Subgroups — #34197974

Order 20224.

450 conjugacy classes of subgroups.

The subgroup lattice has 354 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	#23765154	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14579100	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	C₂×#14579100	—	2	1	no	—
11	C₂×#14579100	—	2	1	no	—
12	C₈×D₃₁₆	C₂²	1	1	yes	characteristic
13	D₄×C₇₉⋊C₈	C₄	1	1	yes	characteristic
14	#19098628	C₄	1	1	yes	characteristic
15	C₈×C₂²⋊D₇₉	—	2	1	no	—
16	(C₂²⋊D₇₉)⋊C₈	—	2	1	no	—
17	C₄⋊C₈×D₇₉	C₄	1	1	yes	characteristic
18	C₂²⋊C₈×D₇₉	—	2	1	no	—
19	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
20	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic, complemented
21	Dic₇₉⋊C₁₆	—	2	1	no	—
22	#19098338	—	2	1	no	—
23	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
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	#14579100	—	2	4	no	—
46	#14579100	—	4	2	no	—
47	C₂×C₇₉⋊C₁₆	—	2	1	no	—
48	C₂×C₇₉⋊C₁₆	—	2	1	no	—
49	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
50	Dic₇₉⋊C₈	—	2	1	no	—
51	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
52	(C₂×D₇₉)⋊C₈	—	2	1	no	—
53	(C₂×D₇₉)⋊C₈	—	2	1	no	—
54	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₇₉	D₄	1	1	yes	characteristic
56	C₈×Dic₇₉	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₇₉	—	2	1	no	—
58	C₄×D₄×C₇₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₁₆×C₇₉	—	2	1	no	—
60	C₂×C₁₆×C₇₉	—	2	1	no	—
61	C₄⋊C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
62	C₂²⋊C₈×C₇₉	—	2	1	no	—
63	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
64	C₂³×D₇₉	—	2	1	no	—
65	C₂×D₃₁₆	C₂×C₈	1	1	yes	characteristic
66	C₂×C₂²⋊D₇₉	—	2	1	no	—
67	D₄×D₇₉	—	2	2	no	—
68	D₄×D₇₉	—	2	2	no	—
69	C₂²×Dic₇₉	—	2	1	no	—
70	C₂×C₄×D₇₉	—	2	1	no	—
71	C₂×C₄×D₇₉	—	2	1	no	—
72	C₂×C₄×D₇₉	—	2	1	no	—
73	C₂×C₄×D₇₉	—	4	2	no	—
74	C₂×C₄×D₇₉	—	4	2	no	—
75	C₂×C₄×D₇₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×D₇₉	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄×D₇₉	M₁₆	1	1	yes	characteristic
78	C₂×C₇₉⋊C₈	—	2	1	no	—
79	C₂×C₇₉⋊C₈	—	4	2	no	—
80	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₈×D₇₉	—	2	2	no	—
83	C₈×D₇₉	—	4	2	no	—
84	C₈×D₇₉	—	4	1	no	—
85	C₈×D₇₉	—	4	1	no	—
86	C₈×D₇₉	—	4	2	no	—
87	C₈×D₇₉	D₈	1	2	yes	—
88	C₈×D₇₉	SD₁₆	1	2	yes	—
89	C₄⋊Dic₇₉	C₂×C₈	1	1	yes	characteristic
90	Dic₇₉⋊C₄	—	2	1	no	—
91	C₂²⋊Dic₇₉	—	2	1	no	—
92	(C₂×D₇₉)⋊C₄	—	2	1	no	—
93	(C₂×D₇₉)⋊C₄	—	2	1	no	—
94	C₄×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×Dic₇₉	C₂×C₈	1	1	yes	characteristic
96	C₄×Dic₇₉	M₁₆	1	1	yes	characteristic
97	C₇₉⋊C₁₆	—	2	2	no	—
98	C₇₉⋊C₁₆	—	4	1	no	—
99	C₂×D₄×C₇₉	C₂×C₈	1	1	yes	characteristic
100	C₂²×C₄×C₇₉	—	2	1	no	—
101	C₂×C₈×C₇₉	—	2	1	no	—
102	C₂×C₈×C₇₉	—	4	1	no	—
103	C₂×C₈×C₇₉	—	4	1	no	—
104	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
106	C₄⋊C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
107	C₂²⋊C₄×C₇₉	—	2	1	no	—
108	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
109	C₁₆×C₇₉	—	2	2	no	—
110	C₁₆×C₇₉	—	4	1	no	—
111	C₂³×C₇₉	—	2	1	no	—
112	C₂²×D₇₉	—	2	4	no	—
113	C₂²×D₇₉	—	2	1	no	—
114	C₂²×D₇₉	—	4	2	no	—
115	C₂²×D₇₉	—	4	2	no	—
116	C₂²×D₇₉	C₂²⋊C₈	1	1	yes	characteristic
117	D₄×C₇₉	—	2	1	no	—
118	D₄×C₇₉	M₃₂	1	2	yes	complemented
119	C₂×C₄×C₇₉	—	2	1	no	—
120	C₂×C₄×C₇₉	—	4	1	no	—
121	C₂×C₄×C₇₉	—	4	1	no	—
122	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
125	C₂²⋊D₇₉	—	2	4	no	—
126	C₂×Dic₇₉	—	2	1	no	—
127	C₂×Dic₇₉	—	4	2	no	—
128	C₂×Dic₇₉	C₂²⋊C₈	1	1	yes	characteristic
129	C₂×Dic₇₉	C₂²⋊C₈	1	1	yes	characteristic
130	C₂×Dic₇₉	C₂²⋊C₈	1	1	yes	characteristic
131	D₃₁₆	—	2	1	no	—
132	D₃₁₆	M₃₂	1	2	yes	complemented
133	C₄×D₇₉	—	2	1	no	—
134	C₄×D₇₉	—	2	1	no	—
135	C₄×D₇₉	—	4	1	no	—
136	C₄×D₇₉	—	4	2	no	—
137	C₄×D₇₉	—	4	1	no	—
138	C₄×D₇₉	—	4	1	no	—
139	C₄×D₇₉	—	4	2	no	—
140	C₄×D₇₉	#102	1	2	yes	—
141	C₄×D₇₉	#104	1	2	yes	—
142	C₈×C₇₉	—	2	1	no	—
143	C₈×C₇₉	—	4	1	no	—
144	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
145	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
146	C₇₉⋊C₈	—	2	1	no	—
147	C₇₉⋊C₈	—	4	1	no	—
148	C₇₉⋊C₈	#102	1	2	yes	—
149	C₂²×C₇₉	—	2	2	no	—
150	C₂²×C₇₉	—	4	1	no	—
151	C₂²×C₇₉	—	4	1	no	—
152	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
153	C₂×D₇₉	—	2	1	no	—
154	C₂×D₇₉	—	2	1	no	—
155	C₂×D₇₉	—	2	2	no	—
156	C₂×D₇₉	—	4	8	no	—
157	C₂×D₇₉	—	4	1	no	—
158	C₂×D₇₉	—	4	1	no	—
159	C₂×D₇₉	D₄⋊C₈	1	2	yes	—
160	C₄×C₇₉	—	2	1	no	—
161	C₄×C₇₉	—	4	1	no	—
162	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
163	C₄×C₇₉	C₂×#104	1	1	yes	characteristic
164	C₄×C₇₉	C₂×M₃₂	1	1	yes	characteristic
165	C₄×C₇₉	C₂×M₃₂	1	1	yes	characteristic
166	Dic₇₉	—	2	1	no	—
167	Dic₇₉	—	2	1	no	—
168	Dic₇₉	—	4	1	no	—
169	Dic₇₉	D₄⋊C₈	1	2	yes	—
170	#11583	—	79	1	no	maximal, Sylow
171	C₂×C₇₉	—	4	2	no	—
172	C₂×C₇₉	C₂×D₄⋊C₈	1	1	yes	characteristic
173	C₂×C₇₉	#2112	1	1	yes	characteristic
174	C₂×C₇₉	#2117	1	1	yes	characteristic
175	D₇₉	—	2	2	no	—
176	D₇₉	—	4	2	no	—
177	C₂×C₈×D₄	—	79	1	no	—
178	#2150	—	79	1	no	—
179	C₄×M₃₂	—	79	1	no	—
180	D₄⋊C₁₆	—	79	1	no	—
181	D₄⋊C₁₆	—	79	1	no	—
182	D₄⋊C₁₆	—	79	1	no	—
183	D₄⋊C₁₆	D₇₉	1	1	yes	characteristic, complemented
184	C₇₉	#11583	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₈	—	158	1	no	—
186	C₂×C₄×D₄	—	79	1	no	—
187	C₂×M₃₂	—	158	1	no	—
188	C₂×M₃₂	—	158	1	no	—
189	C₈×D₄	—	79	1	no	—
190	C₈×D₄	—	79	1	no	—
191	C₈×D₄	—	79	1	no	—
192	C₈×D₄	—	158	1	no	—
193	C₈×D₄	—	158	1	no	—
194	C₈×D₄	C₂×D₇₉	1	1	yes	characteristic
195	C₂×C₄⋊C₈	—	79	1	no	—
196	C₂×C₂²⋊C₈	—	158	1	no	—
197	C₂×C₄×C₈	—	79	1	no	—
198	C₄⋊C₁₆	—	79	1	no	—
199	C₄⋊C₁₆	—	158	1	no	—
200	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic, complemented
201	#346	—	158	1	no	—
202	C₄×C₁₆	—	79	1	no	—
203	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic, complemented
204	C₂²×D₄	—	79	1	no	—
205	C₂³×C₄	—	158	1	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	—	316	2	no	—
208	C₂²×C₈	—	316	2	no	—
209	C₂²×C₈	—	79	1	no	—
210	C₂²×C₈	—	79	1	no	—
211	C₂²×C₈	—	158	1	no	—
212	C₂²×C₈	—	158	1	no	—
213	C₂²×C₈	—	158	1	no	—
214	C₂²×C₈	—	158	1	no	—
215	C₄×D₄	—	79	1	no	—
216	C₄×D₄	—	79	1	no	—
217	C₄×D₄	—	79	1	no	—
218	C₄×D₄	—	158	1	no	—
219	C₄×D₄	—	158	1	no	—
220	C₄×D₄	C₄×D₇₉	1	1	yes	characteristic
221	C₂×C₄⋊C₄	—	79	1	no	—
222	C₂×C₂²⋊C₄	—	158	1	no	—
223	C₂×C₄²	—	79	1	no	—
224	M₃₂	—	316	2	no	—
225	M₃₂	—	158	4	no	—
226	C₂×C₁₆	—	2	1	no	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	158	1	no	—
229	C₂×C₁₆	—	158	1	no	—
230	C₄⋊C₈	—	79	1	no	—
231	C₄⋊C₈	—	158	1	no	—
232	C₄⋊C₈	C₄×D₇₉	1	1	yes	characteristic
233	C₂²⋊C₈	—	2	1	no	—
234	C₂²⋊C₈	—	158	1	no	—
235	C₂²⋊C₈	—	158	1	no	—
236	C₂²⋊C₈	—	158	1	no	—
237	C₄×C₈	—	79	1	no	—
238	C₄×C₈	—	79	1	no	—
239	C₄×C₈	—	79	1	no	—
240	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
241	C₂⁴	—	158	1	no	—
242	C₂×D₄	—	79	1	no	—
243	C₂×D₄	—	158	1	no	—
244	C₂×D₄	—	158	2	no	—
245	C₂×D₄	—	158	2	no	—
246	C₂×D₄	C₈×D₇₉	1	1	yes	characteristic
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	316	2	no	—
249	C₂²×C₄	—	316	2	no	—
250	C₂²×C₄	—	79	1	no	—
251	C₂²×C₄	—	79	1	no	—
252	C₂²×C₄	—	79	1	no	—
253	C₂²×C₄	—	158	1	no	—
254	C₂²×C₄	—	158	1	no	—
255	C₂²×C₄	—	158	1	no	—
256	C₂²×C₄	—	158	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	316	2	no	—
261	C₂×C₈	—	316	2	no	—
262	C₂×C₈	—	316	2	no	—
263	C₂×C₈	—	316	1	no	—
264	C₂×C₈	—	316	1	no	—
265	C₂×C₈	—	79	1	no	—
266	C₂×C₈	—	79	2	no	—
267	C₂×C₈	—	79	2	no	—
268	C₂×C₈	—	79	1	no	—
269	C₂×C₈	—	158	2	no	—
270	C₂×C₈	—	158	1	no	—
271	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
272	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
273	C₄⋊C₄	—	79	1	no	—
274	C₄⋊C₄	—	158	1	no	—
275	C₄⋊C₄	C₈×D₇₉	1	1	yes	characteristic
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	158	1	no	—
278	C₂²⋊C₄	—	158	1	no	—
279	C₂²⋊C₄	—	158	1	no	—
280	C₄²	—	79	1	no	—
281	C₄²	—	79	1	no	—
282	C₄²	—	79	1	no	—
283	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
284	C₁₆	—	2	2	no	—
285	C₁₆	—	4	1	no	—
286	C₁₆	—	316	1	no	—
287	C₁₆	—	158	2	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	316	2	no	—
290	C₂³	—	316	2	no	—
291	C₂³	—	79	1	no	—
292	C₂³	—	158	4	no	—
293	C₂³	—	158	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	79	2	no	—
296	D₄	—	158	4	no	—
297	D₄	—	158	1	no	—
298	D₄	#14579100	1	2	yes	complemented
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	316	1	no	—
303	C₂×C₄	—	316	2	no	—
304	C₂×C₄	—	316	2	no	—
305	C₂×C₄	—	316	2	no	—
306	C₂×C₄	—	316	1	no	—
307	C₂×C₄	—	316	1	no	—
308	C₂×C₄	—	79	1	no	—
309	C₂×C₄	—	79	2	no	—
310	C₂×C₄	—	79	2	no	—
311	C₂×C₄	—	79	1	no	—
312	C₂×C₄	—	79	1	no	—
313	C₂×C₄	—	158	1	no	—
314	C₂×C₄	—	158	1	no	—
315	C₂×C₄	—	158	1	no	—
316	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
317	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
318	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	4	1	no	—
321	C₈	—	316	1	no	—
322	C₈	—	79	2	no	—
323	C₈	—	158	1	no	—
324	C₈	D₈×D₇₉	1	1	yes	characteristic
325	C₈	SD₁₆×D₇₉	1	1	yes	characteristic
326	C₂²	—	2	2	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	316	8	no	—
330	C₂²	—	316	1	no	—
331	C₂²	—	316	1	no	—
332	C₂²	—	79	2	no	—
333	C₂²	—	158	2	no	—
334	C₂²	—	158	1	no	—
335	C₂²	—	158	1	no	—
336	C₂²	C₂²⋊C₈×D₇₉	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	316	1	no	—
340	C₄	—	79	2	no	—
341	C₄	—	158	1	no	—
342	C₄	—	158	1	no	—
343	C₄	#102×D₇₉	1	1	yes	characteristic
344	C₄	#104×D₇₉	1	1	yes	characteristic
345	C₄	M₃₂×D₇₉	1	1	yes	characteristic
346	C₄	C₂×#14579100	1	1	yes	characteristic
347	C₂	—	4	2	no	—
348	C₂	—	316	2	no	—
349	C₂	—	158	2	no	—
350	C₂	D₄⋊C₈×D₇₉	1	1	yes	characteristic
351	C₂	#23765009	1	1	yes	characteristic
352	C₂	#23765026	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
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	#11583
LCS	G ▷ C₂×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. #23765154	C₂	semidirect
3. C₄×#14579100	C₂	semidirect
4. #23761356	C₂	semidirect
5. D₄⋊(C₇₉⋊C₁₆)	C₂	semidirect
6. #23761354	C₂	semidirect
7. C₇₉×D₄⋊C₁₆	C₂	semidirect
20. C₄⋊(C₇₉⋊C₁₆)	C₂²	semidirect
23. C₄×C₇₉⋊C₁₆	C₂²	semidirect
25. C₄⋊C₁₆×C₇₉	C₂²	semidirect
26. C₄×C₁₆×C₇₉	C₂²	semidirect
118. D₄×C₇₉	M₃₂	semidirect
132. D₃₁₆	M₃₂	semidirect
183. D₄⋊C₁₆	D₇₉	semidirect
184. C₇₉	#11583	semidirect
200. C₄⋊C₁₆	C₂×D₇₉	semidirect
203. C₄×C₁₆	C₂×D₇₉	semidirect
298. D₄	#14579100	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
#23765154	2
C₄×#14579100	2
#23761356	2
D₄⋊(C₇₉⋊C₁₆)	2
#23761354	2
C₇₉×D₄⋊C₁₆	2
#11583	79