Subgroups — #34446019

Order 20224.

474 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776389	C₂	1	1	yes	maximal, characteristic
2	#23765639	C₂	1	2	yes	maximal
3	#23765629	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765544	C₂	1	1	yes	maximal, characteristic
5	#23765539	C₂	1	2	yes	maximal, complemented
6	#23765462	C₂	1	2	yes	maximal
7	#23765450	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765309	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765269	C₂	1	2	yes	maximal, complemented
10	C₄×#14579104	C₂	1	1	yes	maximal, characteristic
11	Q₈×Q₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099286	—	2	1	no	—
13	C₂×#14579107	C₂²	1	2	yes	—
14	C₂×#14579104	C₂²	1	1	yes	characteristic
15	#19098947	C₂²	1	2	yes	—
16	#19098943	C₂²	1	1	yes	characteristic
17	#19098786	—	2	2	no	—
18	#19098769	—	2	1	no	—
19	C₄×Dic₃₁₆	C₂²	1	2	yes	—
20	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
21	#19098390	C₂²	1	1	yes	characteristic
22	#19098390	C₂²	1	2	yes	—
23	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
24	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
25	#19098367	C₂²	1	1	yes	characteristic
26	#19098366	C₂²	1	2	yes	—
27	#19098366	C₂²	1	2	yes	—
28	#19098366	C₂²	1	2	yes	—
29	#19098363	C₂²	1	1	yes	characteristic
30	#19098363	C₂²	1	2	yes	—
31	#19098339	C₂²	1	2	yes	—
32	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
33	Q₈²×C₇₉	—	2	1	no	—
34	#500×C₇₉	C₂²	1	2	yes	—
35	#500×C₇₉	C₂²	1	1	yes	characteristic
36	#477×C₇₉	—	2	1	no	—
37	#477×C₇₉	—	2	2	no	—
38	C₈×Q₈×C₇₉	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₇₉	C₂²	1	2	yes	—
41	C₂×Dic₃₁₆	—	2	1	no	—
42	C₂×Dic₃₁₆	C₂³	1	2	yes	—
43	C₂×#11929613	—	2	2	no	—
44	C₂×D₆₃₂	C₂³	1	1	yes	characteristic
45	#14579161	—	2	1	no	—
46	#14579156	—	4	2	no	—
47	#14579149	—	2	2	no	—
48	#14579137	—	4	1	no	—
49	C₄×Dic₁₅₈	—	2	2	no	—
50	C₄×D₃₁₆	—	2	1	no	—
51	#14579119	—	4	1	no	—
52	#14579107	—	2	4	no	—
53	#14579104	—	2	2	no	—
54	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
56	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
57	#14579095	C₂³	1	1	yes	characteristic
58	#14579086	—	2	2	no	—
59	#14579083	—	2	1	no	—
60	C₂×Q₁₆×C₇₉	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₇₉	—	2	1	no	—
62	C₄⋊Q₈×C₇₉	—	2	2	no	—
63	C₄⋊Q₈×C₇₉	—	4	1	no	—
64	C₄⋊Q₈×C₇₉	—	4	2	no	—
65	C₄×Q₈×C₇₉	—	2	1	no	—
66	C₄×Q₈×C₇₉	—	2	2	no	—
67	C₄×Q₈×C₇₉	—	4	1	no	—
68	C₄×Q₈×C₇₉	D₄	1	1	yes	characteristic
69	#107×C₇₉	—	2	1	no	—
70	#107×C₇₉	—	2	2	no	—
71	#107×C₇₉	C₂³	1	1	yes	characteristic
72	#107×C₇₉	C₂³	1	2	yes	—
73	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₇₉	D₄	1	2	yes	—
75	#103×C₇₉	—	2	1	no	—
76	#103×C₇₉	—	2	2	no	—
77	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₇₉	C₂³	1	2	yes	—
79	C₂×Dic₁₅₈	—	2	2	no	—
80	C₂×D₃₁₆	—	2	1	no	—
81	C₂×C₄×D₇₉	—	4	1	no	—
82	Dic₃₁₆	—	2	2	no	—
83	Dic₃₁₆	—	2	2	no	—
84	Dic₃₁₆	—	8	1	no	—
85	#11929613	—	8	2	no	—
86	D₆₃₂	—	2	1	no	—
87	D₆₃₂	—	2	1	no	—
88	C₄⋊Dic₇₉	—	2	1	no	—
89	C₄⋊Dic₇₉	—	4	1	no	—
90	C₄⋊Dic₇₉	—	4	1	no	—
91	Dic₇₉⋊C₄	—	4	2	no	—
92	(C₂×D₇₉)⋊C₄	—	4	2	no	—
93	(C₂×D₇₉)⋊C₄	—	4	1	no	—
94	(C₂×D₇₉)⋊C₄	—	4	2	no	—
95	C₄×Dic₇₉	—	4	2	no	—
96	C₇₉⋊C₁₆	—	2	1	no	—
97	C₇₉⋊C₁₆	—	2	2	no	—
98	C₇₉⋊C₁₆	—	2	1	no	—
99	C₂×Q₈×C₇₉	—	2	1	no	—
100	C₂×Q₈×C₇₉	—	4	2	no	—
101	C₂×Q₈×C₇₉	—	4	1	no	—
102	C₂×Q₈×C₇₉	D₈	1	1	yes	characteristic
103	Q₁₆×C₇₉	—	2	1	no	—
104	Q₁₆×C₇₉	#41	1	2	yes	—
105	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₇₉	—	2	1	no	—
109	C₄⋊C₄×C₇₉	—	2	2	no	—
110	C₄⋊C₄×C₇₉	—	4	2	no	—
111	C₄⋊C₄×C₇₉	—	4	1	no	—
112	C₄⋊C₄×C₇₉	—	4	2	no	—
113	C₄⋊C₄×C₇₉	—	4	1	no	—
114	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
115	C₄⋊C₄×C₇₉	D₈	1	2	yes	—
116	C₄²×C₇₉	—	4	2	no	—
117	C₄²×C₇₉	—	4	1	no	—
118	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₇₉	C₂×D₄	1	2	yes	—
120	C₂²×D₇₉	—	4	1	no	—
121	Q₈×C₇₉	—	2	2	no	—
122	Q₈×C₇₉	—	2	2	no	—
123	Q₈×C₇₉	—	4	1	no	—
124	Q₈×C₇₉	—	16	2	no	—
125	Q₈×C₇₉	—	16	1	no	—
126	C₂×C₄×C₇₉	—	4	1	no	—
127	C₂×C₄×C₇₉	—	4	2	no	—
128	C₂×C₄×C₇₉	—	4	1	no	—
129	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×Dic₇₉	—	4	1	no	—
135	C₂×Dic₇₉	—	4	2	no	—
136	D₃₁₆	—	4	1	no	—
137	D₃₁₆	—	4	1	no	—
138	C₄×D₇₉	—	8	2	no	—
139	Dic₁₅₈	—	4	2	no	—
140	Dic₁₅₈	—	4	2	no	—
141	C₈×C₇₉	—	2	2	no	—
142	C₈×C₇₉	—	2	1	no	—
143	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
144	C₈×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
145	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
146	C₂×D₇₉	—	8	1	no	—
147	C₂×D₇₉	—	8	1	no	—
148	C₂×D₇₉	—	8	1	no	—
149	C₄×C₇₉	—	2	2	no	—
150	C₄×C₇₉	—	2	2	no	—
151	C₄×C₇₉	—	2	1	no	—
152	C₄×C₇₉	—	4	2	no	—
153	C₄×C₇₉	—	8	2	no	—
154	C₄×C₇₉	—	8	1	no	—
155	C₄×C₇₉	#510	1	2	yes	—
156	C₄×C₇₉	#548	1	1	yes	characteristic
157	C₄×C₇₉	#577	1	1	yes	characteristic
158	Dic₇₉	—	8	2	no	—
159	Dic₇₉	—	8	1	no	—
160	#33936	—	79	1	no	maximal, Sylow
161	C₂×C₇₉	#3358	1	1	yes	characteristic
162	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
163	C₂×C₇₉	#3416	1	1	yes	characteristic
164	D₇₉	—	16	1	no	—
165	#3393	—	79	1	no	—
166	Q₈×Q₁₆	D₇₉	1	1	yes	characteristic, complemented
167	#2243	—	79	2	no	—
168	#2241	—	79	1	no	—
169	#2226	—	79	1	no	—
170	#2225	—	79	2	no	—
171	#2211	—	79	2	no	—
172	#2209	—	79	1	no	—
173	#2183	—	79	1	no	—
174	#2176	—	79	2	no	—
175	C₄×SD₃₂	—	79	1	no	—
176	C₇₉	#33936	1	1	yes	characteristic, complemented, Sylow
177	Q₈²	—	2	1	no	—
178	#556	—	158	1	no	—
179	C₂×Q₃₂	—	79	2	no	—
180	C₂×SD₃₂	—	79	1	no	—
181	#500	C₂×D₇₉	1	2	yes	—
182	#500	C₂×D₇₉	1	1	yes	characteristic
183	#495	—	79	2	no	—
184	#494	—	79	1	no	—
185	#477	—	2	1	no	—
186	#477	—	2	2	no	—
187	#464	—	158	2	no	—
188	#461	—	158	1	no	—
189	C₈×Q₈	C₂×D₇₉	1	1	yes	characteristic
190	C₄×Q₁₆	—	79	2	no	—
191	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
192	C₄×Q₁₆	C₂×D₇₉	1	2	yes	—
193	C₄×D₈	—	79	1	no	—
194	#367	—	79	1	no	—
195	#367	—	79	2	no	—
196	C₄⋊C₁₆	—	79	1	no	—
197	C₄⋊C₁₆	—	79	2	no	—
198	#358	—	79	1	no	—
199	#358	—	79	2	no	—
200	#358	—	79	2	no	—
201	#358	—	79	2	no	—
202	#357	—	79	1	no	—
203	#357	—	79	2	no	—
204	#346	—	79	2	no	—
205	C₄×C₁₆	—	79	1	no	—
206	C₂×Q₁₆	—	79	2	no	—
207	C₂×Q₁₆	—	158	1	no	—
208	C₂×Q₁₆	C₂²×D₇₉	1	1	yes	characteristic
209	C₂×SD₁₆	—	158	2	no	—
210	C₂×D₈	—	79	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	2	no	—
213	C₄⋊Q₈	—	4	1	no	—
214	C₄⋊Q₈	—	4	2	no	—
215	C₄⋊Q₈	—	158	1	no	—
216	#126	—	316	2	no	—
217	#124	—	158	2	no	—
218	C₂²⋊Q₈	—	316	1	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×Q₈	—	4	1	no	—
222	C₄×Q₈	—	158	2	no	—
223	C₄×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
224	C₄×D₄	—	158	1	no	—
225	#117	—	316	1	no	—
226	Q₃₂	—	158	4	no	—
227	SD₃₂	—	158	2	no	—
228	C₂×C₁₆	—	79	1	no	—
229	C₂×C₁₆	—	79	1	no	—
230	C₂×C₁₆	—	79	2	no	—
231	#107	—	2	1	no	—
232	#107	—	2	2	no	—
233	#107	—	79	1	no	—
234	#107	C₂²×D₇₉	1	1	yes	characteristic
235	#107	C₂²×D₇₉	1	2	yes	—
236	C₄⋊C₈	C₂²⋊D₇₉	1	1	yes	characteristic
237	C₄⋊C₈	C₂²⋊D₇₉	1	2	yes	—
238	#103	—	2	1	no	—
239	#103	—	2	2	no	—
240	#103	—	158	2	no	—
241	#102	—	158	1	no	—
242	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₇₉	1	2	yes	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	158	2	no	—
248	C₂×Q₈	#11929609	1	1	yes	characteristic
249	C₂×D₄	—	158	1	no	—
250	C₂²×C₄	—	316	1	no	—
251	Q₁₆	—	2	1	no	—
252	Q₁₆	—	632	1	no	—
253	Q₁₆	—	158	2	no	—
254	Q₁₆	—	158	2	no	—
255	Q₁₆	#11929625	1	2	yes	—
256	SD₁₆	—	632	2	no	—
257	D₈	—	158	1	no	—
258	D₈	—	158	1	no	—
259	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
261	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	316	2	no	—
269	C₄⋊C₄	—	316	1	no	—
270	C₄⋊C₄	—	316	1	no	—
271	C₄⋊C₄	—	158	1	no	—
272	C₄⋊C₄	#11929609	1	1	yes	characteristic
273	C₄⋊C₄	#11929609	1	2	yes	—
274	C₂²⋊C₄	—	316	2	no	—
275	C₂²⋊C₄	—	316	1	no	—
276	C₂²⋊C₄	—	316	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	316	2	no	—
280	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
281	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
282	C₁₆	—	158	1	no	—
283	C₁₆	—	158	2	no	—
284	C₁₆	—	158	1	no	—
285	C₂³	—	316	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	16	2	no	—
290	Q₈	—	16	1	no	—
291	Q₈	—	316	2	no	—
292	Q₈	—	316	2	no	—
293	D₄	—	316	1	no	—
294	D₄	—	316	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	316	1	no	—
299	C₂×C₄	—	316	2	no	—
300	C₂×C₄	—	632	2	no	—
301	C₂×C₄	C₂×#11929609	1	2	yes	—
302	C₂×C₄	C₂×#11929609	1	2	yes	—
303	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
304	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	C₂×#11929625	1	1	yes	characteristic
309	C₈	#14579209	1	1	yes	characteristic
310	C₂²	—	632	1	no	—
311	C₂²	—	632	1	no	—
312	C₂²	—	632	1	no	—
313	C₂²	C₂²×#11929609	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	1	no	—
320	C₄	—	632	1	no	—
321	C₄	—	632	2	no	—
322	C₄	#19099004	1	2	yes	—
323	C₄	#19099224	1	1	yes	characteristic
324	C₄	#19099371	1	1	yes	characteristic
325	C₂	—	1264	1	no	—
326	C₂	#23776035	1	1	yes	characteristic
327	C₂	C₂×#19099004	1	1	yes	characteristic
328	C₂	#23776547	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	Q₈×Q₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×D₆₃₂	C₂³
hypercenter	Q₈×Q₁₆	D₇₉
hypocenter	C₇₉	#33936
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #23765629	C₂	semidirect
5. #23765539	C₂	semidirect
7. #23765450	C₂	semidirect
8. #23765309	C₂	semidirect
9. #23765269	C₂	semidirect
11. Q₈×Q₁₆×C₇₉	C₂	semidirect
166. Q₈×Q₁₆	D₇₉	semidirect
176. C₇₉	#33936	semidirect

Maximal subgroups

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

subgroup	index
#23776389	2
#23765639	2
#23765629	2
#23765544	2
#23765539	2
#23765462	2
#23765450	2
#23765309	2
#23765269	2
C₄×#14579104	2
Q₈×Q₁₆×C₇₉	2
#33936	79