Subgroups — #34307278

Order 20224.

622 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₁₅₈	C₂	1	2	yes	maximal
2	C₂×#19098693	C₂	1	1	yes	maximal, characteristic
3	#23764175	C₂	1	2	yes	maximal
4	#23764172	C₂	1	4	yes	maximal
5	#23763539	C₂	1	2	yes	maximal
6	#23763538	C₂	1	1	yes	maximal, characteristic
7	#23763305	C₂	1	2	yes	maximal
8	C₄×#108×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
10	#19098693	—	2	2	no	—
11	#19098693	C₄	1	4	yes	—
12	#19098693	C₄	1	4	yes	—
13	#19098691	—	2	4	no	—
14	C₈×Dic₁₅₈	—	2	2	no	—
15	C₂×#14579096	C₂²	1	4	yes	—
16	#19098587	C₂²	1	2	yes	—
17	#19098586	C₂²	1	4	yes	—
18	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×Dic₇₉⋊C₈	C₂²	1	2	yes	—
20	M₁₆×Dic₇₉	C₂²	1	4	yes	—
21	C₂×#14579071	C₂²	1	2	yes	—
22	#19098328	C₂²	1	2	yes	—
23	#19098328	C₂²	1	2	yes	—
24	#19098327	C₂²	1	4	yes	—
25	C₂×#108×C₇₉	C₂²	1	2	yes	—
26	C₄×M₁₆×C₇₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
28	#341×C₇₉	C₂²	1	1	yes	characteristic
29	#341×C₇₉	C₂²	1	1	yes	characteristic
30	C₂²×Dic₁₅₈	Q₈	1	1	yes	characteristic
31	C₂×#11929608	C₂³	1	4	yes	—
32	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
33	C₄×Dic₁₅₈	C₂×C₄	1	2	yes	—
34	C₄×Dic₁₅₈	C₂×C₄	1	2	yes	—
35	C₄×Dic₁₅₈	D₄	1	2	yes	—
36	C₄×Dic₁₅₈	D₄	1	2	yes	—
37	C₂×C₄⋊Dic₇₉	Q₈	1	1	yes	characteristic
38	C₂×Dic₇₉⋊C₄	Q₈	1	2	yes	—
39	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
40	#14579096	—	2	8	no	—
41	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
42	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
43	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
44	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
45	Dic₇₉⋊C₈	—	2	4	no	—
46	Dic₇₉⋊C₈	—	2	4	no	—
47	Dic₇₉⋊C₈	C₂×C₄	1	8	yes	—
48	#14579071	—	2	4	no	—
49	#14579071	C₂×C₄	1	8	yes	—
50	C₈×Dic₇₉	—	2	4	no	—
51	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
52	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
53	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
56	#108×C₇₉	—	2	4	no	—
57	#97×C₇₉	—	2	2	no	—
58	C₄×C₈×C₇₉	—	2	2	no	—
59	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
61	C₂×Dic₁₅₈	—	4	2	no	—
62	C₂×Dic₁₅₈	C₄⋊C₄	1	2	yes	—
63	C₂×Dic₁₅₈	C₄⋊C₄	1	2	yes	—
64	C₂²×Dic₇₉	C₂×Q₈	1	2	yes	—
65	C₂²×Dic₇₉	C₂×Q₈	1	2	yes	—
66	#11929608	—	2	4	no	—
67	#11929608	—	2	4	no	—
68	C₂×C₇₉⋊C₈	—	2	4	no	—
69	C₂×C₇₉⋊C₈	—	2	4	no	—
70	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
71	C₄⋊Dic₇₉	C₄⋊C₄	1	2	yes	—
72	C₄⋊Dic₇₉	C₄⋊C₄	1	2	yes	—
73	Dic₇₉⋊C₄	C₄⋊C₄	1	4	yes	—
74	Dic₇₉⋊C₄	C₄⋊C₄	1	4	yes	—
75	C₄×Dic₇₉	C₂²×C₄	1	4	yes	—
76	C₄×Dic₇₉	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
80	M₁₆×C₇₉	—	2	2	no	—
81	M₁₆×C₇₉	—	2	2	no	—
82	M₁₆×C₇₉	—	2	2	no	—
83	M₁₆×C₇₉	#41	1	4	yes	—
84	C₂×C₈×C₇₉	—	2	2	no	—
85	C₂×C₈×C₇₉	—	2	2	no	—
86	C₂×C₈×C₇₉	—	2	2	no	—
87	C₂×C₈×C₇₉	—	2	2	no	—
88	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
90	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
94	C₂³×C₇₉	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₇₉	—	2	2	no	—
96	C₂×C₄×C₇₉	—	4	1	no	—
97	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
104	C₂×C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
105	C₂×Dic₇₉	—	4	2	no	—
106	C₂×Dic₇₉	—	4	2	no	—
107	C₂×Dic₇₉	C₂×C₄⋊C₄	1	4	yes	—
108	C₂×Dic₇₉	C₂×C₄⋊C₄	1	4	yes	—
109	Dic₁₅₈	—	2	4	no	—
110	Dic₁₅₈	—	4	2	no	—
111	C₈×C₇₉	—	2	2	no	—
112	C₈×C₇₉	—	2	2	no	—
113	C₈×C₇₉	—	2	4	no	—
114	C₈×C₇₉	—	4	2	no	—
115	C₇₉⋊C₈	—	2	4	no	—
116	C₇₉⋊C₈	—	4	4	no	—
117	C₂²×C₇₉	—	2	1	no	—
118	C₂²×C₇₉	—	2	1	no	—
119	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₇₉	#557	1	2	yes	—
121	C₄×C₇₉	—	2	1	no	—
122	C₄×C₇₉	—	2	1	no	—
123	C₄×C₇₉	—	2	1	no	—
124	C₄×C₇₉	#430	1	2	yes	—
125	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
126	C₄×C₇₉	#519	1	1	yes	characteristic
127	C₄×C₇₉	#548	1	2	yes	—
128	Dic₇₉	—	2	4	no	—
129	Dic₇₉	—	2	4	no	—
130	#22747	—	79	1	no	maximal, Sylow
131	C₂×C₇₉	—	2	2	no	—
132	C₂×C₇₉	#2351	1	1	yes	characteristic
133	C₂×C₇₉	C₂×#430	1	1	yes	characteristic
134	C₂×C₇₉	#2916	1	1	yes	characteristic
135	Q₈×M₁₆	—	79	2	no	—
136	C₂×#446	—	79	1	no	—
137	#1950	—	79	4	no	—
138	#1950	—	79	2	no	—
139	#1838	—	79	2	no	—
140	#1837	—	79	1	no	—
141	#1778	—	79	2	no	—
142	C₄×#108	D₇₉	1	1	yes	characteristic
143	C₇₉	#22747	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄×Q₈	—	79	1	no	—
145	#446	—	79	4	no	—
146	#446	—	79	4	no	—
147	#446	—	158	2	no	—
148	#446	—	158	4	no	—
149	C₈×Q₈	—	158	2	no	—
150	C₂×#108	—	79	4	no	—
151	C₂×#108	C₂×D₇₉	1	2	yes	—
152	#423	—	79	4	no	—
153	#423	—	79	2	no	—
154	C₂×C₄⋊C₈	—	79	2	no	—
155	C₂×C₄⋊C₈	—	79	1	no	—
156	C₄×M₁₆	—	79	4	no	—
157	C₄×M₁₆	C₂×D₇₉	1	2	yes	—
158	C₂×#97	—	79	2	no	—
159	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
160	#341	—	79	4	no	—
161	#341	—	79	2	no	—
162	#341	—	79	2	no	—
163	#341	C₂×D₇₉	1	1	yes	characteristic
164	#341	C₂×D₇₉	1	1	yes	characteristic
165	C₂²×Q₈	—	79	1	no	—
166	C₂×M₁₆	—	79	4	no	—
167	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
168	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
169	C₂²×C₈	—	79	2	no	—
170	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
172	C₄×Q₈	—	79	2	no	—
173	C₄×Q₈	—	79	2	no	—
174	C₄×Q₈	—	79	2	no	—
175	C₄×Q₈	—	79	2	no	—
176	C₂×C₄⋊C₄	—	79	2	no	—
177	C₂×C₄⋊C₄	—	79	1	no	—
178	C₂×C₄²	—	79	2	no	—
179	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
180	#108	—	2	4	no	—
181	#108	—	158	8	no	—
182	C₄⋊C₈	—	79	8	no	—
183	C₄⋊C₈	—	79	2	no	—
184	C₄⋊C₈	—	79	2	no	—
185	C₄⋊C₈	—	158	4	no	—
186	C₄⋊C₈	—	158	4	no	—
187	C₄⋊C₈	—	158	2	no	—
188	C₄⋊C₈	—	158	2	no	—
189	#97	—	2	2	no	—
190	#97	—	79	8	no	—
191	#97	—	158	4	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	—	158	4	no	—
194	C₄×C₈	C₄×D₇₉	1	2	yes	—
195	C₄×C₈	C₄×D₇₉	1	2	yes	—
196	C₂×Q₈	—	316	2	no	—
197	C₂×Q₈	—	79	2	no	—
198	C₂×Q₈	—	79	2	no	—
199	C₂²×C₄	—	79	2	no	—
200	C₂²×C₄	—	79	2	no	—
201	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
202	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
203	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	—	158	4	no	—
208	M₁₆	—	158	4	no	—
209	M₁₆	#11929624	1	4	yes	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	79	4	no	—
215	C₂×C₈	—	158	4	no	—
216	C₂×C₈	—	158	4	no	—
217	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
218	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
219	C₄⋊C₄	—	79	4	no	—
220	C₄⋊C₄	—	79	4	no	—
221	C₄⋊C₄	—	79	2	no	—
222	C₄⋊C₄	—	79	2	no	—
223	C₄²	—	79	4	no	—
224	C₄²	—	79	4	no	—
225	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
226	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
227	C₄²	D₄×D₇₉	1	1	yes	characteristic
228	C₄²	D₄×D₇₉	1	1	yes	characteristic
229	C₂³	C₂×Q₈×D₇₉	1	1	yes	characteristic
230	Q₈	—	316	2	no	—
231	Q₈	—	158	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	316	2	no	—
235	C₂×C₄	—	316	2	no	—
236	C₂×C₄	—	79	4	no	—
237	C₂×C₄	—	79	4	no	—
238	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
239	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
240	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
241	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
243	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
244	C₂×C₄	C₂×#11929624	1	2	yes	—
245	C₂×C₄	#14579208	1	2	yes	—
246	C₈	—	2	4	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	4	2	no	—
250	C₈	—	316	4	no	—
251	C₈	—	158	4	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	C₂×C₄⋊C₄×D₇₉	1	1	yes	characteristic
255	C₂²	#19099291	1	2	yes	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	158	4	no	—
260	C₄	—	158	4	no	—
261	C₄	#19098610	1	2	yes	—
262	C₄	C₄×#11929624	1	1	yes	characteristic
263	C₄	#19099040	1	1	yes	characteristic
264	C₄	#19099222	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	#23766320	1	1	yes	characteristic
267	C₂	C₂×#19098610	1	1	yes	characteristic
268	C₂	#23772008	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₄×#108×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×C₄⋊C₄
prime radical	C₂³×C₇₉	C₂²×Q₈
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₄×#108	D₇₉
hypocenter	C₇₉	#22747
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
143. C₇₉	#22747	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₁₅₈	2
C₂×#19098693	2
#23764175	2
#23764172	2
#23763539	2
#23763538	2
#23763305	2
C₄×#108×C₇₉	2
#22747	79