Subgroups — #34198178

Order 20224.

430 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×D₄×D₇₉	C₂	1	1	yes	maximal, characteristic
2	#23765198	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765197	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761356	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761354	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₇₉	—	2	1	no	—
9	C₄×D₄×D₇₉	C₄	1	1	yes	characteristic
10	C₈×D₃₁₆	C₂²	1	1	yes	characteristic
11	D₄×C₇₉⋊C₈	C₄	1	1	yes	characteristic
12	#19098628	C₄	1	1	yes	characteristic
13	C₈×C₂²⋊D₇₉	—	2	1	no	—
14	(C₂²⋊D₇₉)⋊C₈	—	2	1	no	—
15	C₄⋊C₈×D₇₉	C₄	1	1	yes	characteristic
16	C₂²⋊C₈×D₇₉	—	2	1	no	—
17	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
18	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic, complemented
19	Dic₇₉⋊C₁₆	—	2	1	no	—
20	(C₂×D₇₉)⋊C₁₆	—	2	1	no	—
21	(C₂×D₇₉)⋊C₁₆	—	2	1	no	—
22	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
23	C₁₆×Dic₇₉	—	2	1	no	—
24	C₈×D₄×C₇₉	C₂²	1	1	yes	characteristic
25	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic, complemented
26	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic, complemented
27	C₂×D₄×D₇₉	C₈	1	1	yes	characteristic
28	C₂²×C₄×D₇₉	—	2	1	no	—
29	C₂²×C₇₉⋊C₈	—	2	1	no	—
30	C₂×C₈×D₇₉	—	2	1	no	—
31	C₂×C₈×D₇₉	—	2	1	no	—
32	C₂×C₈×D₇₉	—	2	1	no	—
33	C₂×C₈×D₇₉	—	4	2	no	—
34	C₂×C₈×D₇₉	—	4	2	no	—
35	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
36	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
37	C₄×D₃₁₆	C₂×C₄	1	1	yes	characteristic
38	D₄×Dic₇₉	C₈	1	1	yes	characteristic
39	#14579123	C₈	1	1	yes	characteristic
40	C₄×C₂²⋊D₇₉	—	2	1	no	—
41	(C₂²⋊D₇₉)⋊C₄	—	2	1	no	—
42	C₄⋊C₄×D₇₉	C₈	1	1	yes	characteristic
43	C₂²⋊C₄×D₇₉	—	2	1	no	—
44	C₄²×D₇₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₇₉⋊C₁₆	—	2	1	no	—
46	C₂×C₇₉⋊C₁₆	—	2	1	no	—
47	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
48	Dic₇₉⋊C₈	—	2	1	no	—
49	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
50	(C₂×D₇₉)⋊C₈	—	2	1	no	—
51	(C₂×D₇₉)⋊C₈	—	2	1	no	—
52	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₈×Dic₇₉	D₄	1	1	yes	characteristic
54	C₈×Dic₇₉	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₇₉	—	2	1	no	—
56	C₄×D₄×C₇₉	C₂×C₄	1	1	yes	characteristic
57	C₂×C₁₆×C₇₉	—	2	1	no	—
58	C₂×C₁₆×C₇₉	—	2	1	no	—
59	C₄⋊C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
60	C₂²⋊C₈×C₇₉	—	2	1	no	—
61	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
62	C₂³×D₇₉	—	2	1	no	—
63	C₂×D₃₁₆	C₂×C₈	1	1	yes	characteristic
64	C₂×C₂²⋊D₇₉	—	2	1	no	—
65	D₄×D₇₉	—	2	2	no	—
66	D₄×D₇₉	C₁₆	1	4	yes	complemented
67	C₂²×Dic₇₉	—	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	C₂×C₄×D₇₉	—	4	2	no	—
72	C₂×C₄×D₇₉	—	4	2	no	—
73	C₂×C₄×D₇₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×D₇₉	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄×D₇₉	M₁₆	1	1	yes	characteristic
76	C₂×C₇₉⋊C₈	—	2	1	no	—
77	C₂×C₇₉⋊C₈	—	4	2	no	—
78	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₈×D₇₉	—	2	1	no	—
81	C₈×D₇₉	—	2	1	no	—
82	C₈×D₇₉	—	4	1	no	—
83	C₈×D₇₉	—	4	2	no	—
84	C₈×D₇₉	—	4	2	no	—
85	C₈×D₇₉	—	4	1	no	—
86	C₈×D₇₉	—	4	1	no	—
87	C₄⋊Dic₇₉	C₂×C₈	1	1	yes	characteristic
88	Dic₇₉⋊C₄	—	2	1	no	—
89	C₂²⋊Dic₇₉	—	2	1	no	—
90	(C₂×D₇₉)⋊C₄	—	2	1	no	—
91	(C₂×D₇₉)⋊C₄	—	2	1	no	—
92	C₄×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×Dic₇₉	C₂×C₈	1	1	yes	characteristic
94	C₄×Dic₇₉	M₁₆	1	1	yes	characteristic
95	C₇₉⋊C₁₆	—	4	1	no	—
96	C₇₉⋊C₁₆	—	4	1	no	—
97	C₂×D₄×C₇₉	C₂×C₈	1	1	yes	characteristic
98	C₂²×C₄×C₇₉	—	2	1	no	—
99	C₂×C₈×C₇₉	—	2	1	no	—
100	C₂×C₈×C₇₉	—	4	1	no	—
101	C₂×C₈×C₇₉	—	4	1	no	—
102	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₄⋊C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
105	C₂²⋊C₄×C₇₉	—	2	1	no	—
106	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
107	C₁₆×C₇₉	—	4	1	no	—
108	C₁₆×C₇₉	—	4	1	no	—
109	C₂³×C₇₉	—	2	1	no	—
110	C₂²×D₇₉	—	2	1	no	—
111	C₂²×D₇₉	—	2	4	no	—
112	C₂²×D₇₉	—	4	2	no	—
113	C₂²×D₇₉	—	4	2	no	—
114	C₂²×D₇₉	C₂²⋊C₈	1	1	yes	characteristic
115	D₄×C₇₉	—	2	1	no	—
116	D₄×C₇₉	C₂×C₁₆	1	2	yes	—
117	C₂×C₄×C₇₉	—	2	1	no	—
118	C₂×C₄×C₇₉	—	4	1	no	—
119	C₂×C₄×C₇₉	—	4	1	no	—
120	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
123	C₂²⋊D₇₉	—	2	4	no	—
124	C₂×Dic₇₉	—	2	1	no	—
125	C₂×Dic₇₉	—	4	2	no	—
126	C₂×Dic₇₉	C₂²⋊C₈	1	1	yes	characteristic
127	C₂×Dic₇₉	C₂²⋊C₈	1	1	yes	characteristic
128	C₂×Dic₇₉	C₂²⋊C₈	1	1	yes	characteristic
129	D₃₁₆	—	2	1	no	—
130	D₃₁₆	C₂×C₁₆	1	2	yes	—
131	C₄×D₇₉	—	2	1	no	—
132	C₄×D₇₉	—	2	1	no	—
133	C₄×D₇₉	—	4	1	no	—
134	C₄×D₇₉	—	4	2	no	—
135	C₄×D₇₉	—	4	2	no	—
136	C₄×D₇₉	—	4	1	no	—
137	C₄×D₇₉	—	4	1	no	—
138	C₄×D₇₉	C₂×C₁₆	1	2	yes	—
139	C₄×D₇₉	M₃₂	1	2	yes	—
140	C₈×C₇₉	—	2	1	no	—
141	C₈×C₇₉	—	4	1	no	—
142	C₈×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₈×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₇₉⋊C₈	—	2	1	no	—
145	C₇₉⋊C₈	—	2	1	no	—
146	C₇₉⋊C₈	—	4	1	no	—
147	C₂²×C₇₉	—	2	2	no	—
148	C₂²×C₇₉	—	4	1	no	—
149	C₂²×C₇₉	—	4	1	no	—
150	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
151	C₂×D₇₉	—	2	1	no	—
152	C₂×D₇₉	—	2	1	no	—
153	C₂×D₇₉	—	2	2	no	—
154	C₂×D₇₉	—	4	8	no	—
155	C₂×D₇₉	—	4	1	no	—
156	C₂×D₇₉	—	4	1	no	—
157	C₂×D₇₉	C₂²⋊C₁₆	1	2	yes	—
158	C₄×C₇₉	—	2	1	no	—
159	C₄×C₇₉	—	4	1	no	—
160	C₄×C₇₉	#418	1	1	yes	characteristic
161	C₄×C₇₉	#421	1	1	yes	characteristic
162	C₄×C₇₉	C₂²×C₁₆	1	1	yes	characteristic
163	C₄×C₇₉	C₂×M₃₂	1	1	yes	characteristic
164	Dic₇₉	—	2	1	no	—
165	Dic₇₉	—	2	1	no	—
166	Dic₇₉	—	4	1	no	—
167	Dic₇₉	C₂²⋊C₁₆	1	2	yes	—
168	#11629	—	79	1	no	maximal, Sylow
169	C₂×C₇₉	—	4	2	no	—
170	C₂×C₇₉	#1478	1	1	yes	characteristic
171	C₂×C₇₉	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
172	C₂×C₇₉	#2117	1	1	yes	characteristic
173	D₇₉	—	2	2	no	—
174	D₇₉	—	4	2	no	—
175	C₂×C₈×D₄	—	79	1	no	—
176	#2162	—	79	1	no	—
177	#2162	—	79	1	no	—
178	D₄⋊C₁₆	—	79	1	no	—
179	D₄⋊C₁₆	—	79	1	no	—
180	D₄⋊C₁₆	—	79	1	no	—
181	D₄⋊C₁₆	D₇₉	1	1	yes	characteristic, complemented
182	C₇₉	#11629	1	1	yes	characteristic, complemented, Sylow
183	C₂³×C₈	—	158	1	no	—
184	C₂×C₄×D₄	—	79	1	no	—
185	C₈×D₄	—	79	1	no	—
186	C₈×D₄	—	79	1	no	—
187	C₈×D₄	—	79	1	no	—
188	C₈×D₄	—	158	1	no	—
189	C₈×D₄	—	158	1	no	—
190	C₈×D₄	C₂×D₇₉	1	1	yes	characteristic
191	C₂×C₄⋊C₈	—	79	1	no	—
192	C₂×C₂²⋊C₈	—	158	1	no	—
193	C₂×C₄×C₈	—	79	1	no	—
194	C₄⋊C₁₆	—	79	1	no	—
195	C₄⋊C₁₆	—	158	1	no	—
196	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic, complemented
197	C₂²⋊C₁₆	—	158	1	no	—
198	C₂²⋊C₁₆	—	158	1	no	—
199	C₄×C₁₆	—	79	1	no	—
200	C₄×C₁₆	—	158	1	no	—
201	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic, complemented
202	C₂²×D₄	—	79	1	no	—
203	C₂³×C₄	—	158	1	no	—
204	C₂²×C₈	—	2	1	no	—
205	C₂²×C₈	—	316	2	no	—
206	C₂²×C₈	—	316	2	no	—
207	C₂²×C₈	—	79	1	no	—
208	C₂²×C₈	—	79	1	no	—
209	C₂²×C₈	—	158	1	no	—
210	C₂²×C₈	—	158	1	no	—
211	C₂²×C₈	—	158	1	no	—
212	C₂²×C₈	—	158	1	no	—
213	C₄×D₄	—	79	1	no	—
214	C₄×D₄	—	79	1	no	—
215	C₄×D₄	—	79	1	no	—
216	C₄×D₄	—	158	1	no	—
217	C₄×D₄	—	158	1	no	—
218	C₄×D₄	C₄×D₇₉	1	1	yes	characteristic
219	C₂×C₄⋊C₄	—	79	1	no	—
220	C₂×C₂²⋊C₄	—	158	1	no	—
221	C₂×C₄²	—	79	1	no	—
222	C₂×C₁₆	—	2	1	no	—
223	C₂×C₁₆	—	2	1	no	—
224	C₂×C₁₆	—	158	1	no	—
225	C₂×C₁₆	—	158	1	no	—
226	C₄⋊C₈	—	79	1	no	—
227	C₄⋊C₈	—	158	1	no	—
228	C₄⋊C₈	C₄×D₇₉	1	1	yes	characteristic
229	C₂²⋊C₈	—	2	1	no	—
230	C₂²⋊C₈	—	158	1	no	—
231	C₂²⋊C₈	—	158	1	no	—
232	C₂²⋊C₈	—	158	1	no	—
233	C₄×C₈	—	79	1	no	—
234	C₄×C₈	—	79	1	no	—
235	C₄×C₈	—	79	1	no	—
236	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
237	C₂⁴	—	158	1	no	—
238	C₂×D₄	—	79	4	no	—
239	C₂×D₄	—	79	1	no	—
240	C₂×D₄	—	158	2	no	—
241	C₂×D₄	—	158	1	no	—
242	C₂×D₄	C₈×D₇₉	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	316	2	no	—
245	C₂²×C₄	—	316	2	no	—
246	C₂²×C₄	—	79	1	no	—
247	C₂²×C₄	—	79	1	no	—
248	C₂²×C₄	—	79	1	no	—
249	C₂²×C₄	—	158	1	no	—
250	C₂²×C₄	—	158	1	no	—
251	C₂²×C₄	—	158	1	no	—
252	C₂²×C₄	—	158	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	316	1	no	—
257	C₂×C₈	—	316	2	no	—
258	C₂×C₈	—	316	2	no	—
259	C₂×C₈	—	316	2	no	—
260	C₂×C₈	—	316	1	no	—
261	C₂×C₈	—	316	1	no	—
262	C₂×C₈	—	79	1	no	—
263	C₂×C₈	—	79	1	no	—
264	C₂×C₈	—	158	1	no	—
265	C₂×C₈	—	158	1	no	—
266	C₂×C₈	—	158	1	no	—
267	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
268	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
269	C₄⋊C₄	—	79	1	no	—
270	C₄⋊C₄	—	158	1	no	—
271	C₄⋊C₄	C₈×D₇₉	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	158	1	no	—
274	C₂²⋊C₄	—	158	1	no	—
275	C₂²⋊C₄	—	158	1	no	—
276	C₄²	—	79	1	no	—
277	C₄²	—	79	1	no	—
278	C₄²	—	79	1	no	—
279	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
280	C₁₆	—	4	1	no	—
281	C₁₆	—	4	1	no	—
282	C₁₆	—	316	1	no	—
283	C₁₆	—	316	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	316	2	no	—
286	C₂³	—	316	2	no	—
287	C₂³	—	79	1	no	—
288	C₂³	—	158	4	no	—
289	C₂³	—	158	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	79	2	no	—
292	D₄	—	158	4	no	—
293	D₄	—	158	1	no	—
294	D₄	C₁₆×D₇₉	1	2	yes	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	316	1	no	—
299	C₂×C₄	—	316	2	no	—
300	C₂×C₄	—	316	2	no	—
301	C₂×C₄	—	316	2	no	—
302	C₂×C₄	—	316	1	no	—
303	C₂×C₄	—	316	1	no	—
304	C₂×C₄	—	79	1	no	—
305	C₂×C₄	—	79	2	no	—
306	C₂×C₄	—	79	2	no	—
307	C₂×C₄	—	79	1	no	—
308	C₂×C₄	—	79	1	no	—
309	C₂×C₄	—	158	1	no	—
310	C₂×C₄	—	158	1	no	—
311	C₂×C₄	—	158	1	no	—
312	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
313	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
314	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	316	1	no	—
318	C₈	—	158	1	no	—
319	C₈	—	158	1	no	—
320	C₈	#14579185	1	1	yes	characteristic
321	C₈	#14579188	1	1	yes	characteristic
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	316	8	no	—
326	C₂²	—	316	1	no	—
327	C₂²	—	316	1	no	—
328	C₂²	—	79	2	no	—
329	C₂²	—	158	2	no	—
330	C₂²	—	158	1	no	—
331	C₂²	—	158	1	no	—
332	C₂²	C₂²⋊C₈×D₇₉	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	316	1	no	—
336	C₄	—	79	2	no	—
337	C₄	—	158	1	no	—
338	C₄	—	158	1	no	—
339	C₄	#19098566	1	1	yes	characteristic
340	C₄	#19098578	1	1	yes	characteristic
341	C₄	C₂×C₁₆×D₇₉	1	1	yes	characteristic
342	C₄	M₃₂×D₇₉	1	1	yes	characteristic
343	C₂	—	4	2	no	—
344	C₂	—	316	2	no	—
345	C₂	—	158	2	no	—
346	C₂	#23761789	1	1	yes	characteristic
347	C₂	C₂²⋊C₁₆×D₇₉	1	1	yes	characteristic
348	C₂	#23765026	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₉
commutator	C₄×C₇₉	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×D₄⋊C₁₆	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₈
prime radical	C₂×D₄×D₇₉	C₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	D₄⋊C₁₆	D₇₉
hypocenter	C₇₉	#11629
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ D₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #23765198	C₂	semidirect
3. #23765197	C₂	semidirect
4. #23761356	C₂	semidirect
5. D₄⋊(C₇₉⋊C₁₆)	C₂	semidirect
6. #23761354	C₂	semidirect
7. C₇₉×D₄⋊C₁₆	C₂	semidirect
18. C₄⋊(C₇₉⋊C₁₆)	C₂²	semidirect
22. C₄×C₇₉⋊C₁₆	C₂²	semidirect
25. C₄⋊C₁₆×C₇₉	C₂²	semidirect
26. C₄×C₁₆×C₇₉	C₂²	semidirect
66. D₄×D₇₉	C₁₆	semidirect
181. D₄⋊C₁₆	D₇₉	semidirect
182. C₇₉	#11629	semidirect
196. C₄⋊C₁₆	C₂×D₇₉	semidirect
201. C₄×C₁₆	C₂×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₈×D₄×D₇₉	2
#23765198	2
#23765197	2
#23761356	2
D₄⋊(C₇₉⋊C₁₆)	2
#23761354	2
C₇₉×D₄⋊C₁₆	2
#11629	79