Subgroups — #34188277

Order 20224.

566 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763552	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#23761241	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
6	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₇₉⋊C₈	—	2	1	no	—
9	C₂×C₄×C₇₉⋊C₈	—	2	1	no	—
10	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₇₉	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₇₉	—	2	1	no	—
13	#391×C₇₉	—	2	1	no	—
14	#389×C₇₉	—	2	1	no	—
15	#389×C₇₉	C₄	1	2	yes	—
16	#384×C₇₉	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₇₉	—	2	1	no	—
18	C₄×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
19	C₄³×C₇₉	C₂²	1	1	yes	characteristic
20	C₂²×C₇₉⋊C₈	—	2	2	no	—
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
23	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
24	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
25	C₄×C₇₉⋊C₈	—	2	4	no	—
26	C₄×C₇₉⋊C₈	—	2	4	no	—
27	C₄×C₇₉⋊C₈	—	2	2	no	—
28	C₄×C₇₉⋊C₈	Q₈	1	4	yes	complemented
29	C₂²×Q₈×C₇₉	—	2	1	no	—
30	C₄⋊Q₈×C₇₉	—	2	2	no	—
31	C₄⋊Q₈×C₇₉	C₈	1	4	yes	complemented
32	C₄×Q₈×C₇₉	—	4	2	no	—
33	C₄×Q₈×C₇₉	—	4	2	no	—
34	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₇₉	—	2	2	no	—
41	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
48	#95×C₇₉	—	2	2	no	—
49	#95×C₇₉	—	2	2	no	—
50	C₂×C₇₉⋊C₈	—	2	4	no	—
51	C₂×C₇₉⋊C₈	—	2	4	no	—
52	C₂×C₇₉⋊C₈	—	2	4	no	—
53	C₂×C₇₉⋊C₈	—	4	2	no	—
54	C₂×C₇₉⋊C₈	—	4	2	no	—
55	C₂×C₇₉⋊C₈	—	4	2	no	—
56	C₂×Q₈×C₇₉	—	2	4	no	—
57	C₂×Q₈×C₇₉	—	4	2	no	—
58	C₂×Q₈×C₇₉	—	4	2	no	—
59	C₂²×C₄×C₇₉	—	2	2	no	—
60	C₂²×C₄×C₇₉	—	2	2	no	—
61	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₇₉	—	2	4	no	—
69	C₄⋊C₄×C₇₉	—	2	2	no	—
70	C₄⋊C₄×C₇₉	—	2	4	no	—
71	C₄⋊C₄×C₇₉	—	4	4	no	—
72	C₄⋊C₄×C₇₉	—	4	2	no	—
73	C₄⋊C₄×C₇₉	—	4	2	no	—
74	C₄⋊C₄×C₇₉	C₂×C₈	1	4	yes	—
75	C₄²×C₇₉	—	2	2	no	—
76	C₄²×C₇₉	—	2	2	no	—
77	C₄²×C₇₉	—	2	2	no	—
78	C₄²×C₇₉	—	4	2	no	—
79	C₄²×C₇₉	—	4	2	no	—
80	C₄²×C₇₉	C₂×C₈	1	2	yes	—
81	C₄²×C₇₉	M₁₆	1	2	yes	—
82	C₄²×C₇₉	D₈	1	2	yes	—
83	C₄²×C₇₉	SD₁₆	1	2	yes	—
84	C₄²×C₇₉	SD₁₆	1	2	yes	—
85	C₄²×C₇₉	Q₁₆	1	2	yes	—
86	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
87	C₄²×C₇₉	#41	1	2	yes	—
88	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	Q₈×C₇₉	—	4	8	no	—
90	C₂×C₄×C₇₉	—	2	1	no	—
91	C₂×C₄×C₇₉	—	2	1	no	—
92	C₂×C₄×C₇₉	—	2	1	no	—
93	C₂×C₄×C₇₉	—	2	1	no	—
94	C₂×C₄×C₇₉	—	2	1	no	—
95	C₂×C₄×C₇₉	—	2	1	no	—
96	C₂×C₄×C₇₉	—	2	4	no	—
97	C₂×C₄×C₇₉	—	4	2	no	—
98	C₂×C₄×C₇₉	—	4	2	no	—
99	C₂×C₄×C₇₉	—	4	2	no	—
100	C₂×C₄×C₇₉	—	4	2	no	—
101	C₂×C₄×C₇₉	—	4	2	no	—
102	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₇₉	#102	1	2	yes	—
105	C₂×C₄×C₇₉	#102	1	2	yes	—
106	C₂×C₄×C₇₉	#103	1	2	yes	—
107	C₂×C₄×C₇₉	#103	1	2	yes	—
108	C₂×C₄×C₇₉	#104	1	2	yes	—
109	C₂×C₄×C₇₉	#104	1	2	yes	—
110	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₇₉	#131	1	2	yes	—
117	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
121	C₇₉⋊C₈	—	2	8	no	—
122	C₇₉⋊C₈	—	8	2	no	—
123	C₂²×C₇₉	#391	1	1	yes	characteristic
124	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
125	C₂²×C₇₉	#408	1	1	yes	characteristic
126	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
127	C₂²×C₇₉	C₂×#103	1	1	yes	characteristic
128	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
129	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
130	C₄×C₇₉	—	2	2	no	—
131	C₄×C₇₉	—	2	2	no	—
132	C₄×C₇₉	—	2	2	no	—
133	C₄×C₇₉	—	4	4	no	—
134	C₄×C₇₉	—	8	2	no	—
135	C₄×C₇₉	D₄⋊C₈	1	4	yes	—
136	C₄×C₇₉	Q₈⋊C₈	1	4	yes	—
137	C₄×C₇₉	C₈×Q₈	1	2	yes	—
138	C₄×C₇₉	#446	1	2	yes	—
139	C₄×C₇₉	#486	1	1	yes	characteristic
140	C₄×C₇₉	#487	1	1	yes	characteristic
141	C₄×C₇₉	#498	1	1	yes	characteristic
142	C₄×C₇₉	#500	1	1	yes	characteristic
143	#9870	—	79	1	no	maximal, Sylow
144	C₂×C₇₉	C₂×D₄⋊C₈	1	1	yes	characteristic
145	C₂×C₇₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
146	C₂×C₇₉	#1492	1	1	yes	characteristic
147	C₂×C₇₉	#1493	1	1	yes	characteristic
148	C₂×C₇₉	#1984	1	1	yes	characteristic
149	C₂×C₇₉	#1990	1	1	yes	characteristic
150	C₂×C₇₉	#1994	1	1	yes	characteristic
151	C₄×C₄⋊Q₈	D₇₉	1	1	yes	characteristic
152	#1842	—	79	1	no	—
153	C₄²×C₈	—	79	1	no	—
154	#1279	—	79	4	no	—
155	C₇₉	#9870	1	1	yes	characteristic, complemented, Sylow
156	C₂×C₄⋊Q₈	Dic₇₉	1	1	yes	characteristic
157	C₂×C₄×Q₈	—	2	1	no	—
158	C₂×C₄⋊C₈	—	79	2	no	—
159	C₂×C₄⋊C₈	—	158	1	no	—
160	C₂×C₄⋊C₈	—	158	1	no	—
161	C₂×C₄×C₈	—	79	2	no	—
162	C₂×C₄×C₈	—	158	1	no	—
163	C₂×C₄×C₈	—	158	1	no	—
164	#391	—	2	1	no	—
165	#389	—	2	1	no	—
166	#389	Dic₇₉	1	2	yes	—
167	#384	Dic₇₉	1	1	yes	characteristic
168	C₄×C₄⋊C₄	—	2	1	no	—
169	C₄×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
170	C₄³	C₂×D₇₉	1	1	yes	characteristic
171	C₂²×Q₈	—	2	1	no	—
172	C₂²×C₈	—	158	2	no	—
173	C₂²×C₈	—	158	2	no	—
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	C₇₉⋊C₈	1	4	yes	complemented
176	C₄×Q₈	—	4	2	no	—
177	C₄×Q₈	—	4	2	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
183	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
186	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
188	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
191	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
192	C₄⋊C₈	—	316	2	no	—
193	C₄⋊C₈	—	316	2	no	—
194	C₄⋊C₈	—	158	4	no	—
195	C₄×C₈	—	79	4	no	—
196	C₄×C₈	—	158	4	no	—
197	C₄×C₈	—	158	4	no	—
198	C₄×C₈	—	158	2	no	—
199	#95	—	2	2	no	—
200	#95	—	2	2	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	—	4	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
210	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
213	C₂×C₈	—	316	2	no	—
214	C₂×C₈	—	316	2	no	—
215	C₂×C₈	—	316	2	no	—
216	C₂×C₈	—	158	4	no	—
217	C₂×C₈	—	158	4	no	—
218	C₂×C₈	—	158	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	C₂×C₇₉⋊C₈	1	4	yes	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	4	2	no	—
230	C₄²	—	4	2	no	—
231	C₄²	C₂×C₇₉⋊C₈	1	2	yes	—
232	C₄²	#11929608	1	2	yes	—
233	C₄²	#11929609	1	2	yes	—
234	C₄²	#11929611	1	2	yes	—
235	C₄²	#11929612	1	2	yes	—
236	C₄²	#11929614	1	2	yes	—
237	C₄²	Q₈×D₇₉	1	2	yes	—
238	C₄²	#11929625	1	2	yes	—
239	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
240	Q₈	—	4	8	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
254	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
255	C₂×C₄	#14579082	1	2	yes	—
256	C₂×C₄	#14579082	1	2	yes	—
257	C₂×C₄	#14579085	1	2	yes	—
258	C₂×C₄	#14579085	1	2	yes	—
259	C₂×C₄	#14579088	1	2	yes	—
260	C₂×C₄	#14579088	1	2	yes	—
261	C₂×C₄	Q₈×Dic₇₉	1	2	yes	—
262	C₂×C₄	#14579137	1	2	yes	—
263	C₂×C₄	#14579150	1	1	yes	characteristic
264	C₂×C₄	#14579159	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
266	C₂×C₄	C₂×#11929608	1	1	yes	characteristic
267	C₂×C₄	#14579168	1	2	yes	—
268	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
269	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
270	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
271	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
272	C₈	—	632	2	no	—
273	C₈	—	158	8	no	—
274	C₂²	#19098470	1	1	yes	characteristic
275	C₂²	C₂×C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
276	C₂²	#19098530	1	1	yes	characteristic
277	C₂²	C₂×#14579082	1	1	yes	characteristic
278	C₂²	C₂×#14579085	1	1	yes	characteristic
279	C₂²	C₂×#14579088	1	1	yes	characteristic
280	C₂²	C₂×#14579088	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	4	4	no	—
285	C₄	—	8	2	no	—
286	C₄	D₄⋊(C₇₉⋊C₈)	1	4	yes	—
287	C₄	Q₈⋊(C₇₉⋊C₈)	1	4	yes	—
288	C₄	Q₈×C₇₉⋊C₈	1	2	yes	—
289	C₄	#19098694	1	2	yes	—
290	C₄	#19098911	1	1	yes	characteristic
291	C₄	#19098915	1	1	yes	characteristic
292	C₄	#19098959	1	1	yes	characteristic
293	C₄	#19098966	1	1	yes	characteristic
294	C₂	C₂×D₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
295	C₂	C₂×Q₈⋊(C₇₉⋊C₈)	1	1	yes	characteristic
296	C₂	#23761851	1	1	yes	characteristic
297	C₂	#23761855	1	1	yes	characteristic
298	C₂	#23764337	1	1	yes	characteristic
299	C₂	#23764362	1	1	yes	characteristic
300	C₂	#23764378	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₄×C₄⋊Q₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×C₄⋊Q₈	D₇₉
hypocenter	C₇₉	#9870
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
28. C₄×C₇₉⋊C₈	Q₈	semidirect
31. C₄⋊Q₈×C₇₉	C₈	semidirect
155. C₇₉	#9870	semidirect
175. C₄⋊Q₈	C₇₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#23763552	2
C₄²×C₇₉⋊C₈	2
#23761241	2
C₄×C₄⋊Q₈×C₇₉	2
#9870	79