Subgroups — #35152679

Order 20224.

1810 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772349	C₂	1	16	yes	maximal, complemented
2	#23772341	C₂	1	4	yes	maximal, complemented
3	#23772335	C₂	1	4	yes	maximal, complemented
4	#23771964	C₂	1	1	yes	maximal, characteristic
5	C₂×#19098590	C₂	1	2	yes	maximal
6	#23771819	C₂	1	2	yes	maximal
7	C₂×#19098519	C₂	1	1	yes	maximal, characteristic
8	C₇₉×#3458	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11929608	C₂²	1	2	yes	—
10	C₂²×#11929608	C₂²	1	1	yes	characteristic
11	C₂³×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
12	#19098694	C₂²	1	16	yes	complemented
13	#19098694	C₂²	1	32	yes	complemented
14	Q₈×C₇₉⋊C₈	C₂²	1	16	yes	complemented
15	#19098590	C₂²	1	4	yes	—
16	#19098590	C₂²	1	4	yes	—
17	#19098590	C₂²	1	8	yes	—
18	#19098590	C₂²	1	8	yes	—
19	#19098587	C₂²	1	4	yes	—
20	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	8	yes	—
21	#19098519	C₂²	1	8	yes	—
22	#19098519	C₂²	1	8	yes	—
23	#19098519	C₂²	1	8	yes	—
24	C₄×#11929608	C₂²	1	8	yes	—
25	C₂×#14579072	C₂²	1	2	yes	—
26	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
27	#544×C₇₉	C₄	1	8	yes	—
28	#544×C₇₉	C₄	1	4	yes	—
29	#543×C₇₉	C₄	1	4	yes	—
30	#533×C₇₉	C₂²	1	4	yes	—
31	#533×C₇₉	C₂²	1	8	yes	—
32	C₂×#117×C₇₉	C₂²	1	2	yes	—
33	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
34	C₂×#11929608	—	2	8	no	—
35	C₂×#11929608	Q₈	1	8	yes	complemented
36	C₂×#11929608	C₂³	1	4	yes	—
37	C₂×#11929608	C₂³	1	4	yes	—
38	C₂×#11929608	C₂³	1	4	yes	—
39	C₂²×C₇₉⋊C₈	—	2	4	no	—
40	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₇₉⋊C₈	C₂³	1	4	yes	—
43	C₂²×C₇₉⋊C₈	C₂³	1	4	yes	—
44	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
45	C₄⋊(C₇₉⋊C₈)	C₂³	1	16	yes	—
46	C₄⋊(C₇₉⋊C₈)	C₂³	1	16	yes	—
47	C₄⋊(C₇₉⋊C₈)	C₂³	1	16	yes	—
48	#14579072	C₂³	1	8	yes	—
49	#14579072	C₂³	1	16	yes	—
50	C₄×C₇₉⋊C₈	C₂³	1	8	yes	—
51	C₄×C₇₉⋊C₈	C₂³	1	16	yes	—
52	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₇₉	C₂×C₄	1	8	yes	—
54	C₄⋊Q₈×C₇₉	C₂×C₄	1	16	yes	—
55	#125×C₇₉	C₂×C₄	1	8	yes	—
56	#125×C₇₉	C₂×C₄	1	16	yes	—
57	C₂²⋊Q₈×C₇₉	C₂×C₄	1	16	yes	—
58	C₂²⋊Q₈×C₇₉	C₂×C₄	1	32	yes	—
59	C₄×Q₈×C₇₉	C₂³	1	16	yes	—
60	#117×C₇₉	C₂×C₄	1	4	yes	—
61	#117×C₇₉	C₂×C₄	1	4	yes	—
62	#117×C₇₉	C₂×C₄	1	4	yes	—
63	#117×C₇₉	C₂³	1	4	yes	—
64	#117×C₇₉	C₂³	1	4	yes	—
65	#117×C₇₉	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
67	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
69	C₂×C₂²⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
70	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
71	C₂×C₄²×C₇₉	C₂³	1	4	yes	—
72	#11929608	—	2	8	no	—
73	#11929608	—	2	8	no	—
74	#11929608	C₂×Q₈	1	16	yes	—
75	C₂×C₇₉⋊C₈	—	2	8	no	—
76	C₂×C₇₉⋊C₈	—	2	2	no	—
77	C₂×C₇₉⋊C₈	—	2	2	no	—
78	C₂×C₇₉⋊C₈	—	2	8	no	—
79	C₂×C₇₉⋊C₈	C₂×Q₈	1	8	yes	—
80	C₂×C₇₉⋊C₈	C₂⁴	1	2	yes	—
81	C₂×C₇₉⋊C₈	C₂⁴	1	8	yes	—
82	C₂×C₇₉⋊C₈	C₂⁴	1	2	yes	—
83	C₂×C₇₉⋊C₈	C₂⁴	1	4	yes	—
84	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×Q₈×C₇₉	C₂²×C₄	1	16	yes	—
86	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
87	C₂²×C₄×C₇₉	C₂²×C₄	1	8	yes	—
88	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₉	C₂×Q₈	1	4	yes	—
91	C₂²×C₄×C₇₉	#41	1	4	yes	—
92	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
93	C₂²×C₄×C₇₉	C₂⁴	1	2	yes	—
94	C₄⋊C₄×C₇₉	C₂²×C₄	1	8	yes	—
95	C₄⋊C₄×C₇₉	C₂²×C₄	1	8	yes	—
96	C₄⋊C₄×C₇₉	C₂²×C₄	1	8	yes	—
97	C₄⋊C₄×C₇₉	C₂²×C₄	1	32	yes	—
98	C₄⋊C₄×C₇₉	C₂²×C₄	1	16	yes	—
99	C₂²⋊C₄×C₇₉	C₂²×C₄	1	8	yes	—
100	C₂²⋊C₄×C₇₉	C₂²×C₄	1	8	yes	—
101	C₂²⋊C₄×C₇₉	C₂²×C₄	1	8	yes	—
102	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
103	C₄²×C₇₉	C₂²×C₄	1	8	yes	—
104	C₄²×C₇₉	C₂⁴	1	4	yes	—
105	C₄²×C₇₉	C₂⁴	1	8	yes	—
106	C₂³×C₇₉	—	4	2	no	—
107	C₂³×C₇₉	C₄×Q₈	1	4	yes	—
108	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
109	C₂³×C₇₉	C₂³×C₄	1	2	yes	—
110	Q₈×C₇₉	—	4	16	no	—
111	C₂×C₄×C₇₉	—	4	4	no	—
112	C₂×C₄×C₇₉	—	4	8	no	—
113	C₂×C₄×C₇₉	C₄×Q₈	1	8	yes	—
114	C₂×C₄×C₇₉	C₄×Q₈	1	4	yes	—
115	C₂×C₄×C₇₉	C₂³×C₄	1	8	yes	—
116	C₂×C₄×C₇₉	C₂³×C₄	1	16	yes	—
117	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
119	C₂×C₄×C₇₉	C₂²×Q₈	1	2	yes	—
120	C₂×C₄×C₇₉	C₂²×Q₈	1	4	yes	—
121	C₂×C₄×C₇₉	C₂×#41	1	4	yes	—
122	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
123	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
124	C₇₉⋊C₈	—	2	2	no	—
125	C₇₉⋊C₈	—	2	2	no	—
126	C₇₉⋊C₈	—	2	8	no	—
127	C₇₉⋊C₈	C₂²×Q₈	1	8	yes	—
128	C₂²×C₇₉	—	2	2	no	—
129	C₂²×C₇₉	—	2	4	no	—
130	C₂²×C₇₉	—	4	4	no	—
131	C₂²×C₇₉	C₂×C₄×Q₈	1	2	yes	—
132	C₂²×C₇₉	C₂×C₄×Q₈	1	4	yes	—
133	C₂²×C₇₉	C₂⁴×C₄	1	1	yes	characteristic
134	C₄×C₇₉	—	2	8	no	—
135	C₄×C₇₉	—	2	16	no	—
136	C₄×C₇₉	C₂×C₄×Q₈	1	2	yes	—
137	C₄×C₇₉	C₂×C₄×Q₈	1	4	yes	—
138	C₄×C₇₉	C₂³×Q₈	1	1	yes	characteristic
139	C₄×C₇₉	C₂²×#41	1	1	yes	characteristic
140	#60193	—	79	1	no	maximal, Sylow
141	C₂×C₇₉	—	2	2	no	—
142	C₂×C₇₉	—	2	4	no	—
143	C₂×C₇₉	C₂²×C₄×Q₈	1	1	yes	characteristic
144	C₂×C₇₉	#3573	1	2	yes	—
145	#3458	D₇₉	1	1	yes	characteristic
146	#2962	—	79	16	no	—
147	#2961	—	79	4	no	—
148	#2960	—	79	4	no	—
149	#2905	—	79	1	no	—
150	C₂×#424	—	79	2	no	—
151	#2873	—	79	2	no	—
152	C₂×#405	—	79	1	no	—
153	C₇₉	#60193	1	1	yes	characteristic, complemented, Sylow
154	C₂²×M₁₆	—	79	1	no	—
155	C₂²×M₁₆	—	79	2	no	—
156	C₂³×C₈	—	79	1	no	—
157	#544	Dic₇₉	1	4	yes	—
158	#544	Dic₇₉	1	8	yes	—
159	#543	Dic₇₉	1	4	yes	—
160	#533	C₂×D₇₉	1	4	yes	—
161	#533	C₂×D₇₉	1	8	yes	—
162	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
163	C₂×#117	C₂×D₇₉	1	2	yes	—
164	#446	—	79	16	no	—
165	#446	—	79	32	no	—
166	C₈×Q₈	—	79	16	no	—
167	#424	—	79	4	no	—
168	#424	—	79	4	no	—
169	#424	—	79	8	no	—
170	#424	—	79	8	no	—
171	#423	—	79	4	no	—
172	C₂×C₄⋊C₈	—	79	8	no	—
173	#405	—	79	8	no	—
174	#405	—	79	8	no	—
175	#405	—	79	8	no	—
176	C₄×M₁₆	—	79	8	no	—
177	C₂×#97	—	79	2	no	—
178	C₂×C₄×C₈	—	79	2	no	—
179	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
180	C₂×M₁₆	—	79	8	no	—
181	C₂×M₁₆	—	79	4	no	—
182	C₂×M₁₆	—	79	4	no	—
183	C₂×M₁₆	—	79	4	no	—
184	C₂×M₁₆	—	158	8	no	—
185	C₂²×C₈	—	79	2	no	—
186	C₂²×C₈	—	79	1	no	—
187	C₂²×C₈	—	79	1	no	—
188	C₂²×C₈	—	79	4	no	—
189	C₂²×C₈	—	79	4	no	—
190	C₂²×C₈	—	158	4	no	—
191	C₄⋊Q₈	C₂×Dic₇₉	1	8	yes	—
192	C₄⋊Q₈	C₂×Dic₇₉	1	16	yes	—
193	#125	C₂×Dic₇₉	1	8	yes	—
194	#125	C₂×Dic₇₉	1	16	yes	—
195	C₂²⋊Q₈	C₂×Dic₇₉	1	16	yes	—
196	C₂²⋊Q₈	C₂×Dic₇₉	1	32	yes	—
197	C₄×Q₈	C₂²×D₇₉	1	16	yes	—
198	#117	C₂×Dic₇₉	1	4	yes	—
199	#117	C₂×Dic₇₉	1	4	yes	—
200	#117	C₂×Dic₇₉	1	4	yes	—
201	#117	C₂²×D₇₉	1	4	yes	—
202	#117	C₂²×D₇₉	1	4	yes	—
203	#117	C₂²×D₇₉	1	4	yes	—
204	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
205	C₂×C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
206	C₂×C₂²⋊C₄	C₂×Dic₇₉	1	2	yes	—
207	C₂×C₂²⋊C₄	C₂×Dic₇₉	1	4	yes	—
208	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
209	C₂×C₄²	C₂²×D₇₉	1	4	yes	—
210	C₄⋊C₈	—	79	16	no	—
211	C₄⋊C₈	—	79	16	no	—
212	C₄⋊C₈	—	79	16	no	—
213	#97	—	79	8	no	—
214	#97	—	79	16	no	—
215	C₄×C₈	—	79	8	no	—
216	C₄×C₈	—	79	16	no	—
217	C₂⁴	C₂²×Dic₇₉	1	1	yes	characteristic
218	C₂×Q₈	C₂²×Dic₇₉	1	16	yes	—
219	C₂²×C₄	C₂²×Dic₇₉	1	4	yes	—
220	C₂²×C₄	C₂²×Dic₇₉	1	8	yes	—
221	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
222	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
223	C₂²×C₄	Q₈×D₇₉	1	4	yes	—
224	C₂²×C₄	#11929625	1	4	yes	—
225	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
226	C₂²×C₄	C₂³×D₇₉	1	2	yes	—
227	M₁₆	—	79	16	no	—
228	M₁₆	—	158	8	no	—
229	M₁₆	—	158	8	no	—
230	C₂×C₈	—	79	8	no	—
231	C₂×C₈	—	79	4	no	—
232	C₂×C₈	—	79	2	no	—
233	C₂×C₈	—	79	8	no	—
234	C₂×C₈	—	79	2	no	—
235	C₂×C₈	—	158	8	no	—
236	C₂×C₈	—	158	2	no	—
237	C₂×C₈	—	158	2	no	—
238	C₂×C₈	—	158	8	no	—
239	C₄⋊C₄	C₂²×Dic₇₉	1	8	yes	—
240	C₄⋊C₄	C₂²×Dic₇₉	1	8	yes	—
241	C₄⋊C₄	C₂²×Dic₇₉	1	8	yes	—
242	C₄⋊C₄	C₂²×Dic₇₉	1	32	yes	—
243	C₄⋊C₄	C₂²×Dic₇₉	1	16	yes	—
244	C₂²⋊C₄	C₂²×Dic₇₉	1	8	yes	—
245	C₂²⋊C₄	C₂²×Dic₇₉	1	8	yes	—
246	C₂²⋊C₄	C₂²×Dic₇₉	1	8	yes	—
247	C₄²	C₂²×Dic₇₉	1	4	yes	—
248	C₄²	C₂²×Dic₇₉	1	8	yes	—
249	C₄²	C₂³×D₇₉	1	4	yes	—
250	C₄²	C₂³×D₇₉	1	8	yes	—
251	C₂³	—	4	2	no	—
252	C₂³	Q₈×Dic₇₉	1	4	yes	—
253	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
254	C₂³	C₂³×Dic₇₉	1	2	yes	—
255	Q₈	—	4	16	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	Q₈×Dic₇₉	1	8	yes	—
259	C₂×C₄	Q₈×Dic₇₉	1	4	yes	—
260	C₂×C₄	C₂³×Dic₇₉	1	8	yes	—
261	C₂×C₄	C₂³×Dic₇₉	1	16	yes	—
262	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
263	C₂×C₄	C₂³×Dic₇₉	1	2	yes	—
264	C₂×C₄	C₂×Q₈×D₇₉	1	2	yes	—
265	C₂×C₄	C₂×Q₈×D₇₉	1	4	yes	—
266	C₂×C₄	C₂×#11929625	1	4	yes	—
267	C₂×C₄	C₂×#11929625	1	2	yes	—
268	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
269	C₈	—	79	8	no	—
270	C₈	—	158	2	no	—
271	C₈	—	158	2	no	—
272	C₈	—	158	8	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	4	4	no	—
276	C₂²	C₂×Q₈×Dic₇₉	1	2	yes	—
277	C₂²	C₂×Q₈×Dic₇₉	1	4	yes	—
278	C₂²	C₂⁴×Dic₇₉	1	1	yes	characteristic
279	C₄	—	2	8	no	—
280	C₄	—	2	16	no	—
281	C₄	C₂×Q₈×Dic₇₉	1	2	yes	—
282	C₄	C₂×Q₈×Dic₇₉	1	4	yes	—
283	C₄	C₂²×Q₈×D₇₉	1	1	yes	characteristic
284	C₄	C₂²×#11929625	1	1	yes	characteristic
285	C₂	—	2	2	no	—
286	C₂	—	2	4	no	—
287	C₂	C₂²×Q₈×Dic₇₉	1	1	yes	characteristic
288	C₂	#23778112	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₇₉
commutator, socle	C₂²×C₇₉	C₂⁴×C₄
Fitting	C₇₉×#3458	C₂
prime radical	C₂⁴×C₇₉	C₂²×C₄
cosocle	C₂×C₄×C₇₉	C₂⁵
hypercenter	#3458	D₇₉
hypocenter	C₇₉	#60193
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ #3458

Complements

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

N	H ≅ G/N	type
1. #23772349	C₂	semidirect
2. #23772341	C₂	semidirect
3. #23772335	C₂	semidirect
12. #19098694	C₂²	semidirect
13. #19098694	C₂²	semidirect
14. Q₈×C₇₉⋊C₈	C₂²	semidirect
35. C₂×#11929608	Q₈	semidirect
153. C₇₉	#60193	semidirect

Maximal subgroups

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

subgroup	index
#23772349	2
#23772341	2
#23772335	2
#23771964	2
C₂×#19098590	2
#23771819	2
C₂×#19098519	2
C₇₉×#3458	2
#60193	79