Subgroups — #34444694

Order 20224.

488 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23774059	C₂	1	1	yes	maximal, characteristic, complemented
2	#23774050	C₂	1	1	yes	maximal, characteristic
3	#23765565	C₂	1	2	yes	maximal, complemented
4	#23765558	C₂	1	2	yes	maximal, complemented
5	#23765513	C₂	1	2	yes	maximal, complemented
6	#23765496	C₂	1	2	yes	maximal, complemented
7	#23765446	C₂	1	2	yes	maximal, complemented
8	#23765441	C₂	1	2	yes	maximal, complemented
9	C₇₉×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	#19099175	—	2	1	no	—
11	#19099168	—	2	1	no	—
12	C₂×#14579106	C₂²	1	2	yes	—
13	C₂×#14579106	C₂²	1	2	yes	—
14	#19098965	C₂²	1	1	yes	characteristic
15	#19098965	C₂²	1	1	yes	characteristic
16	#19098943	C₂²	1	1	yes	characteristic
17	#19098940	C₂²	1	1	yes	characteristic
18	#19098932	—	2	1	no	—
19	#19098928	—	2	1	no	—
20	#19098821	—	2	1	no	—
21	#19098816	—	2	1	no	—
22	#19098804	C₂²	1	2	yes	—
23	#19098799	C₂²	1	2	yes	—
24	#19098676	—	2	1	no	—
25	#19098671	—	2	1	no	—
26	C₄×Dic₃₁₆	C₂²	1	2	yes	—
27	C₄×D₆₃₂	C₂²	1	2	yes	—
28	#19098391	C₂²	1	2	yes	—
29	#19098389	C₂²	1	2	yes	—
30	#19098368	C₂²	1	2	yes	—
31	#19098368	C₂²	1	2	yes	—
32	#19098365	C₂²	1	2	yes	—
33	#19098365	C₂²	1	2	yes	—
34	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
36	#405×C₇₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₇₉	C₂²	1	2	yes	—
38	C₄⋊C₁₆×C₇₉	C₂²	1	2	yes	—
39	C₂×Dic₃₁₆	—	2	1	no	—
40	C₂×Dic₃₁₆	C₂³	1	2	yes	—
41	C₂×D₆₃₂	—	2	1	no	—
42	C₂×D₆₃₂	C₂³	1	2	yes	—
43	#14579161	—	2	1	no	—
44	#14579158	—	2	1	no	—
45	#14579154	—	4	2	no	—
46	#14579153	—	2	1	no	—
47	#14579153	—	2	1	no	—
48	#14579151	—	4	2	no	—
49	#14579146	—	4	2	no	—
50	#14579140	—	2	2	no	—
51	#14579135	—	2	2	no	—
52	#14579134	—	4	2	no	—
53	C₄×Dic₁₅₈	—	2	2	no	—
54	C₄×D₃₁₆	—	2	2	no	—
55	#14579106	—	4	2	no	—
56	#14579106	—	4	2	no	—
57	#14579095	C₂³	1	2	yes	—
58	#14579095	C₂³	1	2	yes	—
59	#14579093	—	2	1	no	—
60	#14579093	—	2	1	no	—
61	#14579086	—	2	2	no	—
62	#14579086	—	2	2	no	—
63	#14579083	—	2	2	no	—
64	#14579083	—	2	2	no	—
65	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
67	#117×C₇₉	D₄	1	1	yes	characteristic
68	M₃₂×C₇₉	—	4	1	no	—
69	C₂×C₁₆×C₇₉	—	4	1	no	—
70	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
71	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
72	#97×C₇₉	D₄	1	1	yes	characteristic
73	#97×C₇₉	D₄	1	1	yes	characteristic
74	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
76	C₂×Dic₁₅₈	—	2	2	no	—
77	C₂×D₃₁₆	—	2	2	no	—
78	C₂×D₃₁₆	—	4	2	no	—
79	C₂×C₂²⋊D₇₉	—	4	2	no	—
80	C₂×C₄×D₇₉	—	4	2	no	—
81	Dic₃₁₆	—	2	2	no	—
82	Dic₃₁₆	—	2	2	no	—
83	Dic₃₁₆	—	4	2	no	—
84	D₆₃₂	—	2	2	no	—
85	D₆₃₂	—	2	2	no	—
86	D₆₃₂	—	4	2	no	—
87	C₄⋊Dic₇₉	—	2	2	no	—
88	C₄⋊Dic₇₉	—	2	2	no	—
89	C₄⋊Dic₇₉	—	4	2	no	—
90	Dic₇₉⋊C₄	—	4	2	no	—
91	Dic₇₉⋊C₄	—	4	2	no	—
92	Dic₇₉⋊C₄	—	4	2	no	—
93	Dic₇₉⋊C₄	—	4	2	no	—
94	C₂²⋊Dic₇₉	—	4	2	no	—
95	(C₂×D₇₉)⋊C₄	—	4	2	no	—
96	(C₂×D₇₉)⋊C₄	—	4	2	no	—
97	C₄×Dic₇₉	—	4	2	no	—
98	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
99	M₁₆×C₇₉	—	4	1	no	—
100	C₂×C₈×C₇₉	—	2	1	no	—
101	C₂×C₈×C₇₉	—	2	1	no	—
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	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₇₉	D₈	1	2	yes	—
107	C₂²⋊C₄×C₇₉	D₈	1	2	yes	—
108	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
110	C₁₆×C₇₉	—	2	2	no	—
111	C₁₆×C₇₉	—	2	2	no	—
112	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
113	C₂²×D₇₉	—	4	2	no	—
114	C₂×C₄×C₇₉	—	2	1	no	—
115	C₂×C₄×C₇₉	—	2	1	no	—
116	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
120	C₂²⋊D₇₉	—	16	2	no	—
121	C₂×Dic₇₉	—	4	2	no	—
122	C₂×Dic₇₉	—	4	2	no	—
123	C₂×Dic₇₉	—	4	2	no	—
124	D₃₁₆	—	4	2	no	—
125	D₃₁₆	—	4	2	no	—
126	D₃₁₆	—	16	2	no	—
127	C₄×D₇₉	—	16	2	no	—
128	Dic₁₅₈	—	4	2	no	—
129	Dic₁₅₈	—	4	2	no	—
130	C₈×C₇₉	—	2	1	no	—
131	C₈×C₇₉	—	2	1	no	—
132	C₈×C₇₉	—	2	1	no	—
133	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
134	C₈×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
135	C₂²×C₇₉	—	2	2	no	—
136	C₂²×C₇₉	—	2	1	no	—
137	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
138	C₂×D₇₉	—	8	2	no	—
139	C₂×D₇₉	—	8	2	no	—
140	C₂×D₇₉	—	8	2	no	—
141	C₄×C₇₉	—	2	2	no	—
142	C₄×C₇₉	—	2	2	no	—
143	C₄×C₇₉	—	2	1	no	—
144	C₄×C₇₉	#537	1	1	yes	characteristic
145	C₄×C₇₉	#575	1	1	yes	characteristic
146	Dic₇₉	—	8	2	no	—
147	Dic₇₉	—	8	2	no	—
148	Dic₇₉	—	8	2	no	—
149	#33826	—	79	1	no	maximal, Sylow
150	C₂×C₇₉	—	4	1	no	—
151	C₂×C₇₉	#3245	1	1	yes	characteristic
152	C₂×C₇₉	#3416	1	2	yes	—
153	D₇₉	—	16	2	no	—
154	#3177	—	79	1	no	—
155	#3176	—	79	1	no	—
156	#2229	—	79	2	no	—
157	#2228	—	79	2	no	—
158	#2220	—	79	2	no	—
159	#2217	—	79	2	no	—
160	#2208	—	79	2	no	—
161	#2207	—	79	2	no	—
162	#2151	D₇₉	1	1	yes	characteristic, complemented
163	C₇₉	#33826	1	1	yes	characteristic, complemented, Sylow
164	#541	—	158	1	no	—
165	#540	—	158	1	no	—
166	C₂×SD₃₂	—	79	2	no	—
167	C₂×SD₃₂	—	79	2	no	—
168	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
169	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
170	#499	—	79	1	no	—
171	#499	—	79	1	no	—
172	#494	—	79	1	no	—
173	#493	—	79	1	no	—
174	#491	—	158	1	no	—
175	#490	—	158	1	no	—
176	#470	—	158	1	no	—
177	#469	—	158	1	no	—
178	#467	—	79	2	no	—
179	#466	—	79	2	no	—
180	#442	—	158	1	no	—
181	#441	—	158	1	no	—
182	C₄×Q₁₆	—	79	2	no	—
183	C₄×D₈	—	79	2	no	—
184	#405	C₂×D₇₉	1	1	yes	characteristic
185	#367	—	79	2	no	—
186	#366	—	79	2	no	—
187	C₄⋊C₁₆	C₂×D₇₉	1	2	yes	—
188	C₄⋊C₁₆	C₂×D₇₉	1	2	yes	—
189	#358	—	79	2	no	—
190	#358	—	79	2	no	—
191	#357	—	79	2	no	—
192	#357	—	79	2	no	—
193	C₂×Q₁₆	—	79	2	no	—
194	C₂×Q₁₆	—	158	1	no	—
195	C₂×D₈	—	79	2	no	—
196	C₂×D₈	—	158	1	no	—
197	C₂×M₁₆	D₃₁₆	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
199	C₄⋊Q₈	—	158	1	no	—
200	#127	—	158	1	no	—
201	#126	—	316	2	no	—
202	#125	—	316	2	no	—
203	#125	—	158	1	no	—
204	#125	—	158	1	no	—
205	#123	—	316	2	no	—
206	C₂²⋊Q₈	—	158	2	no	—
207	#121	—	316	2	no	—
208	#121	—	158	2	no	—
209	C₄×Q₈	—	158	2	no	—
210	C₄×D₄	—	158	2	no	—
211	#117	D₃₁₆	1	1	yes	characteristic
212	SD₃₂	—	316	2	no	—
213	SD₃₂	—	316	2	no	—
214	M₃₂	—	4	1	no	—
215	C₂×C₁₆	—	4	1	no	—
216	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
217	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
218	#107	—	79	2	no	—
219	#107	—	79	2	no	—
220	#106	—	158	1	no	—
221	#106	—	158	1	no	—
222	#103	—	158	2	no	—
223	#103	—	158	2	no	—
224	#102	—	158	2	no	—
225	#102	—	158	2	no	—
226	#97	D₃₁₆	1	1	yes	characteristic
227	#97	D₃₁₆	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
230	C₂×Q₈	—	158	2	no	—
231	C₂×D₄	—	316	2	no	—
232	C₂×D₄	—	316	2	no	—
233	C₂×D₄	—	158	2	no	—
234	C₂²×C₄	—	316	2	no	—
235	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
236	Q₁₆	—	316	2	no	—
237	Q₁₆	—	158	2	no	—
238	Q₁₆	—	158	2	no	—
239	D₈	—	316	2	no	—
240	D₈	—	158	2	no	—
241	D₈	—	158	2	no	—
242	M₁₆	—	4	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
246	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
247	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
248	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
249	C₄⋊C₄	—	316	2	no	—
250	C₄⋊C₄	—	316	2	no	—
251	C₄⋊C₄	—	316	2	no	—
252	C₄⋊C₄	—	316	2	no	—
253	C₄⋊C₄	—	316	2	no	—
254	C₄⋊C₄	—	158	2	no	—
255	C₄⋊C₄	—	158	2	no	—
256	C₄⋊C₄	D₆₃₂	1	2	yes	—
257	C₂²⋊C₄	—	316	2	no	—
258	C₂²⋊C₄	—	316	2	no	—
259	C₂²⋊C₄	—	316	2	no	—
260	C₂²⋊C₄	D₆₃₂	1	2	yes	—
261	C₄²	—	316	2	no	—
262	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
263	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
264	C₁₆	—	2	2	no	—
265	C₁₆	—	2	2	no	—
266	C₂³	—	316	2	no	—
267	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
268	Q₈	—	316	2	no	—
269	Q₈	—	316	2	no	—
270	D₄	—	316	2	no	—
271	D₄	—	316	2	no	—
272	D₄	—	1264	2	no	—
273	D₄	—	1264	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	316	2	no	—
277	C₂×C₄	—	316	2	no	—
278	C₂×C₄	—	316	2	no	—
279	C₂×C₄	—	1264	2	no	—
280	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
281	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
282	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
283	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	#14579206	1	1	yes	characteristic
288	C₈	#14579209	1	1	yes	characteristic
289	C₂²	—	2	2	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	632	2	no	—
292	C₂²	—	632	2	no	—
293	C₂²	—	632	2	no	—
294	C₂²	C₂²×D₆₃₂	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	632	2	no	—
299	C₄	—	632	2	no	—
300	C₄	—	632	2	no	—
301	C₄	#19099144	1	1	yes	characteristic
302	C₄	#19099363	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	—	1264	2	no	—
305	C₂	#23774710	1	1	yes	characteristic
306	C₂	#23776548	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₆₃₂
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2151	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	#23774050	C₂
hypercenter	#2151	D₇₉
hypocenter	C₇₉	#33826
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2151

Complements

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

N	H ≅ G/N	type
1. #23774059	C₂	semidirect
3. #23765565	C₂	semidirect
4. #23765558	C₂	semidirect
5. #23765513	C₂	semidirect
6. #23765496	C₂	semidirect
7. #23765446	C₂	semidirect
8. #23765441	C₂	semidirect
9. C₇₉×#2151	C₂	semidirect
162. #2151	D₇₉	semidirect
163. C₇₉	#33826	semidirect

Maximal subgroups

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

subgroup	index
#23774059	2
#23774050	2
#23765565	2
#23765558	2
#23765513	2
#23765496	2
#23765446	2
#23765441	2
C₇₉×#2151	2
#33826	79