1	1	yes	characteristic
351	C₂²	#19099329	1	1	yes	characteristic
352	C₂²	#19099329	1	1	yes	characteristic
353	C₂²	#19099329	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	2	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	316	2	no	—
361	C₄	—	316	2	no	—
362	C₄	—	158	4	no	—
363	C₄	—	158	4	no	—
364	C₄	#19099307	1	1	yes	characteristic
365	C₄	#19099313	1	1	yes	characteristic
366	C₄	#19099317	1	1	yes	characteristic
367	C₄	C₇₉⋊#564	1	1	yes	characteristic
368	C₂	#23766585	1	1	yes	characteristic
369	C₂	#23772480	1	1	yes	characteristic
370	C₂	#23772505	1	1	yes	characteristic
371	C₂	#23772508	1	1	yes	characteristic
372	C₂	#23772509	1	1	yes	characteristic
373	C₂	#23772567	1	1	yes	characteristic
374	C₂	#23772572	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂³×C₈
Fitting	C₇₉×#1831	C₂
socle, prime radical	C₂³×C₇₉	C₂³×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1831	D₇₉
hypocenter	C₇₉	#23369
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ #1831

Complements

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

N	H ≅ G/N	type
39. #14579153	C₈	semidirect
197. C₇₉	#23369	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579153	2
#23764117	2
#23764115	2
#23763960	2
#23763958	2
#23763956	2
#23763954	2
#23763517	2
C₇₉×#1831	2
#23369	79