Subgroups — #34306761

Order 20224.

846 conjugacy classes of subgroups.

The subgroup lattice has 324 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	#23772285	C₂	1	2	yes	maximal
3	#23771997	C₂	1	2	yes	maximal, complemented
4	#23764180	C₂	1	2	yes	maximal, complemented
5	#23763634	C₂	1	4	yes	maximal, complemented
6	#23763564	C₂	1	1	yes	maximal, characteristic, complemented
7	#23763461	C₂	1	2	yes	maximal, complemented
8	C₇₉×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14579183	C₂²	1	1	yes	characteristic
10	C₂×#14579167	C₂²	1	2	yes	—
11	#19099043	C₂²	1	4	yes	—
12	#19098681	C₄	1	8	yes	complemented
13	#19098676	—	2	2	no	—
14	#19098671	—	2	2	no	—
15	C₄×#11929613	—	2	4	no	—
16	C₂×#14579097	C₂²	1	1	yes	characteristic
17	#19098606	C₂²	1	4	yes	—
18	#19098602	—	2	4	no	—
19	#19098600	—	2	2	no	—
20	C₂×#14579095	C₂²	1	2	yes	—
21	#107×D₇₉	—	2	2	no	—
22	C₂×#14579093	C₂²	1	2	yes	—
23	C₂×#14579089	C₂²	1	2	yes	—
24	#19098565	C₂²	1	4	yes	—
25	#19098429	C₂²	1	2	yes	—
26	#19098331	C₂²	1	4	yes	—
27	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
28	#427×C₇₉	C₂²	1	1	yes	characteristic
29	#405×C₇₉	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
31	#339×C₇₉	C₂²	1	2	yes	—
32	C₂×#11929624	C₂³	1	2	yes	—
33	#14579183	C₂×C₄	1	4	yes	—
34	#14579183	Q₈	1	4	yes	—
35	C₂×Dic₃₁₆	D₄	1	1	yes	characteristic
36	C₂×#11929613	D₄	1	2	yes	—
37	C₂×D₆₃₂	D₄	1	1	yes	characteristic
38	#14579167	—	2	4	no	—
39	#14579167	C₂×C₄	1	8	yes	complemented
40	C₂×#11929608	C₂³	1	2	yes	—
41	C₂×#11929607	—	2	2	no	—
42	C₂×C₈×D₇₉	—	2	2	no	—
43	C₄×Dic₁₅₈	—	2	4	no	—
44	C₄×D₃₁₆	—	2	4	no	—
45	(C₂²⋊D₇₉)⋊C₄	—	4	4	no	—
46	#14579119	—	2	4	no	—
47	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
48	C₄⋊C₄×D₇₉	—	2	4	no	—
49	#14579097	C₂×C₄	1	4	yes	complemented
50	#14579095	—	2	2	no	—
51	#14579095	—	2	2	no	—
52	#14579094	—	2	4	no	—
53	#14579093	—	2	2	no	—
54	#14579093	—	2	2	no	—
55	#14579092	—	2	4	no	—
56	#14579089	C₂×C₄	1	8	yes	complemented
57	#14579086	—	2	4	no	—
58	#14579083	—	2	4	no	—
59	C₇₉⋊#95	—	4	2	no	—
60	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
61	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
63	#117×C₇₉	C₂³	1	2	yes	—
64	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
65	#108×C₇₉	—	2	2	no	—
66	#107×C₇₉	D₄	1	2	yes	—
67	#106×C₇₉	D₄	1	2	yes	—
68	#97×C₇₉	—	2	2	no	—
69	C₄×C₈×C₇₉	—	2	2	no	—
70	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	complemented
71	#11929624	—	2	2	no	—
72	#11929624	—	2	2	no	—
73	#11929624	C₂²×C₄	1	4	yes	—
74	#11929624	C₂×Q₈	1	4	yes	—
75	C₂×Dic₁₅₈	C₂×D₄	1	2	yes	—
76	C₂×D₃₁₆	C₂×D₄	1	2	yes	—
77	C₂×C₂²⋊D₇₉	—	2	2	no	—
78	C₂²×Dic₇₉	—	2	4	no	—
79	C₂×C₄×D₇₉	—	2	2	no	—
80	C₂×C₄×D₇₉	—	4	4	no	—
81	Dic₃₁₆	C₄⋊C₄	1	4	yes	—
82	#11929613	C₄⋊C₄	1	8	yes	—
83	D₆₃₂	C₄⋊C₄	1	4	yes	—
84	#11929608	—	2	2	no	—
85	#11929608	C₂²×C₄	1	4	yes	—
86	#11929607	—	2	8	no	—
87	C₂×C₇₉⋊C₈	—	2	2	no	—
88	C₈×D₇₉	—	2	8	no	—
89	C₄⋊Dic₇₉	—	2	4	no	—
90	C₄⋊Dic₇₉	—	2	4	no	—
91	Dic₇₉⋊C₄	—	4	4	no	—
92	(C₂×D₇₉)⋊C₄	—	4	4	no	—
93	C₄×Dic₇₉	—	4	4	no	—
94	C₂²×C₄×C₇₉	—	2	1	no	—
95	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
96	M₁₆×C₇₉	—	2	2	no	—
97	M₁₆×C₇₉	—	2	2	no	—
98	C₂×C₈×C₇₉	—	2	2	no	—
99	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
106	C₂×C₈×C₇₉	#41	1	2	yes	—
107	C₄⋊C₄×C₇₉	C₂×D₄	1	4	yes	—
108	C₂²⋊C₄×C₇₉	—	2	2	no	—
109	C₄²×C₇₉	—	2	2	no	—
110	C₄²×C₇₉	—	2	1	no	—
111	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
112	C₂³×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂²×D₇₉	—	2	2	no	—
114	C₂×C₄×C₇₉	—	2	2	no	—
115	C₂×C₄×C₇₉	—	2	4	no	—
116	C₂×C₄×C₇₉	—	4	2	no	—
117	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂²⋊D₇₉	—	2	4	no	—
124	C₂²⋊D₇₉	—	2	4	no	—
125	C₂×Dic₇₉	—	2	2	no	—
126	C₂×Dic₇₉	—	4	4	no	—
127	C₂×Dic₇₉	—	8	4	no	—
128	D₃₁₆	—	2	2	no	—
129	D₃₁₆	C₂×C₄⋊C₄	1	4	yes	—
130	C₄×D₇₉	—	2	4	no	—
131	C₄×D₇₉	—	2	4	no	—
132	C₄×D₇₉	—	16	4	no	—
133	Dic₁₅₈	—	2	2	no	—
134	Dic₁₅₈	C₂×C₄⋊C₄	1	4	yes	—
135	C₈×C₇₉	—	4	2	no	—
136	C₈×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
137	C₈×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
138	C₈×C₇₉	C₄×D₄	1	2	yes	—
139	C₈×C₇₉	C₄×D₄	1	2	yes	—
140	C₇₉⋊C₈	—	2	4	no	—
141	C₂²×C₇₉	—	2	2	no	—
142	C₂²×C₇₉	#520	1	1	yes	characteristic
143	C₂²×C₇₉	D₄²	1	1	yes	characteristic
144	C₂²×C₇₉	#554	1	1	yes	characteristic
145	C₂×D₇₉	—	2	4	no	—
146	C₂×D₇₉	—	4	2	no	—
147	C₂×D₇₉	—	4	2	no	—
148	C₄×C₇₉	—	4	2	no	—
149	C₄×C₇₉	—	4	4	no	—
150	C₄×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₇₉	#547	1	1	yes	characteristic
153	C₄×C₇₉	D₄×Q₈	1	1	yes	characteristic
154	Dic₇₉	—	2	4	no	—
155	Dic₇₉	—	8	4	no	—
156	#22691	—	79	1	no	maximal, Sylow
157	C₂×C₇₉	—	2	1	no	—
158	C₂×C₇₉	—	2	1	no	—
159	C₂×C₇₉	D₄×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₇₉	#3290	1	2	yes	—
161	D₇₉	—	4	4	no	—
162	C₂×#443	—	79	1	no	—
163	#2952	—	79	2	no	—
164	#2913	—	79	2	no	—
165	#1951	—	79	2	no	—
166	#1860	—	79	4	no	—
167	#1846	—	79	1	no	—
168	#1816	—	79	2	no	—
169	#1772	D₇₉	1	1	yes	characteristic, complemented
170	C₇₉	#22691	1	1	yes	characteristic, complemented, Sylow
171	C₂×#135	—	79	1	no	—
172	C₂×#131	—	79	2	no	—
173	#520	—	79	4	no	—
174	#443	—	79	8	no	—
175	#442	—	158	2	no	—
176	#441	—	158	2	no	—
177	C₄×SD₁₆	—	158	4	no	—
178	C₂×#108	—	79	1	no	—
179	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
180	#428	—	79	4	no	—
181	#428	—	158	4	no	—
182	#427	—	158	2	no	—
183	#427	C₂×D₇₉	1	1	yes	characteristic
184	C₂×#107	—	79	2	no	—
185	C₂×#107	—	158	2	no	—
186	C₂×#106	—	79	2	no	—
187	C₂×#104	—	79	2	no	—
188	#417	—	79	4	no	—
189	#405	C₂×D₇₉	1	2	yes	—
190	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
191	#382	—	79	2	no	—
192	#343	—	79	4	no	—
193	#339	C₂×D₇₉	1	2	yes	—
194	C₂×#41	—	79	2	no	—
195	#135	—	79	4	no	—
196	#135	—	79	4	no	—
197	C₂×Q₁₆	—	79	1	no	—
198	C₂×SD₁₆	—	79	2	no	—
199	C₂×D₈	—	79	1	no	—
200	#131	—	79	8	no	—
201	#131	—	158	4	no	—
202	C₂×M₁₆	—	79	2	no	—
203	C₂×M₁₆	—	158	2	no	—
204	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
205	C₂²×C₈	—	158	2	no	—
206	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
208	C₄×Q₈	—	158	4	no	—
209	C₄×D₄	—	316	4	no	—
210	C₄×D₄	—	158	4	no	—
211	#117	—	158	4	no	—
212	#117	C₂²×D₇₉	1	2	yes	—
213	C₂×C₄⋊C₄	—	79	4	no	—
214	C₂×C₄⋊C₄	—	158	4	no	—
215	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
216	#108	—	2	2	no	—
217	#108	—	79	4	no	—
218	#107	—	158	2	no	—
219	#107	—	158	2	no	—
220	#107	—	158	4	no	—
221	#107	D₃₁₆	1	2	yes	—
222	#106	—	158	2	no	—
223	#106	—	158	2	no	—
224	#106	—	158	4	no	—
225	#106	D₃₁₆	1	2	yes	—
226	#104	—	79	8	no	—
227	#103	—	158	4	no	—
228	#102	—	158	4	no	—
229	#97	—	2	2	no	—
230	C₄×C₈	—	2	2	no	—
231	C₄×C₈	C₄×D₇₉	1	4	yes	complemented
232	#95	—	316	2	no	—
233	#41	—	79	4	no	—
234	#41	—	79	4	no	—
235	#41	—	158	2	no	—
236	#41	—	158	2	no	—
237	C₂×Q₈	—	79	2	no	—
238	C₂×D₄	—	79	2	no	—
239	C₂×D₄	—	158	2	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	316	4	no	—
242	C₂²×C₄	—	158	4	no	—
243	C₂²×C₄	—	158	2	no	—
244	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
245	Q₁₆	—	79	4	no	—
246	SD₁₆	—	79	8	no	—
247	D₈	—	79	4	no	—
248	M₁₆	—	2	2	no	—
249	M₁₆	—	2	2	no	—
250	M₁₆	—	79	4	no	—
251	M₁₆	—	158	2	no	—
252	M₁₆	—	158	8	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	158	2	no	—
255	C₂×C₈	—	158	8	no	—
256	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
257	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
258	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
259	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
260	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
261	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
262	C₂×C₈	Q₈×D₇₉	1	2	yes	—
263	C₂×C₈	#11929624	1	2	yes	—
264	C₄⋊C₄	—	316	4	no	—
265	C₄⋊C₄	—	158	4	no	—
266	C₄⋊C₄	—	158	4	no	—
267	C₄⋊C₄	C₂×D₃₁₆	1	4	yes	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	316	4	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	316	4	no	—
273	C₄²	C₂×C₄×D₇₉	1	2	yes	—
274	C₂³	—	158	2	no	—
275	C₂³	#14579207	1	1	yes	characteristic
276	Q₈	—	79	4	no	—
277	Q₈	—	158	2	no	—
278	D₄	—	79	4	no	—
279	D₄	—	158	4	no	—
280	D₄	—	158	4	no	—
281	D₄	—	158	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	316	4	no	—
286	C₂×C₄	—	632	4	no	—
287	C₂×C₄	—	158	2	no	—
288	C₂×C₄	—	158	4	no	—
289	C₂×C₄	—	158	4	no	—
290	C₂×C₄	—	1264	4	no	—
291	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
295	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
296	C₂×C₄	#14579208	1	1	yes	characteristic
297	C₈	—	4	2	no	—
298	C₈	—	158	4	no	—
299	C₈	C₄⋊C₄×D₇₉	1	2	yes	—
300	C₈	C₄⋊C₄×D₇₉	1	2	yes	—
301	C₈	C₄×D₃₁₆	1	2	yes	—
302	C₈	C₄×D₃₁₆	1	2	yes	—
303	C₂²	—	2	2	no	—
304	C₂²	—	316	2	no	—
305	C₂²	—	316	2	no	—
306	C₂²	—	158	4	no	—
307	C₂²	#19099043	1	1	yes	characteristic
308	C₂²	D₄×D₃₁₆	1	1	yes	characteristic
309	C₂²	#19099273	1	1	yes	characteristic
310	C₄	—	4	2	no	—
311	C₄	—	4	4	no	—
312	C₄	—	632	4	no	—
313	C₄	—	158	4	no	—
314	C₄	C₂×C₄⋊C₄×D₇₉	1	1	yes	characteristic
315	C₄	C₂×C₄×D₃₁₆	1	1	yes	characteristic
316	C₄	#19099218	1	1	yes	characteristic
317	C₄	Q₈×D₃₁₆	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	316	4	no	—
321	C₂	C₄⋊C₄×D₃₁₆	1	1	yes	characteristic
322	C₂	#23775188	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#1772	C₂
socle	C₂²×C₇₉	D₄²
prime radical	C₂×#14579183	C₂²
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1772	D₇₉
hypocenter	C₇₉	#22691
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772

Complements

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

N	H ≅ G/N	type
3. #23771997	C₂	semidirect
4. #23764180	C₂	semidirect
5. #23763634	C₂	semidirect
6. #23763564	C₂	semidirect
7. #23763461	C₂	semidirect
8. C₇₉×#1772	C₂	semidirect
12. #19098681	C₄	semidirect
39. #14579167	C₂×C₄	semidirect
49. #14579097	C₂×C₄	semidirect
56. #14579089	C₂×C₄	semidirect
70. C₄×C₈×C₇₉	C₂×C₄	semidirect
169. #1772	D₇₉	semidirect
170. C₇₉	#22691	semidirect
231. C₄×C₈	C₄×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098681	2
#23772285	2
#23771997	2
#23764180	2
#23763634	2
#23763564	2
#23763461	2
C₇₉×#1772	2
#22691	79