Subgroups — C₂×#23761446 #34199505

Order 20224.

518 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19098316	C₂	1	1	yes	maximal, characteristic, complemented
3	#23761446	C₂	1	4	yes	maximal, direct product factor
4	C₂×#493×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₉⋊C₁₆	—	2	1	no	—
6	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
7	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	complemented
9	#19098316	—	2	1	no	—
10	#19098316	C₂²	1	2	yes	complemented
11	C₂²×D₈×C₇₉	—	2	1	no	—
12	C₂²×D₈×C₇₉	C₄	1	2	yes	—
13	C₂×#127×C₇₉	—	2	1	no	—
14	#493×C₇₉	—	2	1	no	—
15	#493×C₇₉	—	2	1	no	—
16	#493×C₇₉	C₄	1	2	yes	—
17	#493×C₇₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	C₂²×C₇₉⋊C₈	—	4	1	no	—
20	C₂×C₇₉⋊C₁₆	—	2	2	no	—
21	C₂×C₇₉⋊C₁₆	—	4	2	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
24	C₂²×D₄×C₇₉	—	2	2	no	—
25	C₂²×D₄×C₇₉	—	4	1	no	—
26	C₂²×D₄×C₇₉	—	4	1	no	—
27	C₂×D₈×C₇₉	—	2	2	no	—
28	C₂×D₈×C₇₉	—	2	2	no	—
29	C₂×D₈×C₇₉	—	2	2	no	—
30	C₂×D₈×C₇₉	—	4	4	no	—
31	C₂×D₈×C₇₉	—	4	2	no	—
32	C₂×D₈×C₇₉	—	4	2	no	—
33	C₂×D₈×C₇₉	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
36	#127×C₇₉	—	2	2	no	—
37	#127×C₇₉	—	2	2	no	—
38	#127×C₇₉	—	4	1	no	—
39	#127×C₇₉	—	4	1	no	—
40	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
43	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
45	C₂×C₇₉⋊C₈	—	4	2	no	—
46	C₂×C₇₉⋊C₈	—	8	2	no	—
47	C₇₉⋊C₁₆	—	4	2	no	—
48	C₂⁴×C₇₉	—	4	2	no	—
49	C₂×D₄×C₇₉	—	2	4	no	—
50	C₂×D₄×C₇₉	—	4	4	no	—
51	C₂×D₄×C₇₉	—	4	4	no	—
52	C₂×D₄×C₇₉	—	4	2	no	—
53	C₂×D₄×C₇₉	—	4	2	no	—
54	C₂×D₄×C₇₉	—	4	2	no	—
55	C₂×D₄×C₇₉	—	4	2	no	—
56	C₂×D₄×C₇₉	—	4	2	no	—
57	C₂×D₄×C₇₉	—	8	4	no	—
58	C₂×D₄×C₇₉	—	8	2	no	—
59	C₂×D₄×C₇₉	—	8	2	no	—
60	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
63	D₈×C₇₉	—	4	4	no	—
64	D₈×C₇₉	—	4	4	no	—
65	D₈×C₇₉	—	4	2	no	—
66	D₈×C₇₉	—	4	2	no	—
67	D₈×C₇₉	—	4	4	no	—
68	C₂×C₈×C₇₉	—	2	2	no	—
69	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₉	D₈	1	2	yes	—
72	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
73	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₇₉	D₈	1	1	yes	characteristic
77	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
78	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
79	C₂³×C₇₉	—	4	4	no	—
80	C₂³×C₇₉	—	8	4	no	—
81	C₂³×C₇₉	—	8	4	no	—
82	C₂³×C₇₉	—	8	4	no	—
83	C₂³×C₇₉	#102	1	1	yes	characteristic
84	D₄×C₇₉	—	4	4	no	—
85	D₄×C₇₉	—	4	4	no	—
86	D₄×C₇₉	—	8	4	no	—
87	D₄×C₇₉	—	8	4	no	—
88	D₄×C₇₉	—	8	4	no	—
89	D₄×C₇₉	—	8	4	no	—
90	D₄×C₇₉	—	8	2	no	—
91	D₄×C₇₉	—	8	2	no	—
92	C₂×C₄×C₇₉	—	2	2	no	—
93	C₂×C₄×C₇₉	#100	1	2	yes	—
94	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
95	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
96	C₂×C₄×C₇₉	#102	1	2	yes	—
97	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
98	C₂×C₄×C₇₉	D₁₆	1	2	yes	—
99	C₂×C₄×C₇₉	SD₃₂	1	2	yes	—
100	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
103	C₈×C₇₉	—	2	1	no	—
104	C₈×C₇₉	—	2	1	no	—
105	C₈×C₇₉	#102	1	2	yes	—
106	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
107	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
108	C₇₉⋊C₈	—	8	2	no	—
109	C₂²×C₇₉	—	8	4	no	—
110	C₂²×C₇₉	—	8	4	no	—
111	C₂²×C₇₉	—	8	4	no	—
112	C₂²×C₇₉	—	16	8	no	—
113	C₂²×C₇₉	#331	1	2	yes	—
114	C₂²×C₇₉	#357	1	2	yes	—
115	C₂²×C₇₉	#361	1	2	yes	—
116	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
117	C₄×C₇₉	—	2	1	no	—
118	C₄×C₇₉	—	2	1	no	—
119	C₄×C₇₉	#331	1	2	yes	—
120	C₄×C₇₉	#357	1	2	yes	—
121	C₄×C₇₉	C₂×#100	1	1	yes	characteristic
122	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
123	C₄×C₇₉	C₂×D₁₆	1	1	yes	characteristic
124	C₄×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
125	C₂×#1361	—	79	1	no	maximal, Sylow
126	C₂×C₇₉	—	16	4	no	—
127	C₂×C₇₉	#1361	1	4	yes	complemented
128	C₂×C₇₉	C₂×#331	1	1	yes	characteristic
129	C₂×C₇₉	C₂×#357	1	1	yes	characteristic
130	C₂×C₇₉	C₂×#361	1	1	yes	characteristic
131	C₂×#493	D₇₉	1	1	yes	characteristic
132	C₂×C₄⋊C₁₆	—	79	1	no	—
133	C₂×#335	—	79	1	no	—
134	#1361	—	79	4	no	—
135	C₇₉	C₂×#1361	1	1	yes	characteristic, complemented, Sylow
136	C₂²×D₈	—	2	1	no	—
137	C₂²×D₈	Dic₇₉	1	2	yes	—
138	C₂×#127	—	2	1	no	—
139	C₂²×C₁₆	—	158	1	no	—
140	#493	—	2	1	no	—
141	#493	—	2	1	no	—
142	#493	Dic₇₉	1	2	yes	—
143	#493	C₂×D₇₉	1	2	yes	—
144	C₂×C₄⋊C₈	—	158	1	no	—
145	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
146	C₄⋊C₁₆	—	79	2	no	—
147	C₄⋊C₁₆	—	158	1	no	—
148	#335	—	79	2	no	—
149	#335	—	158	1	no	—
150	C₂²×D₄	—	2	2	no	—
151	C₂²×D₄	—	4	1	no	—
152	C₂²×D₄	—	4	1	no	—
153	C₂×D₈	—	2	2	no	—
154	C₂×D₈	—	2	2	no	—
155	C₂×D₈	—	2	2	no	—
156	C₂×D₈	—	4	4	no	—
157	C₂×D₈	—	4	2	no	—
158	C₂×D₈	—	4	2	no	—
159	C₂×D₈	C₂×Dic₇₉	1	4	yes	—
160	C₂²×C₈	—	316	1	no	—
161	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
162	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
163	#127	—	2	2	no	—
164	#127	—	2	2	no	—
165	#127	—	4	1	no	—
166	#127	—	4	1	no	—
167	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
168	C₂×C₁₆	—	316	2	no	—
169	C₂×C₁₆	—	158	2	no	—
170	C₄⋊C₈	—	316	1	no	—
171	C₄⋊C₈	—	158	2	no	—
172	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
173	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
175	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
176	C₂⁴	—	4	2	no	—
177	C₂×D₄	—	2	4	no	—
178	C₂×D₄	—	4	4	no	—
179	C₂×D₄	—	4	4	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂×D₄	—	4	2	no	—
184	C₂×D₄	—	4	2	no	—
185	C₂×D₄	—	8	4	no	—
186	C₂×D₄	—	8	2	no	—
187	C₂×D₄	—	8	2	no	—
188	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
189	C₂²×C₄	#11929609	1	1	yes	characteristic
190	C₂²×C₄	#11929612	1	1	yes	characteristic
191	D₈	—	4	4	no	—
192	D₈	—	4	4	no	—
193	D₈	—	4	4	no	—
194	D₈	—	4	2	no	—
195	D₈	—	4	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	316	2	no	—
198	C₂×C₈	—	632	2	no	—
199	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
201	C₂×C₈	#11929609	1	2	yes	—
202	C₂×C₈	#11929612	1	2	yes	—
203	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
204	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
205	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
206	C₄²	#11929609	1	1	yes	characteristic
207	C₄²	#11929612	1	1	yes	characteristic
208	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
209	C₁₆	—	316	2	no	—
210	C₂³	—	4	4	no	—
211	C₂³	—	8	4	no	—
212	C₂³	—	8	4	no	—
213	C₂³	—	8	4	no	—
214	C₂³	#14579082	1	1	yes	characteristic
215	D₄	—	4	4	no	—
216	D₄	—	4	4	no	—
217	D₄	—	8	4	no	—
218	D₄	—	8	4	no	—
219	D₄	—	8	2	no	—
220	D₄	—	8	2	no	—
221	D₄	—	8	4	no	—
222	D₄	—	8	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	#14579078	1	2	yes	—
225	C₂×C₄	#14579082	1	1	yes	characteristic
226	C₂×C₄	#14579082	1	2	yes	—
227	C₂×C₄	#14579082	1	1	yes	characteristic
228	C₂×C₄	#14579082	1	1	yes	characteristic
229	C₂×C₄	#14579102	1	2	yes	—
230	C₂×C₄	#14579105	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
232	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
233	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	632	2	no	—
237	C₈	#14579082	1	2	yes	—
238	C₈	C₂×#11929609	1	1	yes	characteristic
239	C₈	C₂×#11929612	1	1	yes	characteristic
240	C₂²	—	8	4	no	—
241	C₂²	—	8	4	no	—
242	C₂²	—	8	4	no	—
243	C₂²	—	16	8	no	—
244	C₂²	#19098307	1	2	yes	—
245	C₂²	#19098364	1	2	yes	—
246	C₂²	#19098376	1	2	yes	—
247	C₂²	C₂×#14579082	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	#19098307	1	2	yes	—
251	C₄	#19098364	1	2	yes	—
252	C₄	C₂×#14579078	1	1	yes	characteristic
253	C₄	C₂×#14579082	1	1	yes	characteristic
254	C₄	C₂×#14579102	1	1	yes	characteristic
255	C₄	C₂×#14579105	1	1	yes	characteristic
256	C₂	—	16	4	no	—
257	C₂	#23761446	1	4	yes	direct product factor
258	C₂	C₂×#19098307	1	1	yes	characteristic
259	C₂	C₂×#19098364	1	1	yes	characteristic
260	C₂	C₂×#19098376	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579082
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting, prime radical	C₂×#493×C₇₉	C₂
socle	C₂³×C₇₉	#102
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×#493	D₇₉
hypocenter	C₇₉	C₂×#1361
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	semidirect
2. C₂×#19098316	C₂	semidirect
3. #23761446	C₂	direct (N × H)
8. C₄⋊(C₇₉⋊C₁₆)	C₂²	semidirect
10. #19098316	C₂²	semidirect
127. C₂×C₇₉	#1361	semidirect
135. C₇₉	C₂×#1361	semidirect
257. C₂	#23761446	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₇₉⋊C₁₆)	2
C₂×#19098316	2
#23761446	2
C₂×#493×C₇₉	2
C₂×#1361	79

Subgroups — C2×#23761446 #34199505

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23761446 #34199505