Subgroups — C₂×#23761436 #34199472

Order 20224.

494 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098940	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098384	C₂	1	1	yes	maximal, characteristic, complemented
3	#23761436	C₂	1	4	yes	maximal, direct product factor
4	C₂×#364×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₆₃₂	—	2	1	no	—
6	C₂²×D₆₃₂	—	2	1	no	—
7	C₂×#14579158	C₄	1	2	yes	—
8	C₂×#14579101	—	2	1	no	—
9	#19098940	—	2	1	no	—
10	#19098940	—	2	1	no	—
11	#19098940	C₄	1	2	yes	—
12	#19098940	C₂²	1	2	yes	—
13	#19098384	—	2	1	no	—
14	#19098384	C₂²	1	2	yes	complemented
15	C₂×M₃₂×C₇₉	—	2	1	no	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	#364×C₇₉	—	2	1	no	—
18	#364×C₇₉	C₂²	1	2	yes	complemented
19	C₂²×D₃₁₆	—	2	2	no	—
20	C₂²×D₃₁₆	—	4	2	no	—
21	C₂×D₆₃₂	—	2	2	no	—
22	C₂×D₆₃₂	—	2	2	no	—
23	C₂×D₆₃₂	—	2	2	no	—
24	C₂×D₆₃₂	—	2	2	no	—
25	C₂×D₆₃₂	—	4	1	no	—
26	C₂×D₆₃₂	—	4	1	no	—
27	C₂×D₆₃₂	—	4	1	no	—
28	C₂×D₆₃₂	—	4	1	no	—
29	C₂×D₆₃₂	—	4	1	no	—
30	C₂×D₆₃₂	—	4	1	no	—
31	C₂×D₆₃₂	—	4	1	no	—
32	C₂×D₆₃₂	—	4	1	no	—
33	#14579158	—	2	2	no	—
34	#14579158	—	4	2	no	—
35	#14579158	C₂×C₄	1	4	yes	—
36	#14579101	—	2	2	no	—
37	#14579101	—	4	1	no	—
38	C₂×C₇₉⋊C₁₆	—	4	1	no	—
39	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
42	M₃₂×C₇₉	—	2	2	no	—
43	M₃₂×C₇₉	—	4	1	no	—
44	C₂×C₁₆×C₇₉	—	4	1	no	—
45	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
48	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
49	C₂³×D₇₉	—	4	2	no	—
50	C₂×D₃₁₆	—	2	4	no	—
51	C₂×D₃₁₆	—	4	4	no	—
52	C₂×D₃₁₆	—	4	4	no	—
53	C₂×D₃₁₆	—	4	2	no	—
54	C₂×D₃₁₆	—	4	2	no	—
55	C₂×D₃₁₆	—	4	2	no	—
56	C₂×D₃₁₆	—	4	2	no	—
57	C₂×D₃₁₆	—	4	2	no	—
58	C₂×D₃₁₆	—	8	8	no	—
59	D₆₃₂	—	4	2	no	—
60	D₆₃₂	—	4	2	no	—
61	D₆₃₂	—	4	2	no	—
62	D₆₃₂	—	4	2	no	—
63	D₆₃₂	—	4	2	no	—
64	D₆₃₂	—	4	2	no	—
65	D₆₃₂	—	4	2	no	—
66	D₆₃₂	—	4	2	no	—
67	C₇₉⋊C₁₆	—	4	2	no	—
68	C₂²×C₄×C₇₉	—	2	1	no	—
69	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
73	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
74	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
75	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
76	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
77	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
78	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
79	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
80	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₇₉	—	2	1	no	—
83	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
85	C₁₆×C₇₉	—	4	2	no	—
86	C₂³×C₇₉	#100	1	1	yes	characteristic
87	C₂²×D₇₉	—	4	4	no	—
88	C₂²×D₇₉	—	8	8	no	—
89	C₂²×D₇₉	—	8	2	no	—
90	C₂²×D₇₉	—	8	2	no	—
91	C₂×C₄×C₇₉	—	2	4	no	—
92	C₂×C₄×C₇₉	—	2	1	no	—
93	C₂×C₄×C₇₉	—	2	1	no	—
94	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
95	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
96	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
97	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
98	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	D₃₁₆	—	4	4	no	—
101	D₃₁₆	—	4	4	no	—
102	D₃₁₆	—	8	8	no	—
103	D₃₁₆	—	8	8	no	—
104	D₃₁₆	—	8	2	no	—
105	D₃₁₆	—	8	2	no	—
106	C₈×C₇₉	#102	1	1	yes	characteristic
107	C₈×C₇₉	#102	1	1	yes	characteristic
108	C₈×C₇₉	#102	1	1	yes	characteristic
109	C₈×C₇₉	#102	1	1	yes	characteristic
110	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
111	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
112	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
113	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
114	C₂²×C₇₉	—	2	2	no	—
115	C₂²×C₇₉	#331	1	1	yes	characteristic
116	C₂²×C₇₉	#331	1	1	yes	characteristic
117	C₂²×C₇₉	C₂×#100	1	1	yes	characteristic
118	C₂×D₇₉	—	8	8	no	—
119	C₂×D₇₉	—	8	4	no	—
120	C₂×D₇₉	—	16	4	no	—
121	C₂×D₇₉	—	16	4	no	—
122	C₄×C₇₉	—	2	2	no	—
123	C₄×C₇₉	—	2	2	no	—
124	C₄×C₇₉	#331	1	1	yes	characteristic
125	C₄×C₇₉	#331	1	1	yes	characteristic
126	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
127	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
128	C₂×#1357	—	79	1	no	maximal, Sylow
129	C₂×C₇₉	—	2	1	no	—
130	C₂×C₇₉	—	2	1	no	—
131	C₂×C₇₉	#1357	1	2	yes	complemented
132	C₂×C₇₉	C₂×#331	1	1	yes	characteristic
133	D₇₉	—	16	4	no	—
134	C₂×#493	—	79	1	no	—
135	C₂×#364	—	79	1	no	—
136	C₂×#364	D₇₉	1	1	yes	characteristic, complemented
137	#1357	—	79	4	no	—
138	C₇₉	C₂×#1357	1	1	yes	characteristic, complemented, Sylow
139	C₂²×D₈	—	158	1	no	—
140	C₂²×D₈	—	158	1	no	—
141	C₂×#127	—	79	2	no	—
142	C₂×M₃₂	—	2	1	no	—
143	C₂×M₃₂	—	158	1	no	—
144	#493	—	79	2	no	—
145	#493	—	79	2	no	—
146	#493	—	158	1	no	—
147	#493	—	158	1	no	—
148	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
149	#364	—	2	1	no	—
150	#364	—	79	2	no	—
151	#364	—	158	1	no	—
152	#364	C₂×D₇₉	1	2	yes	complemented
153	C₂²×D₄	—	316	2	no	—
154	C₂²×D₄	—	158	2	no	—
155	C₂×D₈	—	316	1	no	—
156	C₂×D₈	—	316	1	no	—
157	C₂×D₈	—	316	1	no	—
158	C₂×D₈	—	316	1	no	—
159	C₂×D₈	—	316	1	no	—
160	C₂×D₈	—	316	1	no	—
161	C₂×D₈	—	316	1	no	—
162	C₂×D₈	—	316	1	no	—
163	C₂×D₈	—	158	2	no	—
164	C₂×D₈	—	158	2	no	—
165	C₂×D₈	—	158	2	no	—
166	C₂×D₈	—	158	2	no	—
167	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
168	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
169	#127	—	316	2	no	—
170	#127	—	79	4	no	—
171	#127	—	158	2	no	—
172	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
173	M₃₂	—	2	2	no	—
174	M₃₂	—	4	1	no	—
175	M₃₂	—	316	1	no	—
176	M₃₂	—	158	2	no	—
177	C₂×C₁₆	—	4	1	no	—
178	C₂×C₁₆	—	316	1	no	—
179	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
180	C₄×C₈	D₃₁₆	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
182	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
183	C₂⁴	—	316	2	no	—
184	C₂×D₄	—	316	4	no	—
185	C₂×D₄	—	316	4	no	—
186	C₂×D₄	—	316	2	no	—
187	C₂×D₄	—	316	2	no	—
188	C₂×D₄	—	316	2	no	—
189	C₂×D₄	—	316	2	no	—
190	C₂×D₄	—	316	2	no	—
191	C₂×D₄	—	632	8	no	—
192	C₂×D₄	—	158	4	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
195	D₈	—	316	2	no	—
196	D₈	—	316	2	no	—
197	D₈	—	316	2	no	—
198	D₈	—	316	2	no	—
199	D₈	—	316	2	no	—
200	D₈	—	316	2	no	—
201	D₈	—	316	2	no	—
202	D₈	—	316	2	no	—
203	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
204	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
205	C₂×C₈	#11929609	1	1	yes	characteristic
206	C₂×C₈	#11929609	1	1	yes	characteristic
207	C₂×C₈	D₆₃₂	1	1	yes	characteristic
208	C₂×C₈	D₆₃₂	1	1	yes	characteristic
209	C₂×C₈	#11929611	1	1	yes	characteristic
210	C₂×C₈	#11929611	1	1	yes	characteristic
211	C₂×C₈	#11929613	1	1	yes	characteristic
212	C₂×C₈	#11929613	1	1	yes	characteristic
213	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
214	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
215	C₄²	—	2	1	no	—
216	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
217	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
218	C₁₆	—	4	2	no	—
219	C₁₆	—	316	2	no	—
220	C₂³	—	316	4	no	—
221	C₂³	—	632	8	no	—
222	C₂³	—	632	2	no	—
223	C₂³	—	632	2	no	—
224	C₂³	#14579077	1	1	yes	characteristic
225	D₄	—	316	4	no	—
226	D₄	—	316	4	no	—
227	D₄	—	632	8	no	—
228	D₄	—	632	8	no	—
229	D₄	—	632	2	no	—
230	D₄	—	632	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	#14579077	1	1	yes	characteristic
235	C₂×C₄	#14579081	1	1	yes	characteristic
236	C₂×C₄	#14579081	1	1	yes	characteristic
237	C₂×C₄	#14579083	1	1	yes	characteristic
238	C₂×C₄	#14579083	1	1	yes	characteristic
239	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
240	C₈	#14579081	1	1	yes	characteristic
241	C₈	#14579081	1	1	yes	characteristic
242	C₈	#14579083	1	1	yes	characteristic
243	C₈	#14579083	1	1	yes	characteristic
244	C₈	C₂×#11929609	1	1	yes	characteristic
245	C₈	C₂×D₆₃₂	1	1	yes	characteristic
246	C₈	C₂×#11929611	1	1	yes	characteristic
247	C₈	C₂×#11929613	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	632	8	no	—
250	C₂²	—	632	4	no	—
251	C₂²	—	1264	4	no	—
252	C₂²	—	1264	4	no	—
253	C₂²	#19098306	1	1	yes	characteristic
254	C₂²	#19098306	1	1	yes	characteristic
255	C₂²	C₂×#14579077	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	#19098306	1	1	yes	characteristic
259	C₄	#19098306	1	1	yes	characteristic
260	C₄	C₂×#14579081	1	1	yes	characteristic
261	C₄	C₂×#14579083	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	1264	4	no	—
265	C₂	#23761436	1	2	yes	direct product factor
266	C₂	C₂×#19098306	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098306
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₂×#364×C₇₉	C₂
socle	C₂²×C₇₉	#331
prime radical	C₂×#19098940	C₂
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×#364	D₇₉
hypocenter	C₇₉	C₂×#1357
LCS	G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. C₂×#19098384	C₂	semidirect
3. #23761436	C₂	direct (N × H)
4. C₂×#364×C₇₉	C₂	semidirect
14. #19098384	C₂²	semidirect
18. #364×C₇₉	C₂²	semidirect
131. C₂×C₇₉	#1357	semidirect
136. C₂×#364	D₇₉	semidirect
138. C₇₉	C₂×#1357	semidirect
152. #364	C₂×D₇₉	semidirect
265. C₂	#23761436	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×#19098940	2
C₂×#19098384	2
#23761436	2
C₂×#364×C₇₉	2
C₂×#1357	79

Subgroups — C2×#23761436 #34199472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23761436 #34199472