Subgroups — #34179927

Order 20224.

712 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579080	C₂	1	1	yes	maximal, characteristic
2	#23762002	C₂	1	2	yes	maximal
3	#23761847	C₂	1	2	yes	maximal
4	#23761744	C₂	1	1	yes	maximal, characteristic
5	C₂×#557×C₇₉	C₂	1	1	yes	maximal, characteristic
6	#19098620	C₂²	1	2	yes	—
7	C₂×#14579080	—	2	1	no	—
8	C₄×#11929608	C₂²	1	2	yes	—
9	#19098332	—	2	1	no	—
10	#19098305	—	2	4	no	—
11	Q₈⋊(C₇₉⋊C₈)	—	2	4	no	—
12	#19098289	—	2	1	no	—
13	#19098289	—	2	1	no	—
14	#557×C₇₉	—	2	4	no	—
15	#557×C₇₉	—	2	4	no	—
16	C₂×C₄⋊Q₈×C₇₉	—	2	1	no	—
17	C₂×C₄⋊Q₈×C₇₉	C₄	1	1	yes	characteristic
18	C₂×#125×C₇₉	—	2	1	no	—
19	C₂×#125×C₇₉	—	2	1	no	—
20	C₂×#125×C₇₉	C₄	1	1	yes	characteristic
21	C₂×#125×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₇₉	—	2	1	no	—
23	C₂×C₄×Q₈×C₇₉	C₄	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₇₉	C₄	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×#11929608	—	2	2	no	—
28	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
29	#14579080	—	2	4	no	—
30	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
31	#14579072	—	2	2	no	—
32	#14579072	—	2	2	no	—
33	C₄×C₇₉⋊C₈	—	2	2	no	—
34	C₂²×Q₈×C₇₉	—	2	1	no	—
35	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×Q₈×C₇₉	D₄	1	1	yes	characteristic
37	C₄⋊Q₈×C₇₉	—	2	8	no	—
38	C₄⋊Q₈×C₇₉	—	2	2	no	—
39	C₄⋊Q₈×C₇₉	—	2	2	no	—
40	#125×C₇₉	—	2	8	no	—
41	#125×C₇₉	—	2	8	no	—
42	#125×C₇₉	—	2	2	no	—
43	#125×C₇₉	—	2	2	no	—
44	#125×C₇₉	—	2	2	no	—
45	#125×C₇₉	D₄	1	4	yes	—
46	C₄×Q₈×C₇₉	—	2	8	no	—
47	C₄×Q₈×C₇₉	—	2	2	no	—
48	C₄×Q₈×C₇₉	—	2	2	no	—
49	C₄×Q₈×C₇₉	—	2	2	no	—
50	C₄×Q₈×C₇₉	—	2	2	no	—
51	C₄×Q₈×C₇₉	—	2	2	no	—
52	C₄×Q₈×C₇₉	—	2	2	no	—
53	C₄×Q₈×C₇₉	—	2	2	no	—
54	C₄×Q₈×C₇₉	D₄	1	4	yes	—
55	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
57	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
59	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
60	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
61	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
62	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
63	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
69	C₂×C₄²×C₇₉	—	2	1	no	—
70	C₂×C₄²×C₇₉	—	2	1	no	—
71	C₂×C₄²×C₇₉	—	2	1	no	—
72	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
75	#11929608	—	2	4	no	—
76	#11929608	—	4	2	no	—
77	C₂×C₇₉⋊C₈	—	4	2	no	—
78	C₂×C₇₉⋊C₈	—	4	2	no	—
79	C₂×Q₈×C₇₉	—	2	4	no	—
80	C₂×Q₈×C₇₉	—	2	4	no	—
81	C₂×Q₈×C₇₉	—	2	2	no	—
82	C₂×Q₈×C₇₉	—	2	1	no	—
83	C₂×Q₈×C₇₉	—	2	2	no	—
84	C₂×Q₈×C₇₉	—	4	1	no	—
85	C₂×Q₈×C₇₉	—	4	2	no	—
86	C₂×Q₈×C₇₉	—	4	2	no	—
87	C₂×Q₈×C₇₉	C₂²⋊C₄	1	2	yes	—
88	C₂²×C₄×C₇₉	—	2	1	no	—
89	C₂²×C₄×C₇₉	—	2	1	no	—
90	C₂²×C₄×C₇₉	—	2	1	no	—
91	C₂²×C₄×C₇₉	—	2	1	no	—
92	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
99	C₄⋊C₄×C₇₉	—	2	4	no	—
100	C₄⋊C₄×C₇₉	—	2	4	no	—
101	C₄⋊C₄×C₇₉	—	2	4	no	—
102	C₄⋊C₄×C₇₉	—	2	4	no	—
103	C₄⋊C₄×C₇₉	—	2	4	no	—
104	C₄⋊C₄×C₇₉	—	2	1	no	—
105	C₄⋊C₄×C₇₉	—	2	1	no	—
106	C₄⋊C₄×C₇₉	—	2	1	no	—
107	C₄⋊C₄×C₇₉	—	2	4	no	—
108	C₄⋊C₄×C₇₉	—	2	4	no	—
109	C₄⋊C₄×C₇₉	—	2	4	no	—
110	C₄⋊C₄×C₇₉	—	2	2	no	—
111	C₄⋊C₄×C₇₉	—	2	2	no	—
112	C₄⋊C₄×C₇₉	—	2	2	no	—
113	C₄⋊C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
114	C₄⋊C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
115	C₄⋊C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
116	C₄²×C₇₉	—	2	4	no	—
117	C₄²×C₇₉	—	2	4	no	—
118	C₄²×C₇₉	—	2	1	no	—
119	C₄²×C₇₉	—	2	1	no	—
120	C₄²×C₇₉	—	2	1	no	—
121	C₄²×C₇₉	—	2	1	no	—
122	C₄²×C₇₉	—	2	4	no	—
123	C₄²×C₇₉	C₂×D₄	1	2	yes	—
124	C₄²×C₇₉	C₂×D₄	1	2	yes	—
125	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	Q₈×C₇₉	—	2	4	no	—
127	Q₈×C₇₉	—	2	2	no	—
128	Q₈×C₇₉	—	4	4	no	—
129	Q₈×C₇₉	—	4	8	no	—
130	Q₈×C₇₉	#101	1	4	yes	complemented
131	C₂×C₄×C₇₉	—	2	4	no	—
132	C₂×C₄×C₇₉	—	2	2	no	—
133	C₂×C₄×C₇₉	—	2	1	no	—
134	C₂×C₄×C₇₉	—	2	1	no	—
135	C₂×C₄×C₇₉	—	2	1	no	—
136	C₂×C₄×C₇₉	—	2	1	no	—
137	C₂×C₄×C₇₉	—	2	2	no	—
138	C₂×C₄×C₇₉	—	2	1	no	—
139	C₂×C₄×C₇₉	—	2	1	no	—
140	C₂×C₄×C₇₉	—	2	2	no	—
141	C₂×C₄×C₇₉	—	2	1	no	—
142	C₂×C₄×C₇₉	—	2	2	no	—
143	C₂×C₄×C₇₉	—	2	2	no	—
144	C₂×C₄×C₇₉	—	2	1	no	—
145	C₂×C₄×C₇₉	—	2	1	no	—
146	C₂×C₄×C₇₉	—	4	1	no	—
147	C₂×C₄×C₇₉	—	4	2	no	—
148	C₂×C₄×C₇₉	—	4	1	no	—
149	C₂×C₄×C₇₉	—	4	1	no	—
150	C₂×C₄×C₇₉	—	4	1	no	—
151	C₂×C₄×C₇₉	—	4	1	no	—
152	C₂×C₄×C₇₉	—	4	1	no	—
153	C₂×C₄×C₇₉	—	4	1	no	—
154	C₂×C₄×C₇₉	#104	1	2	yes	—
155	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₇₉	#120	1	2	yes	—
158	C₂×C₄×C₇₉	#120	1	2	yes	—
159	C₇₉⋊C₈	—	4	4	no	—
160	C₇₉⋊C₈	—	8	2	no	—
161	C₂²×C₇₉	—	2	1	no	—
162	C₂²×C₇₉	—	2	1	no	—
163	C₂²×C₇₉	#379	1	1	yes	characteristic
164	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
165	C₂²×C₇₉	#421	1	1	yes	characteristic
166	C₄×C₇₉	—	2	2	no	—
167	C₄×C₇₉	—	2	4	no	—
168	C₄×C₇₉	—	2	1	no	—
169	C₄×C₇₉	—	2	2	no	—
170	C₄×C₇₉	—	2	2	no	—
171	C₄×C₇₉	—	4	1	no	—
172	C₄×C₇₉	—	4	1	no	—
173	C₄×C₇₉	—	4	2	no	—
174	C₄×C₇₉	—	4	1	no	—
175	C₄×C₇₉	—	4	2	no	—
176	C₄×C₇₉	—	4	2	no	—
177	C₄×C₇₉	C₂×#101	1	2	yes	—
178	#8374	—	79	1	no	maximal, Sylow
179	C₂×C₇₉	—	2	1	no	—
180	C₂×C₇₉	—	2	1	no	—
181	C₂×C₇₉	#1784	1	1	yes	characteristic
182	C₂×C₇₉	#1789	1	1	yes	characteristic
183	C₂×C₇₉	#1796	1	1	yes	characteristic
184	C₂×#557	D₇₉	1	1	yes	characteristic
185	C₄×#101	—	79	1	no	—
186	#1528	—	79	2	no	—
187	#1491	—	79	2	no	—
188	#1467	—	79	1	no	—
189	C₇₉	#8374	1	1	yes	characteristic, complemented, Sylow
190	#557	—	2	4	no	—
191	#557	—	2	4	no	—
192	C₂×C₄⋊Q₈	—	2	1	no	—
193	C₂×C₄⋊Q₈	Dic₇₉	1	1	yes	characteristic
194	C₂×#125	—	2	1	no	—
195	C₂×#125	—	2	1	no	—
196	C₂×#125	Dic₇₉	1	1	yes	characteristic
197	C₂×#125	C₂×D₇₉	1	1	yes	characteristic
198	C₂×C₄×Q₈	—	2	1	no	—
199	C₂×C₄×Q₈	Dic₇₉	1	1	yes	characteristic
200	C₂×C₄×Q₈	Dic₇₉	1	1	yes	characteristic
201	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
202	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
203	#432	—	79	2	no	—
204	C₂×#101	—	158	1	no	—
205	C₄×M₁₆	—	79	2	no	—
206	#343	—	158	1	no	—
207	#330	—	158	4	no	—
208	Q₈⋊C₈	—	158	4	no	—
209	#324	—	158	1	no	—
210	#324	—	158	1	no	—
211	C₂²×Q₈	—	2	1	no	—
212	C₂²×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
213	C₂²×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
214	C₂×M₁₆	—	158	2	no	—
215	C₄⋊Q₈	—	2	8	no	—
216	C₄⋊Q₈	—	2	2	no	—
217	C₄⋊Q₈	—	2	2	no	—
218	#125	—	2	8	no	—
219	#125	—	2	8	no	—
220	#125	—	2	2	no	—
221	#125	—	2	2	no	—
222	#125	—	2	2	no	—
223	#125	C₂²⋊D₇₉	1	4	yes	—
224	C₄×Q₈	—	2	8	no	—
225	C₄×Q₈	—	2	2	no	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×Q₈	—	2	2	no	—
228	C₄×Q₈	—	2	2	no	—
229	C₄×Q₈	—	2	2	no	—
230	C₄×Q₈	—	2	2	no	—
231	C₄×Q₈	—	2	2	no	—
232	C₄×Q₈	C₂²⋊D₇₉	1	4	yes	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	2	1	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
242	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
243	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
244	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
245	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
246	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
247	C₂×C₄²	—	2	1	no	—
248	C₂×C₄²	—	2	1	no	—
249	C₂×C₄²	—	2	1	no	—
250	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
251	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
252	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
253	C₄⋊C₈	—	158	2	no	—
254	#101	—	158	4	no	—
255	C₂²⋊C₈	—	158	2	no	—
256	#97	—	158	2	no	—
257	#97	—	158	2	no	—
258	C₄×C₈	—	158	2	no	—
259	C₂×Q₈	—	2	4	no	—
260	C₂×Q₈	—	2	4	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	4	2	no	—
266	C₂×Q₈	—	4	2	no	—
267	C₂×Q₈	C₂²⋊Dic₇₉	1	2	yes	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
273	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
274	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
275	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
276	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
279	M₁₆	—	316	2	no	—
280	M₁₆	—	158	4	no	—
281	C₂×C₈	—	316	2	no	—
282	C₂×C₈	—	316	2	no	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	—	2	4	no	—
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	4	no	—
292	C₄⋊C₄	—	2	4	no	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	C₂²⋊Dic₇₉	1	2	yes	—
298	C₄⋊C₄	C₂²⋊Dic₇₉	1	2	yes	—
299	C₄⋊C₄	C₂²⋊Dic₇₉	1	2	yes	—
300	C₄²	—	2	4	no	—
301	C₄²	—	2	4	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	2	1	no	—
304	C₄²	—	2	1	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	2	4	no	—
307	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
308	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
309	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
310	Q₈	—	2	4	no	—
311	Q₈	—	2	2	no	—
312	Q₈	—	4	4	no	—
313	Q₈	—	4	8	no	—
314	Q₈	#14579080	1	4	yes	complemented
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	2	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	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	#14579088	1	2	yes	—
339	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
340	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
341	C₂×C₄	#14579130	1	2	yes	—
342	C₂×C₄	#14579130	1	2	yes	—
343	C₈	—	316	4	no	—
344	C₈	—	632	2	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	#19098419	1	1	yes	characteristic
348	C₂²	C₂×#14579088	1	1	yes	characteristic
349	C₂²	#19098580	1	1	yes	characteristic
350	C₄	—	2	4	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	2	no	—
358	C₄	—	4	2	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	1	no	—
361	C₄	C₂×#14579080	1	2	yes	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	#23763324	1	1	yes	characteristic
365	C₂	#23763344	1	1	yes	characteristic
366	C₂	#23763375	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098419
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#557×C₇₉	C₂
socle	C₂²×C₇₉	#379
prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#557	D₇₉
hypocenter	C₇₉	#8374
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#557

Complements

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

N	H ≅ G/N	type
130. Q₈×C₇₉	#101	semidirect
189. C₇₉	#8374	semidirect
314. Q₈	#14579080	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579080	2
#23762002	2
#23761847	2
#23761744	2
C₂×#557×C₇₉	2
#8374	79