Subgroups — #34198091

Order 20224.

338 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098693	C₂	1	1	yes	maximal, characteristic
2	C₄×#14579101	C₂	1	1	yes	maximal, characteristic
3	#23761381	C₂	1	4	yes	maximal, complemented
4	C₇₉×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₅₈	C₄	1	1	yes	characteristic
6	C₂×#14579101	—	2	1	no	—
7	#19098693	—	2	1	no	—
8	#19098693	—	2	1	no	—
9	#19098693	C₄	1	2	yes	—
10	#19098693	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₇₉⋊C₈)	C₄	1	1	yes	characteristic
12	C₂×Dic₇₉⋊C₈	—	2	1	no	—
13	C₂×#14579071	—	2	1	no	—
14	#19098339	C₂²	1	2	yes	—
15	C₄×C₇₉⋊C₁₆	—	2	1	no	—
16	C₂×M₃₂×C₇₉	—	2	1	no	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₁₆×C₇₉	—	2	1	no	—
19	C₄⋊C₁₆×C₇₉	C₂²	1	2	yes	—
20	C₂²×Dic₁₅₈	C₈	1	1	yes	characteristic
21	C₂²×C₇₉⋊C₈	—	2	1	no	—
22	C₄×Dic₁₅₈	—	2	1	no	—
23	C₄×Dic₁₅₈	—	2	1	no	—
24	C₄×Dic₁₅₈	C₈	1	2	yes	—
25	C₄×Dic₁₅₈	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊Dic₇₉	C₈	1	1	yes	characteristic
27	C₂×Dic₇₉⋊C₄	—	2	1	no	—
28	C₂×C₄×Dic₇₉	—	2	1	no	—
29	#14579101	—	2	4	no	—
30	C₂×C₇₉⋊C₁₆	—	2	2	no	—
31	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
32	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
33	Dic₇₉⋊C₈	—	2	2	no	—
34	Dic₇₉⋊C₈	—	2	2	no	—
35	#14579071	—	2	2	no	—
36	#14579071	—	2	2	no	—
37	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₇₉	—	4	2	no	—
41	C₂×C₁₆×C₇₉	—	2	2	no	—
42	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₅₈	—	2	1	no	—
47	C₂×Dic₁₅₈	—	4	2	no	—
48	C₂×Dic₁₅₈	C₂×C₈	1	2	yes	—
49	C₂²×Dic₇₉	—	2	1	no	—
50	C₂²×Dic₇₉	—	2	1	no	—
51	C₂×C₇₉⋊C₈	—	2	2	no	—
52	C₂×C₇₉⋊C₈	—	4	2	no	—
53	C₄⋊Dic₇₉	—	2	1	no	—
54	C₄⋊Dic₇₉	C₂×C₈	1	2	yes	—
55	Dic₇₉⋊C₄	—	2	2	no	—
56	Dic₇₉⋊C₄	—	2	2	no	—
57	C₄×Dic₇₉	—	2	2	no	—
58	C₄×Dic₇₉	—	2	2	no	—
59	C₇₉⋊C₁₆	—	2	4	no	—
60	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
63	C₂×C₈×C₇₉	—	2	2	no	—
64	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
67	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
68	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
73	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
74	C₁₆×C₇₉	—	4	2	no	—
75	C₂³×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
76	C₂×C₄×C₇₉	—	2	2	no	—
77	C₂×C₄×C₇₉	—	4	1	no	—
78	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₇₉	#103	1	2	yes	—
82	C₂×C₄×C₇₉	#104	1	2	yes	—
83	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
86	C₂×Dic₇₉	—	2	2	no	—
87	C₂×Dic₇₉	—	2	2	no	—
88	C₂×Dic₇₉	—	4	2	no	—
89	C₂×Dic₇₉	—	4	2	no	—
90	Dic₁₅₈	—	4	2	no	—
91	Dic₁₅₈	—	4	2	no	—
92	C₈×C₇₉	—	2	1	no	—
93	C₈×C₇₉	—	2	1	no	—
94	C₈×C₇₉	#103	1	2	yes	—
95	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
96	C₈×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
97	C₇₉⋊C₈	—	4	2	no	—
98	C₂²×C₇₉	—	2	1	no	—
99	C₂²×C₇₉	—	2	1	no	—
100	C₂²×C₇₉	Q₈⋊C₈	1	2	yes	—
101	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
102	C₄×C₇₉	—	2	1	no	—
103	C₄×C₇₉	—	2	1	no	—
104	C₄×C₇₉	—	2	1	no	—
105	C₄×C₇₉	Q₈⋊C₈	1	2	yes	—
106	C₄×C₇₉	#349	1	2	yes	—
107	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
108	C₄×C₇₉	C₂×#104	1	1	yes	characteristic
109	Dic₇₉	—	4	2	no	—
110	Dic₇₉	—	4	2	no	—
111	#11611	—	79	1	no	maximal, Sylow
112	C₂×C₇₉	—	2	2	no	—
113	C₂×C₇₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
114	C₂×C₇₉	C₂×#349	1	1	yes	characteristic
115	C₂×C₇₉	#2117	1	1	yes	characteristic
116	C₂×#446	—	79	1	no	—
117	#2150	D₇₉	1	1	yes	characteristic
118	C₄×M₃₂	—	79	1	no	—
119	#1338	—	79	4	no	—
120	C₇₉	#11611	1	1	yes	characteristic, complemented, Sylow
121	C₂×C₄×Q₈	—	79	1	no	—
122	C₂×M₃₂	—	2	1	no	—
123	C₂×M₃₂	—	158	1	no	—
124	#446	—	79	2	no	—
125	#446	—	79	2	no	—
126	#446	—	158	1	no	—
127	#446	—	158	1	no	—
128	C₂×C₄⋊C₈	—	79	1	no	—
129	C₂×C₄⋊C₈	—	158	1	no	—
130	C₂×#97	—	158	1	no	—
131	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
132	C₄⋊C₁₆	—	2	1	no	—
133	C₄⋊C₁₆	C₂×D₇₉	1	2	yes	—
134	#346	—	79	2	no	—
135	C₄×C₁₆	—	158	1	no	—
136	C₂²×Q₈	—	79	1	no	—
137	C₂²×C₈	—	158	1	no	—
138	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
140	C₄×Q₈	—	79	2	no	—
141	C₄×Q₈	—	79	2	no	—
142	C₄×Q₈	—	158	1	no	—
143	C₄×Q₈	—	158	1	no	—
144	C₂×C₄⋊C₄	—	79	1	no	—
145	C₂×C₄⋊C₄	—	158	1	no	—
146	C₂×C₄²	—	158	1	no	—
147	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
148	M₃₂	—	4	2	no	—
149	M₃₂	—	158	4	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	158	2	no	—
152	C₄⋊C₈	—	79	2	no	—
153	C₄⋊C₈	—	158	2	no	—
154	C₄⋊C₈	—	158	2	no	—
155	C₄⋊C₈	—	158	1	no	—
156	#97	—	158	2	no	—
157	#97	—	158	2	no	—
158	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
159	C₄×C₈	D₃₁₆	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
161	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
162	C₂×Q₈	—	316	2	no	—
163	C₂×Q₈	—	79	2	no	—
164	C₂×Q₈	—	158	1	no	—
165	C₂²×C₄	—	158	1	no	—
166	C₂²×C₄	—	158	1	no	—
167	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
168	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
169	C₂²×C₄	#11929607	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	316	2	no	—
172	C₂×C₈	—	158	2	no	—
173	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
174	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
175	C₂×C₈	#11929612	1	2	yes	—
176	C₂×C₈	#11929614	1	2	yes	—
177	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
178	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
179	C₄⋊C₄	—	79	2	no	—
180	C₄⋊C₄	—	158	2	no	—
181	C₄⋊C₄	—	158	1	no	—
182	C₄⋊C₄	—	158	2	no	—
183	C₄²	—	158	2	no	—
184	C₄²	—	158	2	no	—
185	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
186	C₄²	C₈×D₇₉	1	1	yes	characteristic
187	C₄²	#11929607	1	1	yes	characteristic
188	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
189	C₁₆	—	4	2	no	—
190	C₁₆	—	158	4	no	—
191	C₂³	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
192	Q₈	—	316	2	no	—
193	Q₈	—	316	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	316	2	no	—
197	C₂×C₄	—	316	2	no	—
198	C₂×C₄	—	158	2	no	—
199	C₂×C₄	—	158	2	no	—
200	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
201	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
202	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
203	C₂×C₄	#14579084	1	2	yes	—
204	C₂×C₄	#14579087	1	2	yes	—
205	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
206	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
207	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	316	2	no	—
211	C₈	#14579084	1	2	yes	—
212	C₈	C₂×#11929612	1	1	yes	characteristic
213	C₈	C₂×#11929614	1	1	yes	characteristic
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	#19098293	1	2	yes	—
217	C₂²	C₂×(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	316	2	no	—
222	C₄	—	316	2	no	—
223	C₄	#19098293	1	2	yes	—
224	C₄	#19098344	1	2	yes	—
225	C₄	C₂×#14579084	1	1	yes	characteristic
226	C₄	C₂×#14579087	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₂×#19098293	1	1	yes	characteristic
229	C₂	C₂×#19098344	1	1	yes	characteristic
230	C₂	#23765027	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#2150	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₈
prime radical	C₂³×C₇₉	C₂²⋊C₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#2150	D₇₉
hypocenter	C₇₉	#11611
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #23761381	C₂	semidirect
120. C₇₉	#11611	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098693	2
C₄×#14579101	2
#23761381	2
C₇₉×#2150	2
#11611	79