Subgroups — #34196702

Order 20224.

574 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098612	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098335	C₂	1	2	yes	maximal
3	C₂×#19098327	C₂	1	2	yes	maximal
4	#23763098	C₂	1	1	yes	maximal, characteristic
5	C₂×#338×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	C₂²	1	2	yes	—
7	C₂³×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×#14579120	C₂²	1	1	yes	characteristic
9	#19098612	—	2	4	no	—
10	C₂×#14579097	—	2	2	no	—
11	#19098335	—	2	4	no	—
12	#19098327	—	2	4	no	—
13	#19098317	—	2	2	no	—
14	C₂²×M₁₆×C₇₉	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	#338×C₇₉	C₄	1	4	yes	complemented
17	C₂×#11929608	—	2	2	no	—
18	C₂×#11929608	—	2	2	no	—
19	C₂×#11929608	—	2	2	no	—
20	C₂×#11929608	—	2	2	no	—
21	C₂×#11929608	—	2	2	no	—
22	C₂×#11929608	D₄	1	4	yes	—
23	C₂²×C₇₉⋊C₈	—	2	1	no	—
24	C₂²×C₇₉⋊C₈	—	2	1	no	—
25	C₂²×C₇₉⋊C₈	—	2	2	no	—
26	C₂²×C₇₉⋊C₈	—	2	2	no	—
27	C₂²×C₇₉⋊C₈	—	2	2	no	—
28	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
29	#14579120	—	2	2	no	—
30	#14579120	—	2	2	no	—
31	C₂×C₄⋊Dic₇₉	—	2	1	no	—
32	C₂×C₂²⋊Dic₇₉	Q₈	1	2	yes	—
33	C₂×C₄×Dic₇₉	—	2	1	no	—
34	#14579097	—	2	8	no	—
35	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₇₉	—	2	2	no	—
37	C₂×M₁₆×C₇₉	—	2	2	no	—
38	C₂×M₁₆×C₇₉	—	2	2	no	—
39	C₂×M₁₆×C₇₉	—	2	2	no	—
40	C₂×M₁₆×C₇₉	—	2	2	no	—
41	C₂×M₁₆×C₇₉	—	2	2	no	—
42	C₂×M₁₆×C₇₉	—	2	2	no	—
43	C₂×M₁₆×C₇₉	C₂×C₄	1	4	yes	—
44	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
45	C₂×M₁₆×C₇₉	Q₈	1	2	yes	—
46	C₂²×C₈×C₇₉	—	2	2	no	—
47	C₂²×C₈×C₇₉	D₄	1	2	yes	—
48	C₂²×Dic₇₉	—	2	2	no	—
49	#11929608	—	2	4	no	—
50	#11929608	—	2	4	no	—
51	#11929608	—	4	2	no	—
52	#11929608	—	4	2	no	—
53	C₂×C₇₉⋊C₈	—	2	2	no	—
54	C₂×C₇₉⋊C₈	—	2	4	no	—
55	C₂×C₇₉⋊C₈	—	2	4	no	—
56	C₂×C₇₉⋊C₈	—	4	2	no	—
57	C₂×C₇₉⋊C₈	—	4	1	no	—
58	C₂×C₇₉⋊C₈	—	4	1	no	—
59	C₂×C₇₉⋊C₈	—	4	2	no	—
60	C₂×C₇₉⋊C₈	—	4	2	no	—
61	C₄⋊Dic₇₉	—	4	2	no	—
62	C₂²⋊Dic₇₉	—	2	4	no	—
63	C₄×Dic₇₉	—	4	1	no	—
64	C₄×Dic₇₉	—	4	1	no	—
65	C₂⁴×C₇₉	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	#41	1	2	yes	—
72	C₂²×C₄×C₇₉	#41	1	2	yes	—
73	C₂²×C₄×C₇₉	#41	1	2	yes	—
74	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
75	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
76	M₁₆×C₇₉	—	2	4	no	—
77	M₁₆×C₇₉	—	2	4	no	—
78	M₁₆×C₇₉	—	2	4	no	—
79	M₁₆×C₇₉	—	2	4	no	—
80	M₁₆×C₇₉	—	4	2	no	—
81	M₁₆×C₇₉	—	4	2	no	—
82	C₂×C₈×C₇₉	—	2	4	no	—
83	C₂×C₈×C₇₉	—	2	2	no	—
84	C₂×C₈×C₇₉	—	2	2	no	—
85	C₂×C₈×C₇₉	—	4	2	no	—
86	C₂×C₈×C₇₉	—	4	2	no	—
87	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
88	C₂³×C₇₉	—	4	2	no	—
89	C₂³×C₇₉	C₄×Q₈	1	1	yes	characteristic
90	C₂³×C₇₉	C₂²⋊Q₈	1	2	yes	—
91	C₂³×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂³×C₇₉	#125	1	2	yes	—
93	C₂³×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₇₉	#117	1	2	yes	—
96	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
99	C₂×C₄×C₇₉	#121	1	2	yes	—
100	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
101	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₇₉	#123	1	2	yes	—
103	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
104	C₂×C₄×C₇₉	#123	1	2	yes	—
105	C₂×C₄×C₇₉	#124	1	2	yes	—
106	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
107	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
108	C₂×C₄×C₇₉	#125	1	1	yes	characteristic
109	C₂×C₄×C₇₉	#125	1	2	yes	—
110	C₂×C₄×C₇₉	#126	1	2	yes	—
111	C₂×C₄×C₇₉	#126	1	2	yes	—
112	C₂×Dic₇₉	—	4	4	no	—
113	C₂×Dic₇₉	—	4	2	no	—
114	C₈×C₇₉	—	2	4	no	—
115	C₈×C₇₉	—	4	4	no	—
116	C₇₉⋊C₈	—	4	4	no	—
117	C₇₉⋊C₈	—	4	2	no	—
118	C₂²×C₇₉	—	2	2	no	—
119	C₂²×C₇₉	—	2	4	no	—
120	C₂²×C₇₉	—	4	1	no	—
121	C₂²×C₇₉	—	4	2	no	—
122	C₂²×C₇₉	—	4	1	no	—
123	C₂²×C₇₉	#387	1	2	yes	—
124	C₂²×C₇₉	#389	1	1	yes	characteristic
125	C₂²×C₇₉	#393	1	1	yes	characteristic
126	C₂²×C₇₉	#398	1	2	yes	—
127	C₂²×C₇₉	#400	1	1	yes	characteristic
128	C₄×C₇₉	#382	1	1	yes	characteristic
129	C₄×C₇₉	#385	1	1	yes	characteristic
130	C₄×C₇₉	#388	1	2	yes	—
131	C₄×C₇₉	#394	1	1	yes	characteristic
132	C₄×C₇₉	#397	1	1	yes	characteristic
133	C₄×C₇₉	#400	1	2	yes	—
134	Dic₇₉	—	8	2	no	—
135	#11295	—	79	1	no	maximal, Sylow
136	C₂×C₇₉	—	2	1	no	—
137	C₂×C₇₉	—	2	2	no	—
138	C₂×C₇₉	—	2	1	no	—
139	C₂×C₇₉	—	2	2	no	—
140	C₂×C₇₉	#1442	1	1	yes	characteristic
141	C₂×C₇₉	#2103	1	2	yes	—
142	C₂×#430	—	79	1	no	—
143	C₂×#344	—	79	2	no	—
144	C₂×#341	—	79	2	no	—
145	C₂×#338	D₇₉	1	1	yes	characteristic
146	#1729	—	79	1	no	—
147	C₇₉	#11295	1	1	yes	characteristic, complemented, Sylow
148	C₂²×M₁₆	—	79	2	no	—
149	C₂²×M₁₆	C₂×D₇₉	1	2	yes	—
150	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
151	C₂³×C₈	—	79	1	no	—
152	C₂×#117	—	79	1	no	—
153	#430	—	158	4	no	—
154	C₂×#108	—	158	2	no	—
155	#344	—	158	4	no	—
156	#341	—	158	4	no	—
157	#338	Dic₇₉	1	4	yes	complemented
158	#336	—	158	2	no	—
159	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	79	4	no	—
168	C₂×M₁₆	—	158	2	no	—
169	C₂×M₁₆	—	158	2	no	—
170	C₂×M₁₆	—	158	2	no	—
171	C₂×M₁₆	—	158	2	no	—
172	C₂×M₁₆	—	158	2	no	—
173	C₂×M₁₆	Dic₁₅₈	1	2	yes	—
174	C₂×M₁₆	C₂×Dic₇₉	1	4	yes	—
175	C₂×M₁₆	C₂×Dic₇₉	1	2	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	79	2	no	—
178	C₂²×C₈	—	158	1	no	—
179	C₂²×C₈	—	158	2	no	—
180	C₂²×C₈	—	158	2	no	—
181	C₂²×C₈	—	158	2	no	—
182	C₂²×C₈	—	158	1	no	—
183	C₂²×C₈	D₃₁₆	1	2	yes	—
184	#117	—	158	2	no	—
185	#117	—	158	2	no	—
186	C₂×C₄⋊C₄	—	158	1	no	—
187	C₂×C₂²⋊C₄	—	79	2	no	—
188	C₂×C₄²	—	158	1	no	—
189	#108	—	158	8	no	—
190	C₂⁴	Q₈×D₇₉	1	1	yes	characteristic
191	C₂²×C₄	—	158	2	no	—
192	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
193	C₂²×C₄	D₄×D₇₉	1	2	yes	—
194	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
196	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
197	C₂²×C₄	#11929623	1	2	yes	—
198	C₂²×C₄	#11929623	1	1	yes	characteristic
199	C₂²×C₄	#11929624	1	2	yes	—
200	C₂²×C₄	#11929624	1	2	yes	—
201	C₂²×C₄	#11929625	1	1	yes	characteristic
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	4	2	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	316	2	no	—
209	M₁₆	—	316	2	no	—
210	M₁₆	—	158	4	no	—
211	M₁₆	—	158	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	316	2	no	—
218	C₂×C₈	—	316	1	no	—
219	C₂×C₈	—	316	1	no	—
220	C₂×C₈	—	316	2	no	—
221	C₂×C₈	—	316	2	no	—
222	C₂×C₈	—	158	2	no	—
223	C₂×C₈	—	158	4	no	—
224	C₂×C₈	—	158	4	no	—
225	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
226	C₄⋊C₄	—	316	2	no	—
227	C₂²⋊C₄	—	158	4	no	—
228	C₄²	—	316	1	no	—
229	C₄²	—	316	1	no	—
230	C₂³	—	4	2	no	—
231	C₂³	Q₈×Dic₇₉	1	1	yes	characteristic
232	C₂³	#14579139	1	2	yes	—
233	C₂³	#14579141	1	1	yes	characteristic
234	C₂³	#14579151	1	2	yes	—
235	C₂³	#14579160	1	1	yes	characteristic
236	C₂×C₄	—	316	4	no	—
237	C₂×C₄	—	316	2	no	—
238	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
239	C₂×C₄	#14579120	1	2	yes	—
240	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
241	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
242	C₂×C₄	#14579131	1	1	yes	characteristic
243	C₂×C₄	#14579134	1	2	yes	—
244	C₂×C₄	#14579136	1	1	yes	characteristic
245	C₂×C₄	#14579138	1	2	yes	—
246	C₂×C₄	#14579143	1	2	yes	—
247	C₂×C₄	#14579145	1	1	yes	characteristic
248	C₂×C₄	#14579146	1	2	yes	—
249	C₂×C₄	#14579147	1	2	yes	—
250	C₂×C₄	#14579149	1	1	yes	characteristic
251	C₂×C₄	#14579149	1	1	yes	characteristic
252	C₂×C₄	#14579152	1	1	yes	characteristic
253	C₂×C₄	#14579153	1	2	yes	—
254	C₂×C₄	#14579154	1	2	yes	—
255	C₂×C₄	#14579156	1	2	yes	—
256	C₈	—	2	4	no	—
257	C₈	—	4	4	no	—
258	C₈	—	316	4	no	—
259	C₈	—	316	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	#19098449	1	2	yes	—
266	C₂²	#19098462	1	1	yes	characteristic
267	C₂²	#19098476	1	1	yes	characteristic
268	C₂²	#19098497	1	2	yes	—
269	C₂²	#19098503	1	1	yes	characteristic
270	C₄	—	632	2	no	—
271	C₄	#19098429	1	1	yes	characteristic
272	C₄	#19098438	1	1	yes	characteristic
273	C₄	#19098456	1	2	yes	—
274	C₄	#19098481	1	1	yes	characteristic
275	C₄	#19098490	1	1	yes	characteristic
276	C₄	#19098505	1	2	yes	—
277	C₂	—	2	1	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	2	no	—
281	C₂	#23761657	1	1	yes	characteristic
282	C₂	#23764981	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14579149
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₂×#338×C₇₉	C₂
socle	C₂²×C₇₉	#400
prime radical	C₂⁴×C₇₉	C₂×Q₈
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	C₂×#338	D₇₉
hypocenter	C₇₉	#11295
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#338

Complements

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

N	H ≅ G/N	type
16. #338×C₇₉	C₄	semidirect
147. C₇₉	#11295	semidirect
157. #338	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098612	2
C₂×#19098335	2
C₂×#19098327	2
#23763098	2
C₂×#338×C₇₉	2
#11295	79