Subgroups — #34177708

Order 20224.

714 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098616	C₂	1	2	yes	maximal
2	#23761233	C₂	1	4	yes	maximal, complemented
3	C₄×#117×C₇₉	C₂	1	1	yes	maximal, characteristic
4	#19098616	—	2	2	no	—
5	#19098616	—	2	2	no	—
6	#19098616	C₄	1	4	yes	complemented
7	#19098616	C₂²	1	4	yes	—
8	C₂×C₄⋊(C₇₉⋊C₈)	C₄	1	4	yes	complemented
9	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
10	C₂×C₄×C₇₉⋊C₈	—	2	2	no	—
11	C₂×#117×C₇₉	C₄	1	2	yes	—
12	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₇₉	—	2	2	no	—
14	C₄×C₂²⋊C₄×C₇₉	—	2	2	no	—
15	#376×C₇₉	C₄	1	2	yes	—
16	C₄³×C₇₉	C₂²	1	2	yes	—
17	C₂²×C₇₉⋊C₈	—	2	4	no	—
18	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
19	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	8	yes	—
20	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
21	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
22	C₄×C₇₉⋊C₈	—	2	4	no	—
23	C₄×C₇₉⋊C₈	—	2	4	no	—
24	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₇₉	D₄	1	2	yes	—
26	#117×C₇₉	—	2	4	no	—
27	#117×C₇₉	C₈	1	8	yes	complemented
28	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
29	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
30	C₂×C₄²×C₇₉	—	2	4	no	—
31	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₉	D₄	1	2	yes	—
36	C₂×C₄²×C₇₉	D₄	1	2	yes	—
37	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
38	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
39	#95×C₇₉	—	2	4	no	—
40	C₂×C₇₉⋊C₈	—	2	8	no	—
41	C₂×C₇₉⋊C₈	—	4	8	no	—
42	C₂⁴×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₉	—	2	4	no	—
44	C₂²×C₄×C₇₉	—	4	4	no	—
45	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
46	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₇₉	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₇₉	M₁₆	1	4	yes	—
55	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
57	C₄⋊C₄×C₇₉	—	2	8	no	—
58	C₂²⋊C₄×C₇₉	—	2	8	no	—
59	C₄²×C₇₉	—	2	4	no	—
60	C₄²×C₇₉	—	4	8	no	—
61	C₄²×C₇₉	C₄²	1	1	yes	characteristic
62	C₄²×C₇₉	C₄²	1	1	yes	characteristic
63	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
66	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
67	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
68	C₄²×C₇₉	C₂×C₈	1	8	yes	complemented
69	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
70	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
72	C₂³×C₇₉	—	4	2	no	—
73	C₂³×C₇₉	#95	1	2	yes	—
74	C₂³×C₇₉	C₂²⋊C₈	1	4	yes	—
75	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₇₉	—	2	4	no	—
77	C₂×C₄×C₇₉	—	2	4	no	—
78	C₂×C₄×C₇₉	—	2	4	no	—
79	C₂×C₄×C₇₉	—	2	8	no	—
80	C₂×C₄×C₇₉	—	4	2	no	—
81	C₂×C₄×C₇₉	—	4	2	no	—
82	C₂×C₄×C₇₉	—	4	2	no	—
83	C₂×C₄×C₇₉	—	4	8	no	—
84	C₂×C₄×C₇₉	#95	1	2	yes	—
85	C₂×C₄×C₇₉	#95	1	2	yes	—
86	C₂×C₄×C₇₉	#95	1	2	yes	—
87	C₂×C₄×C₇₉	C₄×C₈	1	4	yes	—
88	C₂×C₄×C₇₉	#97	1	4	yes	—
89	C₂×C₄×C₇₉	C₂²⋊C₈	1	4	yes	—
90	C₂×C₄×C₇₉	C₄⋊C₈	1	8	yes	—
91	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
97	C₇₉⋊C₈	—	4	8	no	—
98	C₂²×C₇₉	—	2	2	no	—
99	C₂²×C₇₉	—	2	2	no	—
100	C₂²×C₇₉	—	2	2	no	—
101	C₂²×C₇₉	—	4	4	no	—
102	C₂²×C₇₉	#336	1	4	yes	—
103	C₂²×C₇₉	C₂×#95	1	1	yes	characteristic
104	C₂²×C₇₉	C₂×C₂²⋊C₈	1	2	yes	—
105	C₄×C₇₉	—	2	2	no	—
106	C₄×C₇₉	—	2	2	no	—
107	C₄×C₇₉	—	2	2	no	—
108	C₄×C₇₉	—	4	8	no	—
109	C₄×C₇₉	#336	1	4	yes	—
110	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
111	C₄×C₇₉	C₂×#97	1	1	yes	characteristic
112	C₄×C₇₉	#421	1	4	yes	—
113	C₄×C₇₉	C₂×C₄⋊C₈	1	2	yes	—
114	#7866	—	79	1	no	maximal, Sylow
115	C₂×C₇₉	—	2	2	no	—
116	C₂×C₇₉	—	2	4	no	—
117	C₂×C₇₉	C₂×#336	1	1	yes	characteristic
118	C₂×C₇₉	#1733	1	2	yes	—
119	C₂×#431	—	79	2	no	—
120	C₄×#117	D₇₉	1	1	yes	characteristic
121	#1276	—	79	4	no	—
122	C₇₉	#7866	1	1	yes	characteristic, complemented, Sylow
123	C₂×#117	Dic₇₉	1	2	yes	—
124	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
125	#431	—	79	4	no	—
126	#431	—	79	4	no	—
127	#431	—	158	2	no	—
128	#431	—	158	2	no	—
129	C₂×C₄⋊C₈	—	79	4	no	—
130	C₂×C₂²⋊C₈	—	158	2	no	—
131	C₂×C₄×C₈	—	158	2	no	—
132	C₄×C₄⋊C₄	—	2	2	no	—
133	C₄×C₂²⋊C₄	—	2	2	no	—
134	#376	Dic₇₉	1	2	yes	—
135	C₄³	C₂×D₇₉	1	2	yes	—
136	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
137	C₂³×C₄	C₂²⋊D₇₉	1	2	yes	—
138	C₂²×C₈	—	158	4	no	—
139	#117	—	2	4	no	—
140	#117	C₇₉⋊C₈	1	8	yes	complemented
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₂²⋊C₄	—	2	2	no	—
143	C₂×C₄²	—	2	4	no	—
144	C₂×C₄²	Dic₁₅₈	1	2	yes	—
145	C₂×C₄²	C₄×D₇₉	1	2	yes	—
146	C₂×C₄²	C₄×D₇₉	1	2	yes	—
147	C₂×C₄²	D₃₁₆	1	2	yes	—
148	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
149	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
150	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
151	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
152	C₄⋊C₈	—	79	8	no	—
153	C₄⋊C₈	—	158	4	no	—
154	C₂²⋊C₈	—	158	4	no	—
155	C₂²⋊C₈	—	158	4	no	—
156	C₄×C₈	—	158	4	no	—
157	C₄×C₈	—	158	4	no	—
158	#95	—	2	4	no	—
159	C₂⁴	C₂²⋊Dic₇₉	1	1	yes	characteristic
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	4	4	no	—
162	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
163	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
164	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
165	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
166	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
167	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
168	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
169	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
170	C₂²×C₄	C₂×C₇₉⋊C₈	1	4	yes	—
171	C₂²×C₄	#11929608	1	4	yes	—
172	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
174	C₂×C₈	—	316	8	no	—
175	C₂×C₈	—	158	8	no	—
176	C₄⋊C₄	—	2	8	no	—
177	C₂²⋊C₄	—	2	8	no	—
178	C₄²	—	2	4	no	—
179	C₄²	—	4	8	no	—
180	C₄²	C₄×Dic₇₉	1	1	yes	characteristic
181	C₄²	C₄×Dic₇₉	1	1	yes	characteristic
182	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
183	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
184	C₄²	Dic₇₉⋊C₄	1	2	yes	—
185	C₄²	Dic₇₉⋊C₄	1	2	yes	—
186	C₄²	C₄⋊Dic₇₉	1	2	yes	—
187	C₄²	C₂×C₇₉⋊C₈	1	8	yes	complemented
188	C₄²	C₂×C₄×D₇₉	1	2	yes	—
189	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
190	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
191	C₂³	—	4	2	no	—
192	C₂³	C₇₉⋊#95	1	2	yes	—
193	C₂³	C₂²⋊(C₇₉⋊C₈)	1	4	yes	—
194	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	8	no	—
203	C₂×C₄	C₇₉⋊#95	1	2	yes	—
204	C₂×C₄	C₇₉⋊#95	1	2	yes	—
205	C₂×C₄	C₇₉⋊#95	1	2	yes	—
206	C₂×C₄	C₄×C₇₉⋊C₈	1	4	yes	—
207	C₂×C₄	#14579072	1	4	yes	—
208	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	4	yes	—
209	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	8	yes	—
210	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
211	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	2	yes	—
212	C₂×C₄	C₂×Dic₇₉⋊C₄	1	2	yes	—
213	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
214	C₂×C₄	C₂²×C₇₉⋊C₈	1	2	yes	—
215	C₂×C₄	C₂×#11929608	1	2	yes	—
216	C₈	—	316	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	4	no	—
221	C₂²	#19098318	1	4	yes	—
222	C₂²	C₂×C₇₉⋊#95	1	1	yes	characteristic
223	C₂²	C₂×C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	4	8	no	—
228	C₄	#19098318	1	4	yes	—
229	C₄	C₂×C₄×C₇₉⋊C₈	1	1	yes	characteristic
230	C₄	C₂×#14579072	1	1	yes	characteristic
231	C₄	#19098581	1	4	yes	—
232	C₄	C₂×C₄⋊(C₇₉⋊C₈)	1	2	yes	—
233	C₂	—	2	2	no	—
234	C₂	—	2	4	no	—
235	C₂	C₂×#19098318	1	1	yes	characteristic
236	C₂	#23763110	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₃₁₆
commutator	C₄×C₇₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₄×#117×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#95
prime radical	C₂⁴×C₇₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×#117	D₇₉
hypocenter	C₇₉	#7866
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #23761233	C₂	semidirect
6. #19098616	C₄	semidirect
8. C₂×C₄⋊(C₇₉⋊C₈)	C₄	semidirect
27. #117×C₇₉	C₈	semidirect
68. C₄²×C₇₉	C₂×C₈	semidirect
122. C₇₉	#7866	semidirect
140. #117	C₇₉⋊C₈	semidirect
187. C₄²	C₂×C₇₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098616	2
#23761233	2
C₄×#117×C₇₉	2
#7866	79