Subgroups — #34196307

Order 20224.

654 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×#19098601	C₂	1	1	yes	maximal, characteristic
2	#23763133	C₂	1	2	yes	maximal
3	#23763117	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098320	C₂	1	2	yes	maximal
5	C₂×#424×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579120	C₂²	1	2	yes	—
7	C₂×#14579120	C₂²	1	2	yes	—
8	#19098601	—	2	4	no	—
9	C₂×#14579095	—	2	1	no	—
10	C₂×#14579093	—	2	1	no	—
11	#19098330	—	2	4	no	—
12	#19098326	—	2	2	no	—
13	#19098320	—	2	4	no	—
14	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
17	#424×C₇₉	—	2	2	no	—
18	#424×C₇₉	—	2	1	no	—
19	#424×C₇₉	—	2	1	no	—
20	C₂×C₄⋊C₈×C₇₉	C₄	1	4	yes	complemented
21	#14579120	—	2	2	no	—
22	#14579120	—	2	2	no	—
23	#14579120	—	2	2	no	—
24	#14579120	—	2	2	no	—
25	#14579120	—	2	2	no	—
26	#14579120	—	2	2	no	—
27	#14579120	D₄	1	4	yes	—
28	#14579120	Q₈	1	4	yes	—
29	C₂×C₄⋊Dic₇₉	—	2	2	no	—
30	C₂×C₄⋊Dic₇₉	—	2	2	no	—
31	C₂×C₂²⋊Dic₇₉	—	2	2	no	—
32	C₂×C₂²⋊Dic₇₉	—	2	2	no	—
33	C₂×C₄×Dic₇₉	—	2	2	no	—
34	C₂×C₄×Dic₇₉	—	2	2	no	—
35	#14579095	—	2	4	no	—
36	#14579093	—	2	4	no	—
37	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₇₉	—	2	1	no	—
39	C₂×M₁₆×C₇₉	—	2	1	no	—
40	C₂×M₁₆×C₇₉	—	2	2	no	—
41	C₂×M₁₆×C₇₉	—	2	2	no	—
42	C₂²×C₈×C₇₉	—	2	2	no	—
43	C₂²×C₈×C₇₉	—	2	2	no	—
44	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₇₉	D₄	1	2	yes	—
47	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
49	#117×C₇₉	—	2	1	no	—
50	#117×C₇₉	—	2	1	no	—
51	#117×C₇₉	—	2	1	no	—
52	#117×C₇₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₇₉	Q₈	1	2	yes	—
54	C₂×C₂²⋊C₄×C₇₉	D₄	1	2	yes	—
55	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
56	C₄⋊C₈×C₇₉	—	2	4	no	—
57	C₄⋊C₈×C₇₉	—	2	4	no	—
58	C₂²×Dic₇₉	—	4	2	no	—
59	C₂²×Dic₇₉	—	4	2	no	—
60	C₄⋊Dic₇₉	—	2	4	no	—
61	C₄⋊Dic₇₉	—	2	4	no	—
62	C₄⋊Dic₇₉	—	4	2	no	—
63	C₄⋊Dic₇₉	—	4	2	no	—
64	C₂²⋊Dic₇₉	—	2	4	no	—
65	C₂²⋊Dic₇₉	—	2	4	no	—
66	C₂²⋊Dic₇₉	—	4	2	no	—
67	C₂²⋊Dic₇₉	—	4	2	no	—
68	C₄×Dic₇₉	—	2	4	no	—
69	C₄×Dic₇₉	—	2	4	no	—
70	C₄×Dic₇₉	—	4	2	no	—
71	C₄×Dic₇₉	—	4	2	no	—
72	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
74	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
82	C₂²×C₄×C₇₉	#41	1	2	yes	—
83	C₂²×C₄×C₇₉	#41	1	2	yes	—
84	M₁₆×C₇₉	—	4	1	no	—
85	M₁₆×C₇₉	—	4	2	no	—
86	M₁₆×C₇₉	—	4	1	no	—
87	C₂×C₈×C₇₉	—	2	2	no	—
88	C₂×C₈×C₇₉	—	2	4	no	—
89	C₂×C₈×C₇₉	—	2	4	no	—
90	C₂×C₈×C₇₉	—	2	1	no	—
91	C₂×C₈×C₇₉	—	2	1	no	—
92	C₂×C₈×C₇₉	—	2	4	no	—
93	C₂×C₈×C₇₉	—	2	1	no	—
94	C₂×C₈×C₇₉	—	2	1	no	—
95	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
96	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
97	C₄⋊C₄×C₇₉	—	2	2	no	—
98	C₄⋊C₄×C₇₉	—	2	2	no	—
99	C₂²⋊C₄×C₇₉	—	2	2	no	—
100	C₂²⋊C₄×C₇₉	—	2	2	no	—
101	C₄²×C₇₉	—	2	2	no	—
102	C₄²×C₇₉	—	2	2	no	—
103	C₂³×C₇₉	—	4	2	no	—
104	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂³×C₇₉	#121	1	1	yes	characteristic
106	C₂³×C₇₉	#121	1	1	yes	characteristic
107	C₂³×C₇₉	#121	1	2	yes	—
108	C₂³×C₇₉	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₇₉	—	2	4	no	—
110	C₂×C₄×C₇₉	—	4	4	no	—
111	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
116	C₂×C₄×C₇₉	#121	1	2	yes	—
117	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
122	C₂×C₄×C₇₉	#123	1	2	yes	—
123	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
124	C₂×C₄×C₇₉	#125	1	2	yes	—
125	C₂×C₄×C₇₉	#125	1	2	yes	—
126	C₂×C₄×C₇₉	#126	1	2	yes	—
127	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
129	C₂×Dic₇₉	—	4	4	no	—
130	C₂×Dic₇₉	—	4	4	no	—
131	C₂×Dic₇₉	—	8	4	no	—
132	C₂×Dic₇₉	—	8	4	no	—
133	C₈×C₇₉	—	2	2	no	—
134	C₈×C₇₉	—	2	2	no	—
135	C₈×C₇₉	—	4	2	no	—
136	C₂²×C₇₉	—	2	2	no	—
137	C₂²×C₇₉	—	2	2	no	—
138	C₂²×C₇₉	—	2	2	no	—
139	C₂²×C₇₉	—	4	2	no	—
140	C₂²×C₇₉	—	4	2	no	—
141	C₂²×C₇₉	#380	1	1	yes	characteristic
142	C₂²×C₇₉	#389	1	2	yes	—
143	C₂²×C₇₉	#394	1	1	yes	characteristic
144	C₂²×C₇₉	#396	1	2	yes	—
145	C₂²×C₇₉	#398	1	1	yes	characteristic
146	C₄×C₇₉	—	4	4	no	—
147	C₄×C₇₉	#386	1	1	yes	characteristic
148	C₄×C₇₉	#387	1	2	yes	—
149	C₄×C₇₉	#391	1	1	yes	characteristic
150	C₄×C₇₉	#395	1	1	yes	characteristic
151	C₄×C₇₉	#398	1	2	yes	—
152	C₄×C₇₉	#400	1	1	yes	characteristic
153	Dic₇₉	—	8	4	no	—
154	Dic₇₉	—	8	4	no	—
155	#11229	—	79	1	no	maximal, Sylow
156	C₂×C₇₉	—	2	1	no	—
157	C₂×C₇₉	—	2	2	no	—
158	C₂×C₇₉	—	2	2	no	—
159	C₂×C₇₉	—	2	1	no	—
160	C₂×C₇₉	#1441	1	1	yes	characteristic
161	C₂×C₇₉	#1812	1	2	yes	—
162	C₂×#427	—	79	1	no	—
163	C₂×#424	D₇₉	1	1	yes	characteristic
164	#1740	—	79	2	no	—
165	#1735	—	79	1	no	—
166	C₂×H(3,ℤ/4)	—	79	2	no	—
167	C₇₉	#11229	1	1	yes	characteristic, complemented, Sylow
168	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
169	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
170	C₂×#117	—	79	2	no	—
171	C₂×#117	—	79	2	no	—
172	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
173	#427	—	158	4	no	—
174	C₂×#107	—	158	1	no	—
175	C₂×#106	—	158	1	no	—
176	#424	—	2	2	no	—
177	#424	—	2	1	no	—
178	#424	—	2	1	no	—
179	C₂×C₄⋊C₈	Dic₇₉	1	4	yes	complemented
180	#342	—	158	4	no	—
181	#340	—	158	2	no	—
182	H(3,ℤ/4)	—	158	4	no	—
183	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	1	no	—
186	C₂×M₁₆	—	2	1	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
191	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
192	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
193	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
194	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
195	#117	—	2	1	no	—
196	#117	—	2	1	no	—
197	#117	—	2	1	no	—
198	#117	—	2	1	no	—
199	#117	—	79	4	no	—
200	#117	—	79	4	no	—
201	#117	—	158	2	no	—
202	#117	—	158	2	no	—
203	#117	—	158	2	no	—
204	#117	—	158	2	no	—
205	#117	—	158	2	no	—
206	#117	—	158	2	no	—
207	C₂×C₄⋊C₄	—	158	2	no	—
208	C₂×C₄⋊C₄	—	158	2	no	—
209	C₂×C₄⋊C₄	Dic₁₅₈	1	2	yes	—
210	C₂×C₂²⋊C₄	—	158	2	no	—
211	C₂×C₂²⋊C₄	—	158	2	no	—
212	C₂×C₂²⋊C₄	D₃₁₆	1	2	yes	—
213	C₂×C₄²	—	158	2	no	—
214	C₂×C₄²	—	158	2	no	—
215	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
216	#107	—	158	4	no	—
217	#106	—	158	4	no	—
218	C₄⋊C₈	—	2	4	no	—
219	C₄⋊C₈	—	2	4	no	—
220	C₂⁴	C₂×D₃₁₆	1	1	yes	characteristic
221	C₂²×C₄	—	316	2	no	—
222	C₂²×C₄	—	316	2	no	—
223	C₂²×C₄	C₄⋊Dic₇₉	1	4	yes	—
224	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
225	C₂²×C₄	D₄×D₇₉	1	2	yes	—
226	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
228	C₂²×C₄	Q₈×D₇₉	1	2	yes	—
229	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
230	C₂²×C₄	#11929623	1	2	yes	—
231	C₂²×C₄	#11929624	1	1	yes	characteristic
232	C₂²×C₄	#11929624	1	1	yes	characteristic
233	C₂²×C₄	#11929625	1	2	yes	—
234	M₁₆	—	4	1	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	4	1	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
246	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	316	2	no	—
250	C₄⋊C₄	—	316	2	no	—
251	C₄⋊C₄	—	158	4	no	—
252	C₄⋊C₄	—	158	4	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	316	2	no	—
256	C₂²⋊C₄	—	316	2	no	—
257	C₂²⋊C₄	—	158	4	no	—
258	C₂²⋊C₄	—	158	4	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	316	2	no	—
262	C₄²	—	316	2	no	—
263	C₄²	—	158	4	no	—
264	C₄²	—	158	4	no	—
265	C₂³	—	4	2	no	—
266	C₂³	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
267	C₂³	#14579135	1	1	yes	characteristic
268	C₂³	#14579135	1	1	yes	characteristic
269	C₂³	#14579136	1	2	yes	—
270	C₂³	#14579141	1	2	yes	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	316	4	no	—
274	C₂×C₄	—	316	4	no	—
275	C₂×C₄	—	632	4	no	—
276	C₂×C₄	—	632	4	no	—
277	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
278	C₂×C₄	#14579120	1	1	yes	characteristic
279	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
280	C₂×C₄	Q₈×Dic₇₉	1	2	yes	—
281	C₂×C₄	#14579129	1	1	yes	characteristic
282	C₂×C₄	#14579133	1	2	yes	—
283	C₂×C₄	#14579137	1	2	yes	—
284	C₂×C₄	#14579139	1	2	yes	—
285	C₂×C₄	#14579140	1	1	yes	characteristic
286	C₂×C₄	#14579140	1	1	yes	characteristic
287	C₂×C₄	#14579142	1	1	yes	characteristic
288	C₂×C₄	#14579146	1	2	yes	—
289	C₂×C₄	#14579150	1	1	yes	characteristic
290	C₂×C₄	#14579151	1	2	yes	—
291	C₂×C₄	#14579152	1	2	yes	—
292	C₂×C₄	#14579156	1	2	yes	—
293	C₂×C₄	#14579159	1	1	yes	characteristic
294	C₂×C₄	#14579160	1	2	yes	—
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	4	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#19098423	1	1	yes	characteristic
304	C₂²	#19098462	1	2	yes	—
305	C₂²	#19098480	1	1	yes	characteristic
306	C₂²	#19098485	1	2	yes	—
307	C₂²	#19098493	1	1	yes	characteristic
308	C₄	—	4	4	no	—
309	C₄	—	632	4	no	—
310	C₄	—	632	4	no	—
311	C₄	#19098443	1	1	yes	characteristic
312	C₄	#19098449	1	2	yes	—
313	C₄	#19098470	1	1	yes	characteristic
314	C₄	#19098483	1	1	yes	characteristic
315	C₄	#19098496	1	2	yes	—
316	C₄	#19098502	1	1	yes	characteristic
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	—	2	2	no	—
320	C₂	—	2	1	no	—
321	C₂	#23761653	1	1	yes	characteristic
322	C₂	#23763446	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14579120
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₂×#424×C₇₉	C₂
socle	C₂²×C₇₉	#380
prime radical	C₂⁴×C₇₉	C₂×D₄
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	C₂×#424	D₇₉
hypocenter	C₇₉	#11229
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#424

Complements

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

N	H ≅ G/N	type
20. C₂×C₄⋊C₈×C₇₉	C₄	semidirect
167. C₇₉	#11229	semidirect
179. C₂×C₄⋊C₈	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098601	2
#23763133	2
#23763117	2
C₂×#19098320	2
C₂×#424×C₇₉	2
#11229	79