Subgroups — C₂×#23765476 #34442649

Order 20224.

790 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098998	C₂	1	2	yes	maximal, complemented
2	C₂×#19098940	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765476	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098384	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098375	C₂	1	2	yes	maximal, complemented
6	C₂×#443×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₆₃₂	—	2	1	no	—
8	C₂²×D₆₃₂	C₂²	1	2	yes	—
9	C₂×#14579158	—	2	1	no	—
10	#19098998	—	2	4	no	—
11	#19098998	C₂²	1	8	yes	complemented
12	C₂×#14579104	—	2	2	no	—
13	C₂×#14579102	—	2	2	no	—
14	C₂×#14579101	C₂²	1	2	yes	—
15	#19098940	—	2	2	no	—
16	#19098940	C₂²	1	4	yes	complemented
17	#19098384	C₂²	1	4	yes	complemented
18	#19098375	C₂²	1	8	yes	complemented
19	C₂×#135×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×#131×C₇₉	—	2	1	no	—
21	#443×C₇₉	—	2	2	no	—
22	#443×C₇₉	C₂²	1	4	yes	complemented
23	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
24	C₂×#104×C₇₉	—	2	1	no	—
25	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂²×D₃₁₆	—	2	2	no	—
27	C₂²×D₃₁₆	—	4	2	no	—
28	C₂×D₆₃₂	—	2	2	no	—
29	C₂×D₆₃₂	—	2	2	no	—
30	C₂×D₆₃₂	—	2	2	no	—
31	C₂×D₆₃₂	—	2	2	no	—
32	C₂×D₆₃₂	—	2	2	no	—
33	C₂×D₆₃₂	—	2	4	no	—
34	C₂×D₆₃₂	—	4	1	no	—
35	C₂×D₆₃₂	—	4	1	no	—
36	C₂×D₆₃₂	—	4	2	no	—
37	C₂×D₆₃₂	C₂³	1	4	yes	—
38	#14579158	—	2	4	no	—
39	#14579158	—	4	2	no	—
40	#14579104	—	2	8	no	—
41	#14579102	—	2	8	no	—
42	#14579101	—	2	2	no	—
43	#14579101	C₂³	1	4	yes	—
44	C₂×C₇₉⋊C₁₆	—	2	2	no	—
45	C₂×#41×C₇₉	—	2	1	no	—
46	#135×C₇₉	—	2	1	no	—
47	#135×C₇₉	—	2	1	no	—
48	#135×C₇₉	D₄	1	2	yes	complemented
49	#135×C₇₉	C₂³	1	2	yes	—
50	C₂×Q₁₆×C₇₉	D₄	1	1	yes	characteristic, complemented
51	C₂×SD₁₆×C₇₉	—	2	1	no	—
52	C₂×D₈×C₇₉	D₄	1	1	yes	characteristic, complemented
53	#131×C₇₉	—	2	4	no	—
54	#131×C₇₉	—	4	2	no	—
55	C₂×M₁₆×C₇₉	—	2	1	no	—
56	C₂×M₁₆×C₇₉	—	4	1	no	—
57	C₂²×C₈×C₇₉	—	4	1	no	—
58	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
61	#108×C₇₉	—	2	1	no	—
62	#108×C₇₉	C₂³	1	2	yes	—
63	#104×C₇₉	—	2	4	no	—
64	C₄×C₈×C₇₉	D₄	1	2	yes	—
65	C₄×C₈×C₇₉	C₂³	1	2	yes	—
66	C₂³×D₇₉	—	4	2	no	—
67	C₂×D₃₁₆	—	2	4	no	—
68	C₂×D₃₁₆	—	4	8	no	—
69	C₂×D₃₁₆	—	4	2	no	—
70	C₂×D₃₁₆	—	4	2	no	—
71	C₂×D₃₁₆	—	4	2	no	—
72	C₂×D₃₁₆	—	4	2	no	—
73	C₂×D₃₁₆	—	4	2	no	—
74	C₂×D₃₁₆	—	8	8	no	—
75	D₆₃₂	—	2	4	no	—
76	D₆₃₂	—	2	4	no	—
77	D₆₃₂	—	4	2	no	—
78	D₆₃₂	—	4	2	no	—
79	D₆₃₂	—	4	4	no	—
80	D₆₃₂	—	4	4	no	—
81	C₇₉⋊C₁₆	—	2	4	no	—
82	#41×C₇₉	—	2	2	no	—
83	#41×C₇₉	—	2	2	no	—
84	#41×C₇₉	—	4	1	no	—
85	#41×C₇₉	—	4	1	no	—
86	C₂×Q₈×C₇₉	—	2	1	no	—
87	C₂×D₄×C₇₉	—	2	1	no	—
88	C₂×D₄×C₇₉	—	4	1	no	—
89	C₂²×C₄×C₇₉	—	2	1	no	—
90	C₂²×C₄×C₇₉	—	4	1	no	—
91	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₇₉	—	2	1	no	—
93	Q₁₆×C₇₉	C₂×D₄	1	2	yes	complemented
94	SD₁₆×C₇₉	—	2	2	no	—
95	SD₁₆×C₇₉	—	2	2	no	—
96	D₈×C₇₉	—	2	1	no	—
97	D₈×C₇₉	C₂×D₄	1	2	yes	complemented
98	M₁₆×C₇₉	—	2	2	no	—
99	M₁₆×C₇₉	—	4	4	no	—
100	M₁₆×C₇₉	—	4	1	no	—
101	C₂×C₈×C₇₉	—	4	4	no	—
102	C₂×C₈×C₇₉	—	4	1	no	—
103	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
104	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
105	C₂×C₈×C₇₉	D₈	1	2	yes	—
106	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
111	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
112	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
113	C₄²×C₇₉	—	2	1	no	—
114	C₄²×C₇₉	C₂×D₄	1	2	yes	—
115	C₂³×C₇₉	—	4	1	no	—
116	C₂³×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂²×D₇₉	—	4	4	no	—
118	C₂²×D₇₉	—	8	8	no	—
119	C₂²×D₇₉	—	8	2	no	—
120	C₂²×D₇₉	—	8	2	no	—
121	Q₈×C₇₉	—	2	2	no	—
122	Q₈×C₇₉	—	4	1	no	—
123	D₄×C₇₉	—	2	2	no	—
124	D₄×C₇₉	—	4	2	no	—
125	D₄×C₇₉	—	4	2	no	—
126	D₄×C₇₉	—	4	1	no	—
127	C₂×C₄×C₇₉	—	2	4	no	—
128	C₂×C₄×C₇₉	—	2	2	no	—
129	C₂×C₄×C₇₉	—	4	1	no	—
130	C₂×C₄×C₇₉	—	4	2	no	—
131	C₂×C₄×C₇₉	—	4	2	no	—
132	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
133	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
134	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
138	D₃₁₆	—	4	4	no	—
139	D₃₁₆	—	4	4	no	—
140	D₃₁₆	—	8	16	no	—
141	D₃₁₆	—	8	2	no	—
142	D₃₁₆	—	8	2	no	—
143	C₈×C₇₉	—	4	2	no	—
144	C₈×C₇₉	#121	1	1	yes	characteristic
145	C₈×C₇₉	#121	1	1	yes	characteristic
146	C₈×C₇₉	C₂×D₈	1	2	yes	—
147	C₈×C₇₉	C₂×D₈	1	2	yes	—
148	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
150	C₂²×C₇₉	—	2	2	no	—
151	C₂²×C₇₉	—	4	2	no	—
152	C₂²×C₇₉	—	8	2	no	—
153	C₂²×C₇₉	#459	1	1	yes	characteristic
154	C₂²×C₇₉	#459	1	1	yes	characteristic
155	C₂²×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×D₇₉	—	8	8	no	—
157	C₂×D₇₉	—	8	4	no	—
158	C₂×D₇₉	—	16	4	no	—
159	C₂×D₇₉	—	16	4	no	—
160	C₄×C₇₉	—	2	4	no	—
161	C₄×C₇₉	—	4	2	no	—
162	C₄×C₇₉	#459	1	1	yes	characteristic
163	C₄×C₇₉	#459	1	1	yes	characteristic
164	C₄×C₇₉	C₂×#121	1	1	yes	characteristic
165	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
166	C₂×#2213	—	79	1	no	maximal, Sylow
167	C₂×C₇₉	—	2	1	no	—
168	C₂×C₇₉	—	2	1	no	—
169	C₂×C₇₉	—	8	2	no	—
170	C₂×C₇₉	#2213	1	2	yes	complemented
171	C₂×C₇₉	C₂×#459	1	1	yes	characteristic
172	D₇₉	—	16	4	no	—
173	C₂×#509	—	79	2	no	—
174	C₂×#493	—	79	1	no	—
175	C₂×#443	D₇₉	1	1	yes	characteristic, complemented
176	#2213	—	79	8	no	—
177	C₂×#364	—	79	1	no	—
178	C₂×#361	—	79	2	no	—
179	C₇₉	C₂×#2213	1	1	yes	characteristic, complemented, Sylow
180	C₂×#135	C₂×D₇₉	1	1	yes	characteristic
181	C₂²×D₈	—	79	2	no	—
182	C₂²×D₈	—	158	1	no	—
183	C₂×#131	—	2	1	no	—
184	C₂×#127	—	158	1	no	—
185	#509	—	79	8	no	—
186	#509	—	158	4	no	—
187	C₂×SD₃₂	—	158	2	no	—
188	C₂×D₁₆	—	158	2	no	—
189	C₂×M₃₂	—	79	2	no	—
190	#493	—	79	4	no	—
191	#493	—	158	2	no	—
192	#443	—	2	2	no	—
193	#443	C₂×D₇₉	1	4	yes	complemented
194	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
195	C₂×#104	—	2	1	no	—
196	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
197	#364	—	79	4	no	—
198	#361	—	79	8	no	—
199	C₂×#41	—	2	1	no	—
200	C₂²×D₄	—	316	2	no	—
201	C₂²×D₄	—	158	2	no	—
202	#135	—	2	1	no	—
203	#135	—	2	1	no	—
204	#135	D₃₁₆	1	2	yes	complemented
205	#135	C₂²×D₇₉	1	2	yes	—
206	C₂×Q₁₆	D₃₁₆	1	1	yes	characteristic, complemented
207	C₂×SD₁₆	—	2	1	no	—
208	C₂×D₈	—	316	1	no	—
209	C₂×D₈	—	316	1	no	—
210	C₂×D₈	—	316	2	no	—
211	C₂×D₈	—	79	4	no	—
212	C₂×D₈	—	158	2	no	—
213	C₂×D₈	—	158	4	no	—
214	C₂×D₈	—	158	2	no	—
215	C₂×D₈	—	158	2	no	—
216	C₂×D₈	—	158	2	no	—
217	C₂×D₈	—	158	2	no	—
218	C₂×D₈	D₃₁₆	1	1	yes	characteristic, complemented
219	#131	—	2	4	no	—
220	#131	—	4	2	no	—
221	C₂×M₁₆	—	2	1	no	—
222	C₂×M₁₆	—	4	1	no	—
223	C₂²×C₈	—	4	1	no	—
224	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
226	#127	—	316	2	no	—
227	#127	—	158	4	no	—
228	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
229	SD₃₂	—	158	8	no	—
230	D₁₆	—	158	8	no	—
231	M₃₂	—	79	4	no	—
232	M₃₂	—	158	2	no	—
233	C₂×C₁₆	—	158	2	no	—
234	#108	—	2	1	no	—
235	#108	C₂²×D₇₉	1	2	yes	—
236	#104	—	2	4	no	—
237	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
238	C₄×C₈	C₂²×D₇₉	1	2	yes	—
239	C₂⁴	—	316	2	no	—
240	#41	—	2	2	no	—
241	#41	—	2	2	no	—
242	#41	—	4	1	no	—
243	#41	—	4	1	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	316	8	no	—
248	C₂×D₄	—	316	2	no	—
249	C₂×D₄	—	316	2	no	—
250	C₂×D₄	—	316	2	no	—
251	C₂×D₄	—	316	2	no	—
252	C₂×D₄	—	316	2	no	—
253	C₂×D₄	—	632	8	no	—
254	C₂×D₄	—	158	4	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
258	Q₁₆	—	2	1	no	—
259	Q₁₆	C₂×D₃₁₆	1	2	yes	complemented
260	SD₁₆	—	2	2	no	—
261	SD₁₆	—	2	2	no	—
262	D₈	—	2	1	no	—
263	D₈	—	316	2	no	—
264	D₈	—	316	2	no	—
265	D₈	—	316	4	no	—
266	D₈	—	316	4	no	—
267	D₈	—	158	4	no	—
268	D₈	—	158	4	no	—
269	D₈	C₂×D₃₁₆	1	2	yes	complemented
270	M₁₆	—	2	2	no	—
271	M₁₆	—	4	1	no	—
272	M₁₆	—	4	4	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	4	no	—
275	C₂×C₈	#11929609	1	1	yes	characteristic
276	C₂×C₈	#11929609	1	1	yes	characteristic
277	C₂×C₈	#11929609	1	2	yes	—
278	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
279	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
280	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
281	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
282	C₂×C₈	#11929624	1	1	yes	characteristic
283	C₂×C₈	#11929624	1	1	yes	characteristic
284	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
285	C₄²	—	2	1	no	—
286	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
287	C₁₆	—	158	4	no	—
288	C₂³	—	4	1	no	—
289	C₂³	—	316	4	no	—
290	C₂³	—	632	8	no	—
291	C₂³	—	632	2	no	—
292	C₂³	—	632	2	no	—
293	C₂³	#14579186	1	1	yes	characteristic
294	Q₈	—	2	2	no	—
295	Q₈	—	4	1	no	—
296	D₄	—	2	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	316	4	no	—
301	D₄	—	316	4	no	—
302	D₄	—	632	16	no	—
303	D₄	—	632	2	no	—
304	D₄	—	632	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	#14579135	1	1	yes	characteristic
311	C₂×C₄	#14579135	1	1	yes	characteristic
312	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
313	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
314	C₂×C₄	#14579186	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
316	C₈	—	4	2	no	—
317	C₈	#14579135	1	1	yes	characteristic
318	C₈	#14579135	1	1	yes	characteristic
319	C₈	C₂×#11929609	1	2	yes	—
320	C₈	C₂×#11929609	1	2	yes	—
321	C₈	C₂²×D₃₁₆	1	1	yes	characteristic
322	C₈	C₂×#11929624	1	1	yes	characteristic
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	632	8	no	—
327	C₂²	—	632	4	no	—
328	C₂²	—	1264	4	no	—
329	C₂²	—	1264	4	no	—
330	C₂²	#19098758	1	1	yes	characteristic
331	C₂²	#19098758	1	1	yes	characteristic
332	C₂²	C₂×#14579186	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	4	2	no	—
335	C₄	#19098758	1	1	yes	characteristic
336	C₄	#19098758	1	1	yes	characteristic
337	C₄	C₂×#14579135	1	1	yes	characteristic
338	C₄	C₂²×#11929609	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	8	2	no	—
342	C₂	—	1264	4	no	—
343	C₂	#23765476	1	2	yes	direct product factor
344	C₂	C₂×#19098758	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098758
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#443×C₇₉	C₂
socle	C₂²×C₇₉	#459
hypercenter	C₂×#443	D₇₉
hypocenter	C₇₉	C₂×#2213
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#19098998	C₂	semidirect
2. C₂×#19098940	C₂	semidirect
3. #23765476	C₂	direct (N × H)
4. C₂×#19098384	C₂	semidirect
5. C₂×#19098375	C₂	semidirect
6. C₂×#443×C₇₉	C₂	semidirect
11. #19098998	C₂²	semidirect
16. #19098940	C₂²	semidirect
17. #19098384	C₂²	semidirect
18. #19098375	C₂²	semidirect
22. #443×C₇₉	C₂²	semidirect
48. #135×C₇₉	D₄	semidirect
50. C₂×Q₁₆×C₇₉	D₄	semidirect
52. C₂×D₈×C₇₉	D₄	semidirect
93. Q₁₆×C₇₉	C₂×D₄	semidirect
97. D₈×C₇₉	C₂×D₄	semidirect
170. C₂×C₇₉	#2213	semidirect
175. C₂×#443	D₇₉	semidirect
179. C₇₉	C₂×#2213	semidirect
193. #443	C₂×D₇₉	semidirect
204. #135	D₃₁₆	semidirect
206. C₂×Q₁₆	D₃₁₆	semidirect
218. C₂×D₈	D₃₁₆	semidirect
259. Q₁₆	C₂×D₃₁₆	semidirect
269. D₈	C₂×D₃₁₆	semidirect
343. C₂	#23765476	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×#19098998	2
C₂×#19098940	2
#23765476	2
C₂×#19098384	2
C₂×#19098375	2
C₂×#443×C₇₉	2
C₂×#2213	79

Subgroups — C2×#23765476 #34442649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765476 #34442649