Subgroups — C₂×#23761374 #34198053

Order 20224.

414 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098681	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098384	C₂	1	1	yes	maximal, characteristic, complemented
3	#23761374	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₈²×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14579183	C₄	1	1	yes	characteristic
6	C₂×#14579167	—	2	1	no	—
7	C₂×#14579101	—	2	1	no	—
8	#19098681	—	2	1	no	—
9	#19098681	—	2	1	no	—
10	#19098681	C₄	1	2	yes	—
11	#19098681	C₂²	1	2	yes	—
12	C₂×#14579097	C₄	1	1	yes	characteristic
13	C₂×#14579089	—	2	1	no	—
14	#19098384	—	2	1	no	—
15	#19098384	C₂²	1	2	yes	complemented
16	C₂×C₄×C₈×C₇₉	—	2	1	no	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₈²×C₇₉	—	2	1	no	—
19	C₈²×C₇₉	C₂²	1	2	yes	complemented
20	C₂×#11929624	—	2	1	no	—
21	#14579183	—	2	1	no	—
22	#14579183	—	2	1	no	—
23	#14579183	C₈	1	2	yes	complemented
24	#14579183	C₂×C₄	1	2	yes	—
25	C₂×Dic₃₁₆	C₈	1	1	yes	characteristic, complemented
26	C₂×#11929613	—	2	1	no	—
27	C₂×D₆₃₂	C₈	1	1	yes	characteristic, complemented
28	#14579167	—	2	2	no	—
29	#14579167	—	2	2	no	—
30	#14579167	—	4	1	no	—
31	#14579167	—	4	1	no	—
32	C₂×#11929608	—	2	1	no	—
33	C₂×#11929607	—	4	1	no	—
34	C₂×C₈×D₇₉	—	4	1	no	—
35	#14579101	—	2	2	no	—
36	#14579101	—	4	1	no	—
37	C₂×C₇₉⋊C₁₆	—	4	1	no	—
38	#14579097	—	2	1	no	—
39	#14579097	C₂×C₄	1	2	yes	—
40	#14579089	—	2	2	no	—
41	#14579089	—	2	2	no	—
42	C₂²×C₈×C₇₉	—	2	2	no	—
43	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₇₉	—	2	2	no	—
47	C₄×C₈×C₇₉	—	2	2	no	—
48	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
52	#11929624	—	2	2	no	—
53	#11929624	—	2	2	no	—
54	#11929624	—	4	1	no	—
55	#11929624	—	4	1	no	—
56	C₂×Dic₁₅₈	—	2	1	no	—
57	C₂×D₃₁₆	—	2	1	no	—
58	C₂×C₂²⋊D₇₉	—	4	1	no	—
59	C₂×C₄×D₇₉	—	4	1	no	—
60	Dic₃₁₆	—	2	1	no	—
61	Dic₃₁₆	C₂×C₈	1	2	yes	complemented
62	#11929613	—	2	2	no	—
63	#11929613	—	2	2	no	—
64	D₆₃₂	—	2	1	no	—
65	D₆₃₂	C₂×C₈	1	2	yes	complemented
66	#11929608	—	2	2	no	—
67	#11929608	—	4	1	no	—
68	#11929607	—	4	2	no	—
69	#11929607	—	4	2	no	—
70	C₂×C₇₉⋊C₈	—	4	1	no	—
71	C₈×D₇₉	—	4	2	no	—
72	C₈×D₇₉	—	4	2	no	—
73	C₇₉⋊C₁₆	—	4	2	no	—
74	C₂²×C₄×C₇₉	—	2	1	no	—
75	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₇₉	—	2	4	no	—
77	C₂×C₈×C₇₉	—	2	8	no	—
78	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
81	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
82	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
83	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
84	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
85	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
86	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
87	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
88	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₇₉	—	2	1	no	—
91	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
93	C₂³×C₇₉	#104	1	1	yes	characteristic
94	C₂²×D₇₉	—	4	1	no	—
95	C₂×C₄×C₇₉	—	2	4	no	—
96	C₂×C₄×C₇₉	—	2	1	no	—
97	C₂×C₄×C₇₉	—	2	1	no	—
98	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
101	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
102	C₂×C₄×C₇₉	#104	1	1	yes	characteristic
103	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂²⋊D₇₉	—	4	2	no	—
105	C₂²⋊D₇₉	—	4	2	no	—
106	C₂×Dic₇₉	—	4	1	no	—
107	D₃₁₆	—	2	2	no	—
108	D₃₁₆	—	4	1	no	—
109	C₄×D₇₉	—	4	2	no	—
110	C₄×D₇₉	—	4	2	no	—
111	Dic₁₅₈	—	2	2	no	—
112	Dic₁₅₈	—	4	1	no	—
113	C₈×C₇₉	—	2	8	no	—
114	C₈×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
115	C₈×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
116	C₈×C₇₉	#102	1	1	yes	characteristic
117	C₈×C₇₉	#102	1	1	yes	characteristic
118	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
119	C₈×C₇₉	C₂×M₁₆	1	1	yes	characteristic
120	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
121	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
122	C₇₉⋊C₈	—	4	2	no	—
123	C₂²×C₇₉	—	2	2	no	—
124	C₂²×C₇₉	D₄⋊C₈	1	1	yes	characteristic
125	C₂²×C₇₉	D₄⋊C₈	1	1	yes	characteristic
126	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
127	C₂×D₇₉	—	4	2	no	—
128	C₂×D₇₉	—	8	2	no	—
129	C₄×C₇₉	—	2	2	no	—
130	C₄×C₇₉	—	2	2	no	—
131	C₄×C₇₉	D₄⋊C₈	1	1	yes	characteristic
132	C₄×C₇₉	D₄⋊C₈	1	1	yes	characteristic
133	C₄×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
135	Dic₇₉	—	4	2	no	—
136	C₂×#1335	—	79	1	no	maximal, Sylow
137	C₂×C₇₉	—	2	1	no	—
138	C₂×C₇₉	—	2	1	no	—
139	C₂×C₇₉	#1335	1	2	yes	complemented
140	C₂×C₇₉	C₂×D₄⋊C₈	1	1	yes	characteristic
141	D₇₉	—	8	2	no	—
142	C₂×#443	—	79	1	no	—
143	C₂×#364	—	79	1	no	—
144	C₂×C₈²	D₇₉	1	1	yes	characteristic, complemented
145	#1335	—	79	4	no	—
146	C₇₉	C₂×#1335	1	1	yes	characteristic, complemented, Sylow
147	C₂×#135	—	79	1	no	—
148	C₂×#131	—	158	1	no	—
149	C₂×M₃₂	—	158	1	no	—
150	#443	—	79	2	no	—
151	#443	—	79	2	no	—
152	#443	—	158	1	no	—
153	#443	—	158	1	no	—
154	C₂×#108	—	79	1	no	—
155	C₂×#104	—	158	1	no	—
156	C₂×C₄×C₈	—	2	1	no	—
157	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
158	#364	—	79	2	no	—
159	#364	—	158	1	no	—
160	C₈²	—	2	1	no	—
161	C₈²	C₂×D₇₉	1	2	yes	complemented
162	C₂×#41	—	158	1	no	—
163	#135	—	79	2	no	—
164	#135	—	79	2	no	—
165	#135	—	158	1	no	—
166	#135	—	158	1	no	—
167	C₂×Q₁₆	—	79	1	no	—
168	C₂×SD₁₆	—	158	1	no	—
169	C₂×D₈	—	79	1	no	—
170	#131	—	316	1	no	—
171	#131	—	316	1	no	—
172	#131	—	158	2	no	—
173	#131	—	158	2	no	—
174	C₂×M₁₆	—	316	1	no	—
175	C₂×M₁₆	—	158	1	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	316	1	no	—
178	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
179	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
181	M₃₂	—	316	1	no	—
182	M₃₂	—	158	2	no	—
183	C₂×C₁₆	—	316	1	no	—
184	#108	—	79	2	no	—
185	#108	—	158	1	no	—
186	#104	—	158	2	no	—
187	#104	—	158	2	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
191	C₄×C₈	D₃₁₆	1	1	yes	characteristic
192	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
194	#41	—	316	1	no	—
195	#41	—	316	1	no	—
196	#41	—	158	2	no	—
197	#41	—	158	2	no	—
198	C₂×Q₈	—	158	1	no	—
199	C₂×D₄	—	316	1	no	—
200	C₂×D₄	—	158	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	316	1	no	—
203	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
204	Q₁₆	—	79	2	no	—
205	Q₁₆	—	158	1	no	—
206	SD₁₆	—	158	2	no	—
207	SD₁₆	—	158	2	no	—
208	D₈	—	79	2	no	—
209	D₈	—	158	1	no	—
210	M₁₆	—	316	1	no	—
211	M₁₆	—	316	2	no	—
212	M₁₆	—	316	2	no	—
213	M₁₆	—	158	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	8	no	—
216	C₂×C₈	—	316	1	no	—
217	C₂×C₈	—	316	2	no	—
218	C₂×C₈	—	316	2	no	—
219	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
220	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
221	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
222	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
223	C₂×C₈	#11929607	1	1	yes	characteristic
224	C₂×C₈	#11929607	1	1	yes	characteristic
225	C₂×C₈	D₆₃₂	1	1	yes	characteristic
226	C₂×C₈	D₆₃₂	1	1	yes	characteristic
227	C₂×C₈	#11929613	1	1	yes	characteristic
228	C₂×C₈	#11929613	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
230	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
231	C₄²	—	2	1	no	—
232	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
233	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
234	C₁₆	—	316	2	no	—
235	C₂³	—	316	1	no	—
236	C₂³	#14579089	1	1	yes	characteristic
237	Q₈	—	316	1	no	—
238	Q₈	—	158	2	no	—
239	D₄	—	316	2	no	—
240	D₄	—	316	2	no	—
241	D₄	—	316	1	no	—
242	D₄	—	158	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	316	1	no	—
247	C₂×C₄	—	316	2	no	—
248	C₂×C₄	—	316	2	no	—
249	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
250	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
251	C₂×C₄	#14579083	1	1	yes	characteristic
252	C₂×C₄	#14579083	1	1	yes	characteristic
253	C₂×C₄	#14579089	1	1	yes	characteristic
254	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
255	C₈	—	2	8	no	—
256	C₈	—	316	2	no	—
257	C₈	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
258	C₈	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
259	C₈	#14579083	1	1	yes	characteristic
260	C₈	#14579083	1	1	yes	characteristic
261	C₈	C₂×C₈×D₇₉	1	1	yes	characteristic
262	C₈	C₂×#11929607	1	1	yes	characteristic
263	C₈	C₂×D₆₃₂	1	1	yes	characteristic
264	C₈	C₂×#11929613	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	316	2	no	—
267	C₂²	—	632	2	no	—
268	C₂²	#19098292	1	1	yes	characteristic
269	C₂²	#19098292	1	1	yes	characteristic
270	C₂²	C₂×#14579089	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	316	2	no	—
274	C₄	#19098292	1	1	yes	characteristic
275	C₄	#19098292	1	1	yes	characteristic
276	C₄	C₂×(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
277	C₄	C₂×#14579083	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	632	2	no	—
281	C₂	#23761374	1	2	yes	direct product factor
282	C₂	C₂×#19098292	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₆₃₂
commutator	C₈×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₂×C₈²×C₇₉	C₂
socle	C₂²×C₇₉	D₄⋊C₈
prime radical	C₂×#14579183	C₄
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×C₈²	D₇₉
hypocenter	C₇₉	C₂×#1335
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. C₂×#19098384	C₂	semidirect
3. #23761374	C₂	direct (N × H)
4. C₂×C₈²×C₇₉	C₂	semidirect
15. #19098384	C₂²	semidirect
19. C₈²×C₇₉	C₂²	semidirect
23. #14579183	C₈	semidirect
25. C₂×Dic₃₁₆	C₈	semidirect
27. C₂×D₆₃₂	C₈	semidirect
61. Dic₃₁₆	C₂×C₈	semidirect
65. D₆₃₂	C₂×C₈	semidirect
139. C₂×C₇₉	#1335	semidirect
144. C₂×C₈²	D₇₉	semidirect
146. C₇₉	C₂×#1335	semidirect
161. C₈²	C₂×D₇₉	semidirect
281. C₂	#23761374	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×#19098681	2
C₂×#19098384	2
#23761374	2
C₂×C₈²×C₇₉	2
C₂×#1335	79

Subgroups — C2×#23761374 #34198053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23761374 #34198053