Subgroups — #34178755

Order 20224.

618 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11929608	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098285	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19098284	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11929608	C₄	1	2	yes	—
9	C₂×C₂²⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
10	C₄×#11929608	—	2	2	no	—
11	C₄×#11929608	—	2	2	no	—
12	C₂×#14579072	C₄	1	2	yes	—
13	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
17	#19098318	C₄	1	2	yes	—
18	#19098285	—	2	2	no	—
19	#19098284	—	2	2	no	—
20	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₈×C₇₉	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	#336×C₇₉	C₄	1	2	yes	—
25	C₂×#11929608	—	2	4	no	—
26	C₂×#11929608	—	2	4	no	—
27	C₂×#11929608	—	2	4	no	—
28	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
32	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
33	#14579072	—	2	4	no	—
34	C₄×C₇₉⋊C₈	—	2	1	no	—
35	C₄×C₇₉⋊C₈	—	2	1	no	—
36	C₄×C₇₉⋊C₈	—	2	1	no	—
37	C₄×C₇₉⋊C₈	—	2	1	no	—
38	C₄×C₇₉⋊C₈	—	2	1	no	—
39	C₄×C₇₉⋊C₈	—	2	1	no	—
40	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
41	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
42	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₇₉	—	2	1	no	—
48	C₂×C₄²×C₇₉	—	2	1	no	—
49	C₂×C₄²×C₇₉	—	2	1	no	—
50	C₂×C₄²×C₇₉	—	2	1	no	—
51	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
55	C₂²⋊C₈×C₇₉	—	2	4	no	—
56	C₄×C₈×C₇₉	—	2	1	no	—
57	C₄×C₈×C₇₉	—	2	1	no	—
58	C₄×C₈×C₇₉	—	2	1	no	—
59	C₄×C₈×C₇₉	—	2	1	no	—
60	#11929608	—	2	16	no	—
61	C₂×C₇₉⋊C₈	—	2	2	no	—
62	C₂×C₇₉⋊C₈	—	2	2	no	—
63	C₂×C₇₉⋊C₈	—	2	2	no	—
64	C₂×C₇₉⋊C₈	—	2	2	no	—
65	C₂×C₇₉⋊C₈	—	2	2	no	—
66	C₂×C₇₉⋊C₈	—	2	2	no	—
67	C₂×C₇₉⋊C₈	—	2	2	no	—
68	C₂×C₇₉⋊C₈	—	2	2	no	—
69	C₂×C₇₉⋊C₈	—	2	2	no	—
70	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	4	yes	—
71	C₂×C₇₉⋊C₈	M₁₆	1	4	yes	—
72	C₂×C₇₉⋊C₈	M₁₆	1	4	yes	complemented
73	C₂⁴×C₇₉	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	—	2	1	no	—
75	C₂²×C₄×C₇₉	—	2	1	no	—
76	C₂²×C₄×C₇₉	—	2	2	no	—
77	C₂²×C₄×C₇₉	—	2	1	no	—
78	C₂²×C₄×C₇₉	—	2	1	no	—
79	C₂²×C₄×C₇₉	—	2	1	no	—
80	C₂²×C₄×C₇₉	—	2	1	no	—
81	C₂²×C₄×C₇₉	—	2	2	no	—
82	C₂²×C₄×C₇₉	—	2	2	no	—
83	C₂²×C₄×C₇₉	—	2	2	no	—
84	C₂²×C₄×C₇₉	—	2	2	no	—
85	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
86	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
87	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
88	C₂²×C₄×C₇₉	M₁₆	1	4	yes	—
89	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₇₉	—	2	2	no	—
93	C₂×C₈×C₇₉	—	2	2	no	—
94	C₂×C₈×C₇₉	—	2	2	no	—
95	C₂×C₈×C₇₉	—	2	2	no	—
96	C₂×C₈×C₇₉	—	2	2	no	—
97	C₂×C₈×C₇₉	M₁₆	1	4	yes	complemented
98	C₄²×C₇₉	—	2	1	no	—
99	C₄²×C₇₉	—	2	1	no	—
100	C₄²×C₇₉	—	2	1	no	—
101	C₄²×C₇₉	—	2	1	no	—
102	C₄²×C₇₉	—	2	1	no	—
103	C₄²×C₇₉	—	2	1	no	—
104	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₇₉	#41	1	1	yes	characteristic
107	C₄²×C₇₉	#41	1	1	yes	characteristic
108	C₂³×C₇₉	—	2	1	no	—
109	C₂³×C₇₉	—	2	1	no	—
110	C₂³×C₇₉	—	2	1	no	—
111	C₂³×C₇₉	—	2	1	no	—
112	C₂³×C₇₉	—	2	1	no	—
113	C₂³×C₇₉	—	2	1	no	—
114	C₂³×C₇₉	#97	1	2	yes	—
115	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₇₉	—	2	1	no	—
117	C₂×C₄×C₇₉	—	2	1	no	—
118	C₂×C₄×C₇₉	—	2	4	no	—
119	C₂×C₄×C₇₉	—	2	4	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	—	4	2	no	—
127	C₂×C₄×C₇₉	—	4	2	no	—
128	C₂×C₄×C₇₉	—	4	2	no	—
129	C₂×C₄×C₇₉	—	4	2	no	—
130	C₂×C₄×C₇₉	#97	1	2	yes	—
131	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
133	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
138	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
139	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
140	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
141	C₂×C₄×C₇₉	#131	1	2	yes	—
142	C₂×C₄×C₇₉	#131	1	2	yes	—
143	C₈×C₇₉	—	2	4	no	—
144	C₈×C₇₉	—	4	2	no	—
145	C₇₉⋊C₈	—	2	4	no	—
146	C₇₉⋊C₈	—	2	4	no	—
147	C₇₉⋊C₈	—	2	4	no	—
148	C₇₉⋊C₈	—	4	2	no	—
149	C₂²×C₇₉	—	2	2	no	—
150	C₂²×C₇₉	—	2	2	no	—
151	C₂²×C₇₉	—	2	2	no	—
152	C₂²×C₇₉	—	4	1	no	—
153	C₂²×C₇₉	—	4	1	no	—
154	C₂²×C₇₉	—	4	1	no	—
155	C₂²×C₇₉	—	4	1	no	—
156	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₇₉	C₂×#97	1	1	yes	characteristic
158	C₂²×C₇₉	C₄×M₁₆	1	1	yes	characteristic
159	C₂²×C₇₉	C₄×M₁₆	1	1	yes	characteristic
160	C₂²×C₇₉	C₄×M₁₆	1	1	yes	characteristic
161	C₂²×C₇₉	#405	1	1	yes	characteristic
162	C₂²×C₇₉	#405	1	1	yes	characteristic
163	C₄×C₇₉	—	2	2	no	—
164	C₄×C₇₉	—	4	1	no	—
165	C₄×C₇₉	—	4	1	no	—
166	C₄×C₇₉	#407	1	2	yes	—
167	C₄×C₇₉	#432	1	2	yes	—
168	C₄×C₇₉	#435	1	2	yes	—
169	C₄×C₇₉	#435	1	2	yes	—
170	C₄×C₇₉	#436	1	2	yes	—
171	C₄×C₇₉	#436	1	2	yes	—
172	#8147	—	79	1	no	maximal, Sylow
173	C₂×C₇₉	—	4	2	no	—
174	C₂×C₇₉	#1451	1	1	yes	characteristic
175	C₂×C₇₉	#1451	1	1	yes	characteristic
176	C₂×C₇₉	#1453	1	1	yes	characteristic
177	C₂×C₇₉	#1455	1	1	yes	characteristic
178	C₂×C₇₉	#1749	1	1	yes	characteristic
179	C₂×C₇₉	#1753	1	1	yes	characteristic
180	C₂×C₇₉	#1753	1	1	yes	characteristic
181	C₂×C₄×M₁₆	—	79	1	no	—
182	C₄×C₂²⋊C₈	—	79	1	no	—
183	C₄×C₂²⋊C₈	D₇₉	1	1	yes	characteristic
184	C₂×#322	—	79	1	no	—
185	C₂×#322	—	79	1	no	—
186	#1273	—	79	1	no	—
187	#1273	—	79	1	no	—
188	C₇₉	#8147	1	1	yes	characteristic, complemented, Sylow
189	C₂²×M₁₆	—	79	2	no	—
190	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
191	C₂×C₂²⋊C₈	—	79	2	no	—
192	C₂×C₂²⋊C₈	Dic₇₉	1	2	yes	—
193	C₄×M₁₆	—	158	2	no	—
194	C₄×M₁₆	—	158	2	no	—
195	C₂×#97	—	79	2	no	—
196	C₂×C₄×C₈	—	79	1	no	—
197	C₂×C₄×C₈	—	79	1	no	—
198	C₂×C₄×C₈	—	79	1	no	—
199	C₂×C₄×C₈	—	79	1	no	—
200	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
201	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
202	#336	—	79	2	no	—
203	#336	Dic₇₉	1	2	yes	—
204	#322	—	158	2	no	—
205	#322	—	158	2	no	—
206	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
207	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
208	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
209	C₂×M₁₆	—	158	4	no	—
210	C₂×M₁₆	—	158	4	no	—
211	C₂×M₁₆	—	158	4	no	—
212	C₂²×C₈	—	79	2	no	—
213	C₂²×C₈	—	79	2	no	—
214	C₂²×C₈	—	79	2	no	—
215	C₂²×C₈	—	79	2	no	—
216	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
217	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
218	C₂×C₄²	—	2	1	no	—
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	—	2	1	no	—
221	C₂×C₄²	—	2	1	no	—
222	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
223	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
224	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
225	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
226	C₂²⋊C₈	—	2	4	no	—
227	C₂²⋊C₈	—	158	4	no	—
228	#97	—	158	4	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	79	2	no	—
234	C₄×C₈	—	79	2	no	—
235	C₄×C₈	—	158	1	no	—
236	C₄×C₈	—	158	1	no	—
237	C₄×C₈	—	158	1	no	—
238	C₄×C₈	—	158	1	no	—
239	C₄×C₈	—	158	1	no	—
240	C₄×C₈	—	158	1	no	—
241	C₂⁴	C₄×Dic₇₉	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
254	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
255	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
256	C₂²×C₄	#11929607	1	4	yes	—
257	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
259	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
260	M₁₆	—	158	16	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	79	4	no	—
267	C₂×C₈	—	79	4	no	—
268	C₂×C₈	—	79	4	no	—
269	C₂×C₈	—	158	2	no	—
270	C₂×C₈	—	158	2	no	—
271	C₂×C₈	—	158	2	no	—
272	C₂×C₈	—	158	2	no	—
273	C₂×C₈	—	158	2	no	—
274	C₂×C₈	—	158	2	no	—
275	C₂×C₈	—	158	2	no	—
276	C₂×C₈	—	158	2	no	—
277	C₂×C₈	—	158	2	no	—
278	C₂×C₈	#11929608	1	4	yes	complemented
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	2	1	no	—
285	C₄²	D₄×D₇₉	1	1	yes	characteristic
286	C₄²	D₄×D₇₉	1	1	yes	characteristic
287	C₄²	#11929623	1	1	yes	characteristic
288	C₄²	#11929623	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	#14579071	1	2	yes	—
296	C₂³	C₂×C₄×Dic₇₉	1	1	yes	characteristic
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	#14579071	1	2	yes	—
312	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
313	C₂×C₄	#14579117	1	1	yes	characteristic
314	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
315	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
316	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
317	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
318	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
319	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
320	C₂×C₄	C₂×#11929607	1	2	yes	—
321	C₂×C₄	C₂×#11929608	1	2	yes	—
322	C₂×C₄	#14579167	1	2	yes	—
323	C₂×C₄	#14579168	1	2	yes	—
324	C₈	—	2	4	no	—
325	C₈	—	4	2	no	—
326	C₈	—	316	2	no	—
327	C₈	—	158	4	no	—
328	C₈	—	158	4	no	—
329	C₈	—	158	4	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	C₂²⋊C₄×Dic₇₉	1	1	yes	characteristic
338	C₂²	C₂×#14579071	1	1	yes	characteristic
339	C₂²	M₁₆×Dic₇₉	1	1	yes	characteristic
340	C₂²	M₁₆×Dic₇₉	1	1	yes	characteristic
341	C₂²	C₄×#11929608	1	1	yes	characteristic
342	C₂²	#19098518	1	1	yes	characteristic
343	C₂²	#19098519	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	#19098525	1	2	yes	—
348	C₄	#19098617	1	2	yes	—
349	C₄	#19098635	1	2	yes	—
350	C₄	#19098637	1	2	yes	—
351	C₄	#19098638	1	2	yes	—
352	C₄	#19098641	1	2	yes	—
353	C₂	—	4	2	no	—
354	C₂	#23761689	1	1	yes	characteristic
355	C₂	#23761690	1	1	yes	characteristic
356	C₂	#23761695	1	1	yes	characteristic
357	C₂	#23761703	1	1	yes	characteristic
358	C₂	#23763164	1	1	yes	characteristic
359	C₂	#23763184	1	1	yes	characteristic
360	C₂	#23763186	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₉
commutator, socle	C₂³×C₇₉	C₂×C₄²
Fitting	C₄×C₂²⋊C₈×C₇₉	C₂
prime radical	C₂⁴×C₇₉	C₄²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×C₂²⋊C₈	D₇₉
hypocenter	C₇₉	#8147
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#19098285	C₂	semidirect
4. C₂×#19098284	C₂	semidirect
5. C₇₉⋊#1273	C₂	semidirect
6. C₇₉⋊#1273	C₂	semidirect
72. C₂×C₇₉⋊C₈	M₁₆	semidirect
97. C₂×C₈×C₇₉	M₁₆	semidirect
188. C₇₉	#8147	semidirect
278. C₂×C₈	#11929608	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11929608	2
C₄×C₂²⋊(C₇₉⋊C₈)	2
C₂×#19098285	2
C₂×#19098284	2
C₇₉⋊#1273	2
C₄×C₂²⋊C₈×C₇₉	2
#8147	79