Subgroups — C₂×#23765478 #34442652

Order 20224.

686 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099004	C₂	1	2	yes	maximal
2	C₂×#19098681	C₂	1	1	yes	maximal, characteristic
3	#23765478	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098384	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098379	C₂	1	2	yes	maximal, complemented
6	C₂×#494×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14579183	C₂²	1	1	yes	characteristic
8	C₂×#14579167	—	2	1	no	—
9	#19099004	—	2	4	no	—
10	#19099004	C₂²	1	8	yes	—
11	C₂×#14579107	—	2	2	no	—
12	C₂×#14579104	—	2	2	no	—
13	C₂×#14579101	C₂²	1	2	yes	—
14	#19098681	—	2	2	no	—
15	#19098681	C₂²	1	4	yes	—
16	C₂×#14579097	C₂²	1	1	yes	characteristic
17	C₂×#14579089	—	2	1	no	—
18	#19098384	C₂²	1	4	yes	complemented
19	#19098379	C₂²	1	8	yes	complemented
20	C₂²×Q₁₆×C₇₉	—	2	1	no	—
21	C₂²×Q₁₆×C₇₉	C₂²	1	2	yes	—
22	C₂×C₄⋊Q₈×C₇₉	—	2	1	no	—
23	#494×C₇₉	—	2	2	no	—
24	#494×C₇₉	C₂²	1	4	yes	complemented
25	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂×#11929624	—	2	1	no	—
27	#14579183	—	2	1	no	—
28	#14579183	—	2	1	no	—
29	#14579183	D₄	1	2	yes	—
30	#14579183	C₂³	1	2	yes	—
31	C₂×Dic₃₁₆	D₄	1	1	yes	characteristic
32	C₂×#11929613	—	2	1	no	—
33	C₂×D₆₃₂	D₄	1	1	yes	characteristic
34	#14579167	—	2	4	no	—
35	#14579167	—	4	2	no	—
36	C₂×#11929608	—	2	1	no	—
37	C₂×#11929607	—	4	1	no	—
38	C₂×C₈×D₇₉	—	4	1	no	—
39	#14579107	—	2	8	no	—
40	#14579104	—	2	8	no	—
41	#14579101	—	2	2	no	—
42	#14579101	C₂³	1	4	yes	—
43	C₂×C₇₉⋊C₁₆	—	2	2	no	—
44	#14579097	—	2	1	no	—
45	#14579097	C₂³	1	2	yes	—
46	#14579089	—	2	4	no	—
47	C₂²×Q₈×C₇₉	—	2	2	no	—
48	C₂×Q₁₆×C₇₉	—	2	2	no	—
49	C₂×Q₁₆×C₇₉	—	2	2	no	—
50	C₂×Q₁₆×C₇₉	—	2	2	no	—
51	C₂×Q₁₆×C₇₉	—	2	4	no	—
52	C₂×Q₁₆×C₇₉	—	2	2	no	—
53	C₂×Q₁₆×C₇₉	—	2	2	no	—
54	C₂×Q₁₆×C₇₉	—	4	1	no	—
55	C₂×Q₁₆×C₇₉	—	4	1	no	—
56	C₂×Q₁₆×C₇₉	—	4	2	no	—
57	C₂×Q₁₆×C₇₉	C₂³	1	4	yes	—
58	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₇₉	—	2	4	no	—
61	C₄⋊Q₈×C₇₉	—	4	2	no	—
62	C₂×C₄⋊C₄×C₇₉	—	4	2	no	—
63	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
64	C₄×C₈×C₇₉	D₄	1	2	yes	—
65	C₄×C₈×C₇₉	C₂³	1	2	yes	—
66	#11929624	—	2	2	no	—
67	#11929624	—	2	2	no	—
68	#11929624	—	4	1	no	—
69	#11929624	—	4	1	no	—
70	C₂×Dic₁₅₈	—	2	1	no	—
71	C₂×D₃₁₆	—	2	1	no	—
72	C₂×C₂²⋊D₇₉	—	4	1	no	—
73	C₂×C₄×D₇₉	—	4	1	no	—
74	Dic₃₁₆	—	2	1	no	—
75	Dic₃₁₆	C₂×D₄	1	2	yes	—
76	#11929613	—	2	2	no	—
77	#11929613	—	2	2	no	—
78	D₆₃₂	—	2	1	no	—
79	D₆₃₂	C₂×D₄	1	2	yes	—
80	#11929608	—	2	2	no	—
81	#11929608	—	4	1	no	—
82	#11929607	—	4	4	no	—
83	C₂×C₇₉⋊C₈	—	4	1	no	—
84	C₈×D₇₉	—	4	4	no	—
85	C₇₉⋊C₁₆	—	2	4	no	—
86	C₂×Q₈×C₇₉	—	2	4	no	—
87	C₂×Q₈×C₇₉	—	4	2	no	—
88	C₂×Q₈×C₇₉	—	4	2	no	—
89	C₂×Q₈×C₇₉	—	4	2	no	—
90	C₂×Q₈×C₇₉	—	4	2	no	—
91	C₂×Q₈×C₇₉	—	4	2	no	—
92	C₂²×C₄×C₇₉	—	2	1	no	—
93	C₂²×C₄×C₇₉	—	4	2	no	—
94	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
95	Q₁₆×C₇₉	—	2	4	no	—
96	Q₁₆×C₇₉	—	2	4	no	—
97	Q₁₆×C₇₉	—	4	2	no	—
98	Q₁₆×C₇₉	—	4	2	no	—
99	Q₁₆×C₇₉	—	4	4	no	—
100	Q₁₆×C₇₉	—	4	4	no	—
101	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
102	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
103	C₂×C₈×C₇₉	D₈	1	2	yes	—
104	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
109	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
110	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₇₉	—	4	8	no	—
112	C₄²×C₇₉	—	2	1	no	—
113	C₄²×C₇₉	C₂×D₄	1	2	yes	—
114	C₂³×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂²×D₇₉	—	4	1	no	—
116	Q₈×C₇₉	—	4	4	no	—
117	Q₈×C₇₉	—	4	4	no	—
118	Q₈×C₇₉	—	8	2	no	—
119	Q₈×C₇₉	—	8	2	no	—
120	C₂×C₄×C₇₉	—	2	4	no	—
121	C₂×C₄×C₇₉	—	2	2	no	—
122	C₂×C₄×C₇₉	—	4	4	no	—
123	C₂×C₄×C₇₉	—	8	2	no	—
124	C₂×C₄×C₇₉	—	8	2	no	—
125	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
126	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
131	C₂²⋊D₇₉	—	4	2	no	—
132	C₂²⋊D₇₉	—	4	2	no	—
133	C₂×Dic₇₉	—	4	1	no	—
134	D₃₁₆	—	2	2	no	—
135	D₃₁₆	—	4	1	no	—
136	C₄×D₇₉	—	4	2	no	—
137	C₄×D₇₉	—	4	2	no	—
138	Dic₁₅₈	—	2	2	no	—
139	Dic₁₅₈	—	4	1	no	—
140	C₈×C₇₉	#121	1	1	yes	characteristic
141	C₈×C₇₉	#121	1	1	yes	characteristic
142	C₈×C₇₉	C₂×D₈	1	2	yes	—
143	C₈×C₇₉	C₂×D₈	1	2	yes	—
144	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
145	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
146	C₇₉⋊C₈	—	4	2	no	—
147	C₂²×C₇₉	—	2	2	no	—
148	C₂²×C₇₉	#459	1	1	yes	characteristic
149	C₂²×C₇₉	#459	1	1	yes	characteristic
150	C₂²×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×D₇₉	—	4	2	no	—
152	C₂×D₇₉	—	8	2	no	—
153	C₄×C₇₉	—	2	4	no	—
154	C₄×C₇₉	—	8	4	no	—
155	C₄×C₇₉	#459	1	1	yes	characteristic
156	C₄×C₇₉	#459	1	1	yes	characteristic
157	C₄×C₇₉	C₂×#121	1	1	yes	characteristic
158	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
159	Dic₇₉	—	4	2	no	—
160	C₂×#2214	—	79	1	no	maximal, Sylow
161	C₂×C₇₉	—	2	1	no	—
162	C₂×C₇₉	—	2	1	no	—
163	C₂×C₇₉	#2214	1	2	yes	complemented
164	C₂×C₇₉	C₂×#459	1	1	yes	characteristic
165	D₇₉	—	8	2	no	—
166	C₂×#510	—	79	2	no	—
167	C₂×#494	D₇₉	1	1	yes	characteristic, complemented
168	C₂×#443	—	79	1	no	—
169	#2214	—	79	8	no	—
170	C₂×#364	—	79	1	no	—
171	C₂×#362	—	79	2	no	—
172	C₇₉	C₂×#2214	1	1	yes	characteristic, complemented, Sylow
173	C₂×#135	—	79	1	no	—
174	C₂²×Q₁₆	—	2	1	no	—
175	C₂²×Q₁₆	C₂×D₇₉	1	2	yes	—
176	C₂×#131	—	158	1	no	—
177	C₂×C₄⋊Q₈	—	2	1	no	—
178	#510	—	79	8	no	—
179	#510	—	158	4	no	—
180	C₂×Q₃₂	—	158	2	no	—
181	C₂×SD₃₂	—	158	2	no	—
182	C₂×M₃₂	—	79	2	no	—
183	#494	—	2	2	no	—
184	#494	C₂×D₇₉	1	4	yes	complemented
185	#443	—	79	4	no	—
186	#443	—	158	2	no	—
187	C₂×#108	—	79	1	no	—
188	C₂×#104	—	158	1	no	—
189	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
190	#364	—	79	4	no	—
191	#362	—	79	8	no	—
192	C₂×#41	—	158	1	no	—
193	C₂²×Q₈	—	2	2	no	—
194	#135	—	79	2	no	—
195	#135	—	79	2	no	—
196	#135	—	158	1	no	—
197	#135	—	158	1	no	—
198	C₂×Q₁₆	—	2	2	no	—
199	C₂×Q₁₆	—	2	2	no	—
200	C₂×Q₁₆	—	2	4	no	—
201	C₂×Q₁₆	—	2	2	no	—
202	C₂×Q₁₆	—	2	2	no	—
203	C₂×Q₁₆	—	2	2	no	—
204	C₂×Q₁₆	—	4	1	no	—
205	C₂×Q₁₆	—	4	1	no	—
206	C₂×Q₁₆	—	4	2	no	—
207	C₂×Q₁₆	—	79	1	no	—
208	C₂×Q₁₆	C₂²×D₇₉	1	4	yes	—
209	C₂×SD₁₆	—	158	1	no	—
210	C₂×D₈	—	79	1	no	—
211	#131	—	316	2	no	—
212	#131	—	158	4	no	—
213	C₂×M₁₆	—	316	1	no	—
214	C₂×M₁₆	—	158	1	no	—
215	C₂²×C₈	—	316	1	no	—
216	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
217	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
218	C₄⋊Q₈	—	2	4	no	—
219	C₄⋊Q₈	—	4	2	no	—
220	C₂×C₄⋊C₄	—	4	2	no	—
221	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
222	Q₃₂	—	158	8	no	—
223	SD₃₂	—	158	8	no	—
224	M₃₂	—	79	4	no	—
225	M₃₂	—	158	2	no	—
226	C₂×C₁₆	—	158	2	no	—
227	#108	—	79	2	no	—
228	#108	—	158	1	no	—
229	#104	—	158	4	no	—
230	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
231	C₄×C₈	C₂²×D₇₉	1	2	yes	—
232	#41	—	316	1	no	—
233	#41	—	316	1	no	—
234	#41	—	158	2	no	—
235	#41	—	158	2	no	—
236	C₂×Q₈	—	2	4	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	158	1	no	—
243	C₂×D₄	—	316	1	no	—
244	C₂×D₄	—	158	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	—	316	1	no	—
248	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
249	Q₁₆	—	2	4	no	—
250	Q₁₆	—	2	4	no	—
251	Q₁₆	—	4	2	no	—
252	Q₁₆	—	4	2	no	—
253	Q₁₆	—	4	4	no	—
254	Q₁₆	—	4	4	no	—
255	Q₁₆	—	79	2	no	—
256	Q₁₆	—	158	1	no	—
257	SD₁₆	—	158	2	no	—
258	SD₁₆	—	158	2	no	—
259	D₈	—	79	2	no	—
260	D₈	—	158	1	no	—
261	M₁₆	—	316	1	no	—
262	M₁₆	—	316	4	no	—
263	M₁₆	—	158	2	no	—
264	C₂×C₈	—	316	1	no	—
265	C₂×C₈	—	316	4	no	—
266	C₂×C₈	#11929609	1	1	yes	characteristic
267	C₂×C₈	#11929609	1	1	yes	characteristic
268	C₂×C₈	#11929609	1	2	yes	—
269	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
270	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
271	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
273	C₂×C₈	#11929623	1	1	yes	characteristic
274	C₂×C₈	#11929623	1	1	yes	characteristic
275	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
276	C₄⋊C₄	—	4	8	no	—
277	C₄²	—	2	1	no	—
278	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
279	C₁₆	—	158	4	no	—
280	C₂³	—	316	1	no	—
281	C₂³	#14579187	1	1	yes	characteristic
282	Q₈	—	4	4	no	—
283	Q₈	—	4	4	no	—
284	Q₈	—	8	2	no	—
285	Q₈	—	8	2	no	—
286	Q₈	—	316	1	no	—
287	Q₈	—	158	2	no	—
288	D₄	—	316	2	no	—
289	D₄	—	316	2	no	—
290	D₄	—	316	1	no	—
291	D₄	—	158	2	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	8	2	no	—
296	C₂×C₄	—	8	2	no	—
297	C₂×C₄	—	316	1	no	—
298	C₂×C₄	—	316	2	no	—
299	C₂×C₄	—	316	2	no	—
300	C₂×C₄	#14579132	1	1	yes	characteristic
301	C₂×C₄	#14579132	1	1	yes	characteristic
302	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
303	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
304	C₂×C₄	#14579187	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
306	C₈	—	316	2	no	—
307	C₈	#14579132	1	1	yes	characteristic
308	C₈	#14579132	1	1	yes	characteristic
309	C₈	C₂×#11929609	1	2	yes	—
310	C₈	C₂×#11929609	1	2	yes	—
311	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
312	C₈	C₂×#11929623	1	1	yes	characteristic
313	C₂²	—	2	2	no	—
314	C₂²	—	316	2	no	—
315	C₂²	—	632	2	no	—
316	C₂²	#19098755	1	1	yes	characteristic
317	C₂²	#19098755	1	1	yes	characteristic
318	C₂²	C₂×#14579187	1	1	yes	characteristic
319	C₄	—	2	4	no	—
320	C₄	—	8	4	no	—
321	C₄	—	316	2	no	—
322	C₄	#19098755	1	1	yes	characteristic
323	C₄	#19098755	1	1	yes	characteristic
324	C₄	C₂×#14579132	1	1	yes	characteristic
325	C₄	C₂²×#11929609	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	632	2	no	—
329	C₂	#23765478	1	2	yes	direct product factor
330	C₂	C₂×#19098755	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098755
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#494×C₇₉	C₂
socle	C₂²×C₇₉	#459
prime radical	C₂×#14579183	C₂²
hypercenter	C₂×#494	D₇₉
hypocenter	C₇₉	C₂×#2214
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #23765478	C₂	direct (N × H)
4. C₂×#19098384	C₂	semidirect
5. C₂×#19098379	C₂	semidirect
6. C₂×#494×C₇₉	C₂	semidirect
18. #19098384	C₂²	semidirect
19. #19098379	C₂²	semidirect
24. #494×C₇₉	C₂²	semidirect
163. C₂×C₇₉	#2214	semidirect
167. C₂×#494	D₇₉	semidirect
172. C₇₉	C₂×#2214	semidirect
184. #494	C₂×D₇₉	semidirect
329. C₂	#23765478	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×#19099004	2
C₂×#19098681	2
#23765478	2
C₂×#19098384	2
C₂×#19098379	2
C₂×#494×C₇₉	2
C₂×#2214	79

Subgroups — C2×#23765478 #34442652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765478 #34442652