Subgroups — #34178898

Order 20224.

578 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
2	#23763255	C₂	1	1	yes	maximal, characteristic
3	#23763145	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#23761240	C₂	1	1	yes	maximal, characteristic
6	#23761239	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₇₉	C₄	1	2	yes	—
9	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#14579071	—	2	1	no	—
11	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×Dic₇₉	—	2	2	no	—
13	C₄⋊C₄×Dic₇₉	C₄	1	2	yes	—
14	C₄×C₄⋊Dic₇₉	—	2	1	no	—
15	C₄×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
16	C₄²×Dic₇₉	—	2	1	no	—
17	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
18	#19098317	—	2	1	no	—
19	#19098317	—	2	2	no	—
20	Q₈⋊(C₇₉⋊C₈)	C₄	1	4	yes	complemented
21	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	Q₈⋊C₈×C₇₉	C₄	1	4	yes	complemented
25	C₂²×C₇₉⋊C₈	—	2	1	no	—
26	C₂²×C₇₉⋊C₈	—	2	1	no	—
27	Q₈×Dic₇₉	—	2	4	no	—
28	Q₈×Dic₇₉	—	2	4	no	—
29	C₂×C₄⋊Dic₇₉	—	2	2	no	—
30	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₇₉	—	2	2	no	—
32	C₂×C₄×Dic₇₉	—	2	1	no	—
33	C₂×C₄×Dic₇₉	—	2	2	no	—
34	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₇₉	D₄	1	2	yes	—
37	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
38	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
39	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
40	#14579071	—	4	2	no	—
41	C₄×C₇₉⋊C₈	—	2	1	no	—
42	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
43	C₇₉⋊#95	—	2	1	no	—
44	C₇₉⋊#95	—	2	2	no	—
45	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₉	—	2	1	no	—
47	C₂²×C₈×C₇₉	—	2	1	no	—
48	C₄×Q₈×C₇₉	—	2	1	no	—
49	C₄×Q₈×C₇₉	—	2	1	no	—
50	C₄×Q₈×C₇₉	C₂×C₄	1	2	yes	—
51	C₄×Q₈×C₇₉	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₇₉	—	2	1	no	—
55	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₇₉	—	2	1	no	—
57	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₇₉	—	2	1	no	—
59	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
60	C₂²×Dic₇₉	—	2	2	no	—
61	C₂²×Dic₇₉	C₂²⋊C₄	1	2	yes	—
62	C₂²×Dic₇₉	C₂²⋊C₄	1	2	yes	—
63	C₂×C₇₉⋊C₈	—	2	2	no	—
64	C₂×C₇₉⋊C₈	—	2	2	no	—
65	C₂×C₇₉⋊C₈	—	4	2	no	—
66	C₂×C₇₉⋊C₈	—	4	1	no	—
67	C₂×C₇₉⋊C₈	—	4	1	no	—
68	C₄⋊Dic₇₉	—	2	8	no	—
69	C₄⋊Dic₇₉	—	2	2	no	—
70	C₄⋊Dic₇₉	M₁₆	1	4	yes	—
71	C₄×Dic₇₉	—	2	8	no	—
72	C₄×Dic₇₉	—	2	2	no	—
73	C₄×Dic₇₉	—	2	2	no	—
74	C₄×Dic₇₉	—	2	1	no	—
75	C₄×Dic₇₉	—	2	1	no	—
76	C₄×Dic₇₉	—	4	2	no	—
77	C₄×Dic₇₉	—	4	4	no	—
78	C₄×Dic₇₉	—	4	1	no	—
79	C₄×Dic₇₉	—	4	1	no	—
80	C₄×Dic₇₉	SD₁₆	1	2	yes	—
81	C₄×Dic₇₉	Q₁₆	1	2	yes	—
82	C₂×Q₈×C₇₉	—	2	1	no	—
83	C₂×Q₈×C₇₉	—	2	2	no	—
84	C₂×Q₈×C₇₉	C₄²	1	2	yes	—
85	C₂×Q₈×C₇₉	M₁₆	1	4	yes	—
86	C₂²×C₄×C₇₉	—	2	1	no	—
87	C₂²×C₄×C₇₉	—	2	1	no	—
88	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₇₉	—	2	2	no	—
92	C₂×C₈×C₇₉	—	2	2	no	—
93	C₂×C₈×C₇₉	—	4	2	no	—
94	C₂×C₈×C₇₉	—	4	1	no	—
95	C₂×C₈×C₇₉	—	4	1	no	—
96	C₄⋊C₄×C₇₉	—	2	1	no	—
97	C₄⋊C₄×C₇₉	—	2	2	no	—
98	C₄⋊C₄×C₇₉	—	2	2	no	—
99	C₄⋊C₄×C₇₉	C₄²	1	2	yes	—
100	C₄²×C₇₉	—	2	2	no	—
101	C₄²×C₇₉	—	2	2	no	—
102	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₇₉	#41	1	1	yes	characteristic
105	C₄²×C₇₉	#41	1	1	yes	characteristic
106	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	Q₈×C₇₉	—	2	2	no	—
108	Q₈×C₇₉	—	2	4	no	—
109	Q₈×C₇₉	#97	1	4	yes	complemented
110	C₂×C₄×C₇₉	—	2	2	no	—
111	C₂×C₄×C₇₉	—	2	4	no	—
112	C₂×C₄×C₇₉	—	2	2	no	—
113	C₂×C₄×C₇₉	—	2	2	no	—
114	C₂×C₄×C₇₉	—	4	2	no	—
115	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
122	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
125	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
126	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
127	C₂×Dic₇₉	—	2	2	no	—
128	C₂×Dic₇₉	—	2	2	no	—
129	C₂×Dic₇₉	—	2	2	no	—
130	C₂×Dic₇₉	—	2	4	no	—
131	C₂×Dic₇₉	—	2	2	no	—
132	C₂×Dic₇₉	—	2	2	no	—
133	C₂×Dic₇₉	—	4	2	no	—
134	C₂×Dic₇₉	—	4	2	no	—
135	C₂×Dic₇₉	#103	1	4	yes	—
136	C₈×C₇₉	—	4	2	no	—
137	C₈×C₇₉	—	4	2	no	—
138	C₇₉⋊C₈	—	4	2	no	—
139	C₇₉⋊C₈	—	4	2	no	—
140	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₇₉	#407	1	2	yes	—
142	C₂²×C₇₉	C₂×#103	1	1	yes	characteristic
143	C₂²×C₇₉	#416	1	1	yes	characteristic
144	C₂²×C₇₉	#421	1	2	yes	—
145	C₄×C₇₉	—	2	4	no	—
146	C₄×C₇₉	—	2	1	no	—
147	C₄×C₇₉	—	2	1	no	—
148	C₄×C₇₉	—	4	2	no	—
149	C₄×C₇₉	C₂×#97	1	1	yes	characteristic
150	C₄×C₇₉	C₄×M₁₆	1	1	yes	characteristic
151	C₄×C₇₉	#432	1	2	yes	—
152	C₄×C₇₉	C₄×SD₁₆	1	1	yes	characteristic
153	C₄×C₇₉	C₄×SD₁₆	1	1	yes	characteristic
154	C₄×C₇₉	C₄×Q₁₆	1	1	yes	characteristic
155	C₄×C₇₉	C₄×Q₁₆	1	1	yes	characteristic
156	Dic₇₉	—	2	4	no	—
157	Dic₇₉	—	4	2	no	—
158	Dic₇₉	—	4	4	no	—
159	#8175	—	79	1	no	maximal, Sylow
160	C₂×C₇₉	#1487	1	2	yes	—
161	C₂×C₇₉	#1488	1	2	yes	—
162	C₂×C₇₉	#1749	1	1	yes	characteristic
163	C₂×C₇₉	#1759	1	1	yes	characteristic
164	C₂×C₇₉	C₄×#103	1	1	yes	characteristic
165	C₄²×Q₈	—	79	1	no	—
166	#1767	—	79	1	no	—
167	#1744	—	79	1	no	—
168	C₂×Q₈⋊C₈	—	79	1	no	—
169	C₂×Q₈⋊C₈	D₇₉	1	1	yes	characteristic
170	#1279	—	79	1	no	—
171	#1279	—	79	1	no	—
172	C₇₉	#8175	1	1	yes	characteristic, complemented, Sylow
173	C₂×C₄×Q₈	—	79	2	no	—
174	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
175	C₂×C₄⋊C₈	—	79	1	no	—
176	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
177	C₂×#97	—	158	1	no	—
178	C₂×C₄×C₈	—	79	1	no	—
179	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
180	C₄×C₄⋊C₄	—	79	2	no	—
181	C₄×C₄⋊C₄	—	79	1	no	—
182	C₄×C₄⋊C₄	—	158	1	no	—
183	C₄×C₄⋊C₄	—	158	2	no	—
184	C₄³	—	79	1	no	—
185	C₄³	—	158	1	no	—
186	#336	—	158	2	no	—
187	#336	—	158	1	no	—
188	Q₈⋊C₈	—	79	4	no	—
189	Q₈⋊C₈	Dic₇₉	1	4	yes	complemented
190	C₂²×Q₈	C₄×D₇₉	1	1	yes	characteristic
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	158	1	no	—
194	C₂²×C₈	—	158	1	no	—
195	C₄×Q₈	—	2	1	no	—
196	C₄×Q₈	—	2	1	no	—
197	C₄×Q₈	—	158	4	no	—
198	C₄×Q₈	—	158	4	no	—
199	C₄×Q₈	C₄×D₇₉	1	2	yes	—
200	C₄×Q₈	C₂×Dic₇₉	1	2	yes	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	79	2	no	—
203	C₂×C₄⋊C₄	—	158	2	no	—
204	C₂×C₄⋊C₄	C₄×D₇₉	1	1	yes	characteristic
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	—	79	2	no	—
207	C₂×C₄²	—	79	1	no	—
208	C₂×C₄²	—	79	2	no	—
209	C₂×C₄²	—	79	1	no	—
210	C₂×C₄²	—	158	2	no	—
211	C₂×C₄²	—	158	1	no	—
212	C₂×C₄²	—	158	2	no	—
213	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	79	2	no	—
216	C₄⋊C₈	—	158	1	no	—
217	C₄⋊C₈	C₂×Dic₇₉	1	2	yes	—
218	#97	—	316	2	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	79	2	no	—
221	C₄×C₈	—	158	1	no	—
222	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
223	#95	—	158	2	no	—
224	#95	—	158	1	no	—
225	C₂×Q₈	—	2	1	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	C₄×Dic₇₉	1	2	yes	—
228	C₂×Q₈	#11929607	1	4	yes	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	79	2	no	—
232	C₂²×C₄	—	79	2	no	—
233	C₂²×C₄	—	158	2	no	—
234	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
235	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
236	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	316	2	no	—
243	C₂×C₈	—	316	1	no	—
244	C₂×C₈	—	316	1	no	—
245	C₂×C₈	—	158	2	no	—
246	C₂×C₈	—	158	2	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	79	4	no	—
251	C₄⋊C₄	—	158	8	no	—
252	C₄⋊C₄	—	158	2	no	—
253	C₄⋊C₄	C₄×Dic₇₉	1	2	yes	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	316	2	no	—
257	C₄²	—	316	4	no	—
258	C₄²	—	316	1	no	—
259	C₄²	—	316	1	no	—
260	C₄²	—	79	2	no	—
261	C₄²	—	79	2	no	—
262	C₄²	—	158	8	no	—
263	C₄²	—	158	2	no	—
264	C₄²	—	158	2	no	—
265	C₄²	—	158	1	no	—
266	C₄²	—	158	1	no	—
267	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
268	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
269	C₄²	#11929623	1	1	yes	characteristic
270	C₄²	#11929623	1	1	yes	characteristic
271	C₂³	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
272	Q₈	—	2	2	no	—
273	Q₈	—	2	4	no	—
274	Q₈	#14579071	1	4	yes	complemented
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	316	2	no	—
281	C₂×C₄	—	316	2	no	—
282	C₂×C₄	—	79	4	no	—
283	C₂×C₄	—	158	4	no	—
284	C₂×C₄	—	158	2	no	—
285	C₂×C₄	—	158	2	no	—
286	C₂×C₄	—	158	2	no	—
287	C₂×C₄	—	158	2	no	—
288	C₂×C₄	—	158	2	no	—
289	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
290	C₂×C₄	#14579117	1	1	yes	characteristic
291	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
292	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
293	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
294	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
295	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
296	C₂×C₄	C₂×#11929607	1	2	yes	—
297	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
298	C₂×C₄	Q₁₆×D₇₉	1	1	yes	characteristic
299	C₂×C₄	#14579182	1	1	yes	characteristic
300	C₂×C₄	#14579184	1	1	yes	characteristic
301	C₈	—	4	2	no	—
302	C₈	—	4	2	no	—
303	C₈	—	316	2	no	—
304	C₈	—	316	2	no	—
305	C₂²	C₂²⋊C₄×Dic₇₉	1	1	yes	characteristic
306	C₂²	#19098525	1	2	yes	—
307	C₂²	#103×D₇₉	1	1	yes	characteristic
308	C₂²	#19098558	1	1	yes	characteristic
309	C₂²	#19098578	1	2	yes	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	4	no	—
313	C₄	—	4	2	no	—
314	C₄	—	316	4	no	—
315	C₄	—	316	2	no	—
316	C₄	—	158	4	no	—
317	C₄	C₂×#14579071	1	1	yes	characteristic
318	C₄	M₁₆×Dic₇₉	1	1	yes	characteristic
319	C₄	#19098617	1	2	yes	—
320	C₄	#19098648	1	1	yes	characteristic
321	C₄	SD₁₆×Dic₇₉	1	1	yes	characteristic
322	C₄	#19098655	1	1	yes	characteristic
323	C₄	Q₁₆×Dic₇₉	1	1	yes	characteristic
324	C₂	#23761827	1	2	yes	—
325	C₂	#23761833	1	2	yes	—
326	C₂	#23763164	1	1	yes	characteristic
327	C₂	#23763212	1	1	yes	characteristic
328	C₂	#103×Dic₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×Q₈⋊C₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×Q₈⋊C₈	D₇₉
hypocenter	C₇₉	#8175
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
20. Q₈⋊(C₇₉⋊C₈)	C₄	semidirect
24. Q₈⋊C₈×C₇₉	C₄	semidirect
109. Q₈×C₇₉	#97	semidirect
172. C₇₉	#8175	semidirect
189. Q₈⋊C₈	Dic₇₉	semidirect
274. Q₈	#14579071	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×Dic₇₉	2
#23763255	2
#23763145	2
C₂×Q₈⋊(C₇₉⋊C₈)	2
#23761240	2
#23761239	2
C₂×Q₈⋊C₈×C₇₉	2
#8175	79