Subgroups — #34203112

Order 20224.

614 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579101	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#23761478	C₂	1	4	yes	maximal, complemented
4	C₂×#405×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579101	—	2	2	no	—
6	C₂×#14579101	—	2	2	no	—
7	C₂×#14579101	C₄	1	4	yes	complemented
8	C₂²×C₇₉⋊C₁₆	—	2	2	no	—
9	C₂²×C₇₉⋊C₁₆	—	2	2	no	—
10	C₂²×C₇₉⋊C₁₆	C₄	1	4	yes	complemented
11	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
12	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
13	C₂²×M₁₆×C₇₉	C₄	1	1	yes	characteristic
14	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#117×C₇₉	C₄	1	1	yes	characteristic
16	#405×C₇₉	—	2	2	no	—
17	#405×C₇₉	—	2	2	no	—
18	C₂×#97×C₇₉	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
20	#14579101	—	2	8	no	—
21	C₂×C₇₉⋊C₁₆	—	2	8	no	—
22	C₂×C₇₉⋊C₁₆	—	2	2	no	—
23	C₂×C₇₉⋊C₁₆	—	2	2	no	—
24	C₂×C₇₉⋊C₁₆	—	2	2	no	—
25	C₂×C₇₉⋊C₁₆	—	2	2	no	—
26	C₂×C₇₉⋊C₁₆	C₂×C₄	1	4	yes	—
27	C₂×C₇₉⋊C₁₆	C₂×C₄	1	4	yes	—
28	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂×M₁₆×C₇₉	—	2	2	no	—
30	C₂×M₁₆×C₇₉	—	2	2	no	—
31	C₂×M₁₆×C₇₉	C₈	1	4	yes	—
32	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₉	D₄	1	2	yes	—
37	C₂²×C₈×C₇₉	D₄	1	2	yes	—
38	C₂²×C₈×C₇₉	D₄	1	2	yes	—
39	C₂²×C₈×C₇₉	Q₈	1	2	yes	—
40	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
41	#117×C₇₉	—	2	2	no	—
42	#117×C₇₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₇₉	C₈	1	2	yes	—
44	C₂×C₂²⋊C₄×C₇₉	C₈	1	2	yes	—
45	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
46	#97×C₇₉	—	2	4	no	—
47	C₄×C₈×C₇₉	—	2	2	no	—
48	C₄×C₈×C₇₉	—	2	2	no	—
49	C₇₉⋊C₁₆	—	2	4	no	—
50	C₇₉⋊C₁₆	—	2	4	no	—
51	C₂⁴×C₇₉	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
53	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇₉	C₂×C₈	1	4	yes	—
56	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₇₉	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
59	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
60	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
61	M₁₆×C₇₉	—	2	8	no	—
62	C₂×C₈×C₇₉	—	2	2	no	—
63	C₂×C₈×C₇₉	—	2	2	no	—
64	C₂×C₈×C₇₉	C₄²	1	2	yes	—
65	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
66	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
67	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
69	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
70	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
71	C₂×C₈×C₇₉	C₂×C₈	1	4	yes	—
72	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
78	C₄⋊C₄×C₇₉	—	2	4	no	—
79	C₂²⋊C₄×C₇₉	—	2	4	no	—
80	C₄²×C₇₉	—	2	2	no	—
81	C₄²×C₇₉	—	2	2	no	—
82	C₂³×C₇₉	—	2	4	no	—
83	C₂³×C₇₉	C₄×C₈	1	2	yes	—
84	C₂³×C₇₉	C₂²⋊C₈	1	2	yes	—
85	C₂³×C₇₉	C₄⋊C₈	1	2	yes	—
86	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₇₉	—	2	4	no	—
88	C₂×C₄×C₇₉	—	2	8	no	—
89	C₂×C₄×C₇₉	#95	1	4	yes	—
90	C₂×C₄×C₇₉	C₄×C₈	1	2	yes	—
91	C₂×C₄×C₇₉	#97	1	4	yes	—
92	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
93	C₂×C₄×C₇₉	C₂²⋊C₈	1	4	yes	—
94	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
95	C₂×C₄×C₇₉	C₄⋊C₈	1	4	yes	—
96	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
102	C₈×C₇₉	—	2	4	no	—
103	C₈×C₇₉	#95	1	4	yes	—
104	C₈×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₈×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₇₉	—	2	4	no	—
109	C₂²×C₇₉	—	2	4	no	—
110	C₂²×C₇₉	—	2	4	no	—
111	C₂²×C₇₉	#336	1	4	yes	—
112	C₂²×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
113	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₂²×C₇₉	#504	1	4	yes	—
116	C₄×C₇₉	—	4	4	no	—
117	C₄×C₇₉	#336	1	4	yes	—
118	C₄×C₇₉	C₂×#95	1	1	yes	characteristic
119	C₄×C₇₉	C₂×#97	1	1	yes	characteristic
120	C₄×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₄×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
122	#12545	—	79	1	no	maximal, Sylow
123	C₂×C₇₉	—	2	4	no	—
124	C₂×C₇₉	C₂×#336	1	1	yes	characteristic
125	C₂×C₇₉	#2108	1	2	yes	—
126	C₂×C₇₉	#2113	1	2	yes	—
127	C₂×C₇₉	#2151	1	2	yes	—
128	C₂²×M₃₂	—	79	1	no	—
129	C₂³×C₁₆	—	79	1	no	—
130	C₂×#405	D₇₉	1	1	yes	characteristic
131	#1374	—	79	4	no	—
132	C₇₉	#12545	1	1	yes	characteristic, complemented, Sylow
133	C₂²×M₁₆	Dic₇₉	1	1	yes	characteristic
134	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
135	C₂×#117	Dic₇₉	1	1	yes	characteristic
136	C₂×M₃₂	—	79	4	no	—
137	C₂×M₃₂	—	158	2	no	—
138	C₂×M₃₂	—	158	2	no	—
139	C₂²×C₁₆	—	79	2	no	—
140	C₂²×C₁₆	—	79	2	no	—
141	C₂²×C₁₆	—	79	4	no	—
142	C₂²×C₁₆	—	158	2	no	—
143	C₂²×C₁₆	—	158	2	no	—
144	#405	—	2	2	no	—
145	#405	—	2	2	no	—
146	C₂×#97	Dic₇₉	1	2	yes	—
147	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
148	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	C₇₉⋊C₈	1	4	yes	—
152	C₂²×C₈	Dic₁₅₈	1	2	yes	—
153	C₂²×C₈	C₄×D₇₉	1	2	yes	—
154	C₂²×C₈	C₄×D₇₉	1	2	yes	—
155	C₂²×C₈	D₃₁₆	1	2	yes	—
156	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
157	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
159	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
160	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
161	#117	—	2	2	no	—
162	#117	—	2	2	no	—
163	C₂×C₄⋊C₄	C₇₉⋊C₈	1	2	yes	—
164	C₂×C₂²⋊C₄	C₇₉⋊C₈	1	2	yes	—
165	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
166	M₃₂	—	158	8	no	—
167	C₂×C₁₆	—	79	4	no	—
168	C₂×C₁₆	—	79	4	no	—
169	C₂×C₁₆	—	158	8	no	—
170	C₂×C₁₆	—	158	2	no	—
171	C₂×C₁₆	—	158	2	no	—
172	C₂×C₁₆	—	158	2	no	—
173	C₂×C₁₆	—	158	2	no	—
174	#97	—	2	4	no	—
175	C₄×C₈	—	2	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₂⁴	C₂×C₇₉⋊C₈	1	1	yes	characteristic
178	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
179	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
180	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
181	C₂²×C₄	C₂×C₇₉⋊C₈	1	4	yes	—
182	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₇₉⋊C₈	1	2	yes	—
184	C₂²×C₄	#11929608	1	2	yes	—
185	C₂²×C₄	#11929608	1	2	yes	—
186	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
187	M₁₆	—	2	8	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	C₄×Dic₇₉	1	2	yes	—
191	C₂×C₈	(C₂×D₇₉)⋊C₄	1	4	yes	—
192	C₂×C₈	(C₂×D₇₉)⋊C₄	1	4	yes	—
193	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
194	C₂×C₈	Dic₇₉⋊C₄	1	4	yes	—
195	C₂×C₈	Dic₇₉⋊C₄	1	4	yes	—
196	C₂×C₈	C₄⋊Dic₇₉	1	2	yes	—
197	C₂×C₈	C₂×C₇₉⋊C₈	1	4	yes	—
198	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
200	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
201	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
202	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
203	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
204	C₄⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₁₆	—	158	4	no	—
209	C₁₆	—	158	4	no	—
210	C₂³	—	2	4	no	—
211	C₂³	C₄×C₇₉⋊C₈	1	2	yes	—
212	C₂³	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
213	C₂³	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
214	C₂³	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	C₇₉⋊#95	1	4	yes	—
218	C₂×C₄	C₄×C₇₉⋊C₈	1	2	yes	—
219	C₂×C₄	#14579072	1	4	yes	—
220	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	4	yes	—
221	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
222	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
223	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	4	yes	—
224	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
225	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
228	C₂×C₄	C₂×#11929608	1	1	yes	characteristic
229	C₂×C₄	C₂×#11929608	1	1	yes	characteristic
230	C₈	—	2	4	no	—
231	C₈	C₇₉⋊#95	1	4	yes	—
232	C₈	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
233	C₈	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
234	C₈	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
235	C₈	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	#19098318	1	4	yes	—
240	C₂²	C₂×C₄×C₇₉⋊C₈	1	1	yes	characteristic
241	C₂²	C₂×C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
242	C₂²	C₂×C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
243	C₂²	#19098980	1	4	yes	—
244	C₄	—	4	4	no	—
245	C₄	#19098318	1	4	yes	—
246	C₄	C₂×C₇₉⋊#95	1	1	yes	characteristic
247	C₄	C₂×#14579072	1	1	yes	characteristic
248	C₄	C₂×C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
249	C₄	C₂×C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
250	C₂	—	2	4	no	—
251	C₂	C₂×#19098318	1	1	yes	characteristic
252	C₂	#23764997	1	2	yes	—
253	C₂	#23765014	1	2	yes	—
254	C₂	#23765159	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂×C₄×C₈
Fitting	C₂×#405×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×C₈
prime radical	C₂⁴×C₇₉	C₂×C₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#405	D₇₉
hypocenter	C₇₉	#12545
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #23761478	C₂	semidirect
7. C₂×#14579101	C₄	semidirect
10. C₂²×C₇₉⋊C₁₆	C₄	semidirect
132. C₇₉	#12545	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14579101	2
C₂³×C₇₉⋊C₁₆	2
#23761478	2
C₂×#405×C₇₉	2
#12545	79