Subgroups — #34308021

Order 20224.

782 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic
2	#23766355	C₂	1	2	yes	maximal
3	#23764261	C₂	1	2	yes	maximal
4	#23764042	C₂	1	4	yes	maximal
5	#23763560	C₂	1	1	yes	maximal, characteristic
6	#23763406	C₂	1	2	yes	maximal
7	C₄×#14579095	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₅₈	C₂²	1	2	yes	—
11	C₄×Dic₃₁₆	C₄	1	8	yes	complemented
12	C₂×#14579095	C₂²	1	2	yes	—
13	C₂×#14579095	C₂²	1	2	yes	—
14	C₂×#14579095	C₂²	1	1	yes	characteristic
15	C₂×#14579086	C₂²	1	4	yes	—
16	C₂×#14579086	C₂²	1	2	yes	—
17	#19098469	C₂²	1	4	yes	—
18	#19098450	—	2	4	no	—
19	#19098433	C₂²	1	2	yes	—
20	#19098428	—	2	2	no	—
21	C₄⋊C₄×Dic₇₉	—	2	2	no	—
22	C₄×C₄⋊Dic₇₉	C₂²	1	4	yes	—
23	#19098326	C₂²	1	4	yes	—
24	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
28	#336×C₇₉	C₂²	1	2	yes	—
29	C₂²×Dic₁₅₈	C₂³	1	2	yes	—
30	C₂×Dic₃₁₆	—	2	2	no	—
31	C₂×Dic₃₁₆	—	2	2	no	—
32	C₂×Dic₃₁₆	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₅₈	—	2	4	no	—
34	C₄×Dic₁₅₈	C₂×C₄	1	8	yes	—
35	C₂×C₄⋊Dic₇₉	—	2	2	no	—
36	C₂×C₄⋊Dic₇₉	—	2	2	no	—
37	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
39	C₂×Dic₇₉⋊C₄	—	2	2	no	—
40	C₂×C₄×Dic₇₉	—	2	2	no	—
41	C₂×C₄×Dic₇₉	—	2	4	no	—
42	#14579095	—	2	4	no	—
43	#14579095	—	4	2	no	—
44	#14579095	C₂×C₄	1	4	yes	—
45	#14579086	—	4	4	no	—
46	#14579086	C₂×C₄	1	8	yes	—
47	C₇₉⋊#95	—	2	4	no	—
48	C₇₉⋊#95	—	2	4	no	—
49	C₇₉⋊#95	—	2	4	no	—
50	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
51	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
56	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₇₉	—	2	2	no	—
58	C₄×C₈×C₇₉	—	2	4	no	—
59	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
60	#95×C₇₉	D₄	1	2	yes	—
61	C₂×Dic₁₅₈	—	2	2	no	—
62	C₂×Dic₁₅₈	—	2	2	no	—
63	C₂×Dic₁₅₈	—	2	2	no	—
64	C₂×Dic₁₅₈	—	2	2	no	—
65	C₂×Dic₁₅₈	—	2	2	no	—
66	C₂×Dic₁₅₈	C₂²×C₄	1	4	yes	—
67	C₂²×Dic₇₉	—	2	2	no	—
68	C₂²×Dic₇₉	—	2	2	no	—
69	C₂²×Dic₇₉	—	2	4	no	—
70	Dic₃₁₆	—	2	8	no	—
71	C₄⋊Dic₇₉	—	2	2	no	—
72	C₄⋊Dic₇₉	—	2	4	no	—
73	C₄⋊Dic₇₉	—	2	4	no	—
74	C₄⋊Dic₇₉	—	4	4	no	—
75	C₄⋊Dic₇₉	—	8	2	no	—
76	C₄⋊Dic₇₉	C₂²×C₄	1	4	yes	—
77	Dic₇₉⋊C₄	—	2	8	no	—
78	C₄×Dic₇₉	—	2	8	no	—
79	C₄×Dic₇₉	—	8	4	no	—
80	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₇₉	—	2	2	no	—
86	C₂×C₈×C₇₉	—	2	2	no	—
87	C₂×C₈×C₇₉	—	2	1	no	—
88	C₂×C₈×C₇₉	—	2	2	no	—
89	C₂×C₈×C₇₉	—	2	1	no	—
90	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₇₉	#41	1	2	yes	—
93	C₂×C₈×C₇₉	#41	1	2	yes	—
94	C₄⋊C₄×C₇₉	—	2	2	no	—
95	C₄⋊C₄×C₇₉	Q₁₆	1	4	yes	complemented
96	C₄²×C₇₉	—	2	2	no	—
97	C₄²×C₇₉	—	2	2	no	—
98	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
99	C₄²×C₇₉	#41	1	2	yes	—
100	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₇₉	—	2	2	no	—
102	C₂×C₄×C₇₉	—	2	2	no	—
103	C₂×C₄×C₇₉	—	2	1	no	—
104	C₂×C₄×C₇₉	—	2	1	no	—
105	C₂×C₄×C₇₉	—	2	2	no	—
106	C₂×C₄×C₇₉	—	2	2	no	—
107	C₂×C₄×C₇₉	—	2	2	no	—
108	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₇₉	C₂×Q₁₆	1	4	yes	—
111	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
112	C₂×C₄×C₇₉	#135	1	2	yes	—
113	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
119	C₂×Dic₇₉	—	2	4	no	—
120	C₂×Dic₇₉	—	2	4	no	—
121	C₂×Dic₇₉	—	4	2	no	—
122	C₂×Dic₇₉	—	4	2	no	—
123	C₂×Dic₇₉	—	4	2	no	—
124	C₂×Dic₇₉	—	4	2	no	—
125	C₂×Dic₇₉	—	4	4	no	—
126	C₂×Dic₇₉	—	4	4	no	—
127	C₂×Dic₇₉	—	4	4	no	—
128	Dic₁₅₈	—	2	4	no	—
129	Dic₁₅₈	—	2	4	no	—
130	Dic₁₅₈	—	4	2	no	—
131	Dic₁₅₈	—	4	2	no	—
132	C₈×C₇₉	—	2	2	no	—
133	C₈×C₇₉	—	4	2	no	—
134	C₈×C₇₉	#117	1	4	yes	—
135	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₇₉	#547	1	1	yes	characteristic
137	C₂²×C₇₉	#550	1	1	yes	characteristic
138	C₂²×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
139	C₂²×C₇₉	C₂×#135	1	1	yes	characteristic
140	C₂²×C₇₉	#576	1	1	yes	characteristic
141	C₂²×C₇₉	#578	1	1	yes	characteristic
142	C₄×C₇₉	—	2	4	no	—
143	C₄×C₇₉	—	2	2	no	—
144	C₄×C₇₉	—	4	2	no	—
145	C₄×C₇₉	C₄×Q₁₆	1	4	yes	—
146	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
147	C₄×C₇₉	#520	1	1	yes	characteristic
148	C₄×C₇₉	#555	1	1	yes	characteristic
149	C₄×C₇₉	#556	1	1	yes	characteristic
150	Dic₇₉	—	4	4	no	—
151	Dic₇₉	—	4	4	no	—
152	Dic₇₉	—	8	4	no	—
153	#22808	—	79	1	no	maximal, Sylow
154	C₂×C₇₉	#2354	1	1	yes	characteristic
155	C₂×C₇₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
156	C₂×C₇₉	#2951	1	1	yes	characteristic
157	C₂×C₇₉	#3338	1	1	yes	characteristic
158	C₂×C₇₉	#3342	1	1	yes	characteristic
159	C₂×C₇₉	#3395	1	1	yes	characteristic
160	C₂×C₇₉	#3396	1	1	yes	characteristic
161	C₂×C₄×Q₁₆	—	79	1	no	—
162	#2355	—	79	2	no	—
163	#1969	—	79	2	no	—
164	#1928	—	79	4	no	—
165	#1845	—	79	1	no	—
166	#1803	—	79	2	no	—
167	C₄×#107	—	79	2	no	—
168	C₈×C₄⋊C₄	D₇₉	1	1	yes	characteristic
169	C₇₉	#22808	1	1	yes	characteristic, complemented, Sylow
170	C₂²×Q₁₆	—	79	1	no	—
171	C₂×C₄×Q₈	—	79	2	no	—
172	C₄×Q₁₆	—	79	8	no	—
173	C₂×#107	—	79	2	no	—
174	C₂×#107	—	79	1	no	—
175	C₂×#107	—	79	2	no	—
176	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
177	C₂×#103	—	79	4	no	—
178	C₂×#103	—	79	2	no	—
179	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
180	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
181	#391	—	79	4	no	—
182	#387	—	158	4	no	—
183	#384	—	79	2	no	—
184	#382	—	158	2	no	—
185	C₄×C₄⋊C₄	—	79	4	no	—
186	C₄×C₄⋊C₄	—	158	2	no	—
187	C₄×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
188	#340	—	79	4	no	—
189	#336	C₂×D₇₉	1	2	yes	—
190	C₂²×Q₈	—	79	2	no	—
191	C₂×Q₁₆	—	79	4	no	—
192	C₂×Q₁₆	—	158	2	no	—
193	C₂×Q₁₆	—	158	2	no	—
194	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
195	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
197	C₄×Q₈	—	79	8	no	—
198	C₄×Q₈	—	158	4	no	—
199	C₂×C₄⋊C₄	—	79	2	no	—
200	C₂×C₄⋊C₄	—	79	4	no	—
201	C₂×C₄⋊C₄	—	158	2	no	—
202	C₂×C₄⋊C₄	—	158	2	no	—
203	C₂×C₄⋊C₄	—	158	2	no	—
204	C₂×C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
205	C₂×C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
206	C₂×C₄²	—	158	4	no	—
207	C₂×C₄²	—	158	2	no	—
208	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
209	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
210	#107	—	316	2	no	—
211	#107	—	79	4	no	—
212	#107	—	158	4	no	—
213	C₄⋊C₈	—	2	2	no	—
214	#103	—	316	4	no	—
215	#103	—	79	8	no	—
216	C₄×C₈	—	2	4	no	—
217	C₄×C₈	C₄×D₇₉	1	4	yes	—
218	#95	—	158	4	no	—
219	#95	—	158	4	no	—
220	#95	—	158	4	no	—
221	#95	D₃₁₆	1	2	yes	—
222	C₂×Q₈	—	79	4	no	—
223	C₂×Q₈	—	158	2	no	—
224	C₂×Q₈	—	158	2	no	—
225	C₂×Q₈	—	158	2	no	—
226	C₂×Q₈	—	158	2	no	—
227	C₂×Q₈	—	158	2	no	—
228	C₂²×C₄	—	158	4	no	—
229	C₂²×C₄	—	158	2	no	—
230	C₂²×C₄	—	158	2	no	—
231	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
232	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
234	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
235	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
236	Q₁₆	—	158	8	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
243	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
244	C₂×C₈	#11929623	1	2	yes	—
245	C₂×C₈	#11929625	1	2	yes	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	316	4	no	—
248	C₄⋊C₄	—	79	4	no	—
249	C₄⋊C₄	—	632	2	no	—
250	C₄⋊C₄	—	158	4	no	—
251	C₄⋊C₄	—	158	4	no	—
252	C₄⋊C₄	—	158	2	no	—
253	C₄⋊C₄	—	158	8	no	—
254	C₄⋊C₄	Dic₃₁₆	1	4	yes	complemented
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	632	4	no	—
258	C₄²	—	158	8	no	—
259	C₄²	C₂×C₄×D₇₉	1	2	yes	—
260	C₄²	#11929624	1	2	yes	—
261	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
262	Q₈	—	316	2	no	—
263	Q₈	—	316	2	no	—
264	Q₈	—	158	4	no	—
265	Q₈	—	158	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	316	2	no	—
274	C₂×C₄	—	316	2	no	—
275	C₂×C₄	—	316	2	no	—
276	C₂×C₄	—	316	2	no	—
277	C₂×C₄	—	316	4	no	—
278	C₂×C₄	—	316	4	no	—
279	C₂×C₄	—	316	4	no	—
280	C₂×C₄	—	158	4	no	—
281	C₂×C₄	—	158	4	no	—
282	C₂×C₄	C₄×D₃₁₆	1	2	yes	—
283	C₂×C₄	C₄×D₃₁₆	1	2	yes	—
284	C₂×C₄	C₂×Dic₃₁₆	1	4	yes	—
285	C₂×C₄	C₂×Dic₃₁₆	1	2	yes	—
286	C₂×C₄	#14579183	1	2	yes	—
287	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
288	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
289	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
290	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
291	C₂×C₄	#14579207	1	1	yes	characteristic
292	C₂×C₄	#14579208	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	4	2	no	—
295	C₈	#14579119	1	4	yes	—
296	C₂²	C₂×C₄×D₃₁₆	1	1	yes	characteristic
297	C₂²	#19099217	1	1	yes	characteristic
298	C₂²	#19099235	1	1	yes	characteristic
299	C₂²	C₂²×Dic₃₁₆	1	1	yes	characteristic
300	C₂²	C₂×#14579183	1	1	yes	characteristic
301	C₂²	#19099368	1	1	yes	characteristic
302	C₂²	#19099380	1	1	yes	characteristic
303	C₄	—	2	4	no	—
304	C₄	—	2	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	316	4	no	—
307	C₄	—	316	4	no	—
308	C₄	—	632	4	no	—
309	C₄	C₄×Dic₃₁₆	1	4	yes	—
310	C₄	C₂×#14579119	1	1	yes	characteristic
311	C₄	#19099043	1	1	yes	characteristic
312	C₄	#19099278	1	1	yes	characteristic
313	C₄	#19099286	1	1	yes	characteristic
314	C₂	#23766347	1	1	yes	characteristic
315	C₂	C₂×C₄×Dic₃₁₆	1	1	yes	characteristic
316	C₂	#23772277	1	1	yes	characteristic
317	C₂	#23775793	1	1	yes	characteristic
318	C₂	#23775831	1	1	yes	characteristic
319	C₂	#23776406	1	1	yes	characteristic
320	C₂	#23776413	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₈×C₄⋊C₄×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₇₉
hypocenter	C₇₉	#22808
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
11. C₄×Dic₃₁₆	C₄	semidirect
95. C₄⋊C₄×C₇₉	Q₁₆	semidirect
169. C₇₉	#22808	semidirect
254. C₄⋊C₄	Dic₃₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₃₁₆	2
#23766355	2
#23764261	2
#23764042	2
#23763560	2
#23763406	2
C₄×#14579095	2
C₈×C₄⋊C₄×C₇₉	2
#22808	79