Subgroups — #34193920

Order 20224.

674 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098334	C₂	1	2	yes	maximal
2	C₂×#19098323	C₂	1	2	yes	maximal
3	C₂×C₇₉⋊#338	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098320	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×M₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	C₂²	1	2	yes	—
7	C₂×#14579120	C₂²	1	2	yes	—
8	#19098334	C₄	1	8	yes	complemented
9	#19098323	—	2	4	no	—
10	C₇₉⋊#338	—	2	2	no	—
11	#19098320	—	2	2	no	—
12	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₇₉	—	2	2	no	—
16	C₄×M₁₆×C₇₉	—	2	2	no	—
17	C₂×#97×C₇₉	C₄	1	2	yes	complemented
18	C₂×C₄×C₈×C₇₉	C₄	1	2	yes	complemented
19	C₂×#11929608	—	2	2	no	—
20	C₂×#11929608	—	2	2	no	—
21	C₂×#11929608	—	2	2	no	—
22	C₂×#11929608	—	2	2	no	—
23	C₂×#11929608	—	2	2	no	—
24	C₂×#11929608	C₂×C₄	1	4	yes	—
25	C₂²×C₇₉⋊C₈	—	2	2	no	—
26	#14579120	—	2	2	no	—
27	#14579120	—	2	2	no	—
28	#14579120	—	2	2	no	—
29	#14579120	C₂×C₄	1	4	yes	—
30	C₂×C₄⋊Dic₇₉	—	2	2	no	—
31	C₂×C₂²⋊Dic₇₉	—	2	2	no	—
32	C₂×C₄×Dic₇₉	—	2	2	no	—
33	C₂³×C₄×C₇₉	—	2	1	no	—
34	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₇₉	—	2	1	no	—
36	C₂×M₁₆×C₇₉	—	2	1	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	C₂×C₄	1	4	yes	—
41	C₂×M₁₆×C₇₉	D₄	1	2	yes	—
42	C₂×M₁₆×C₇₉	D₄	1	2	yes	—
43	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇₉	—	2	1	no	—
46	C₂×C₄²×C₇₉	—	2	1	no	—
47	C₂×C₄²×C₇₉	—	2	1	no	—
48	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₇₉	D₄	1	2	yes	—
51	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
52	#97×C₇₉	—	2	4	no	—
53	C₄×C₈×C₇₉	—	2	4	no	—
54	C₂²×Dic₇₉	—	4	2	no	—
55	#11929608	—	2	4	no	—
56	#11929608	—	2	4	no	—
57	#11929608	—	4	2	no	—
58	#11929608	—	4	2	no	—
59	C₂×C₇₉⋊C₈	—	2	4	no	—
60	C₂×C₇₉⋊C₈	—	4	2	no	—
61	C₂×C₇₉⋊C₈	—	4	2	no	—
62	C₄⋊Dic₇₉	—	2	4	no	—
63	C₄⋊Dic₇₉	—	4	2	no	—
64	C₂²⋊Dic₇₉	—	2	4	no	—
65	C₂²⋊Dic₇₉	—	4	2	no	—
66	C₄×Dic₇₉	—	2	4	no	—
67	C₄×Dic₇₉	—	4	2	no	—
68	C₂⁴×C₇₉	#41	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	—	2	1	no	—
70	C₂²×C₄×C₇₉	—	2	1	no	—
71	C₂²×C₄×C₇₉	—	2	2	no	—
72	C₂²×C₄×C₇₉	—	2	2	no	—
73	C₂²×C₄×C₇₉	—	4	4	no	—
74	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
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	2	yes	—
82	C₂²×C₄×C₇₉	#41	1	2	yes	—
83	C₂²×C₄×C₇₉	#41	1	2	yes	—
84	M₁₆×C₇₉	—	2	4	no	—
85	M₁₆×C₇₉	—	2	4	no	—
86	M₁₆×C₇₉	—	2	4	no	—
87	M₁₆×C₇₉	—	2	4	no	—
88	C₂×C₈×C₇₉	—	2	2	no	—
89	C₂×C₈×C₇₉	—	2	2	no	—
90	C₂×C₈×C₇₉	—	2	2	no	—
91	C₂×C₈×C₇₉	—	2	2	no	—
92	C₂×C₈×C₇₉	C₄²	1	4	yes	—
93	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
94	C₄²×C₇₉	—	2	1	no	—
95	C₄²×C₇₉	—	2	1	no	—
96	C₄²×C₇₉	—	2	2	no	—
97	C₄²×C₇₉	—	2	2	no	—
98	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
99	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
100	C₂³×C₇₉	—	4	2	no	—
101	C₂³×C₇₉	#117	1	2	yes	—
102	C₂³×C₇₉	#117	1	1	yes	characteristic
103	C₂³×C₇₉	#121	1	1	yes	characteristic
104	C₂³×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂³×C₇₉	#126	1	2	yes	—
106	C₂×C₄×C₇₉	—	2	4	no	—
107	C₂×C₄×C₇₉	—	2	2	no	—
108	C₂×C₄×C₇₉	—	2	2	no	—
109	C₂×C₄×C₇₉	—	4	4	no	—
110	C₂×C₄×C₇₉	—	4	4	no	—
111	C₂×C₄×C₇₉	—	4	2	no	—
112	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₉	#117	1	2	yes	—
116	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
118	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₇₉	#123	1	2	yes	—
126	C₂×C₄×C₇₉	#123	1	2	yes	—
127	C₂×C₄×C₇₉	#124	1	2	yes	—
128	C₂×C₄×C₇₉	#125	1	2	yes	—
129	C₂×C₄×C₇₉	#126	1	2	yes	—
130	C₂×Dic₇₉	—	4	4	no	—
131	C₂×Dic₇₉	—	8	4	no	—
132	C₈×C₇₉	—	2	4	no	—
133	C₈×C₇₉	—	2	4	no	—
134	C₇₉⋊C₈	—	4	4	no	—
135	C₂²×C₇₉	—	2	2	no	—
136	C₂²×C₇₉	—	2	4	no	—
137	C₂²×C₇₉	—	4	1	no	—
138	C₂²×C₇₉	—	4	1	no	—
139	C₂²×C₇₉	—	4	2	no	—
140	C₂²×C₇₉	#376	1	1	yes	characteristic
141	C₂²×C₇₉	#380	1	1	yes	characteristic
142	C₂²×C₇₉	#383	1	1	yes	characteristic
143	C₂²×C₇₉	#387	1	2	yes	—
144	C₂²×C₇₉	#388	1	2	yes	—
145	C₄×C₇₉	—	2	4	no	—
146	C₄×C₇₉	—	4	2	no	—
147	C₄×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
148	C₄×C₇₉	C₄×C₄⋊C₄	1	1	yes	characteristic
149	C₄×C₇₉	#381	1	1	yes	characteristic
150	C₄×C₇₉	#382	1	1	yes	characteristic
151	C₄×C₇₉	#387	1	2	yes	—
152	C₄×C₇₉	#388	1	2	yes	—
153	Dic₇₉	—	8	4	no	—
154	#10805	—	79	1	no	maximal, Sylow
155	C₂×C₇₉	—	2	2	no	—
156	C₂×C₇₉	—	2	1	no	—
157	C₂×C₇₉	—	2	1	no	—
158	C₂×C₇₉	—	2	2	no	—
159	C₂×C₇₉	#1434	1	1	yes	characteristic
160	C₂×C₇₉	#1841	1	2	yes	—
161	C₂×C₄×M₁₆	D₇₉	1	1	yes	characteristic
162	C₂×#344	—	79	2	no	—
163	C₂×#339	—	79	2	no	—
164	C₂×#338	—	79	1	no	—
165	C₂×H(3,ℤ/4)	—	79	1	no	—
166	C₇₉	#10805	1	1	yes	characteristic, complemented, Sylow
167	C₂²×M₁₆	—	79	2	no	—
168	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
169	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
170	C₂×#117	—	79	2	no	—
171	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
172	C₄×M₁₆	—	2	2	no	—
173	C₄×M₁₆	—	2	2	no	—
174	C₂×#97	Dic₇₉	1	2	yes	complemented
175	C₂×C₄×C₈	Dic₇₉	1	2	yes	complemented
176	#344	—	79	8	no	—
177	#339	—	158	4	no	—
178	#338	—	158	2	no	—
179	H(3,ℤ/4)	—	158	2	no	—
180	C₂³×C₄	—	2	1	no	—
181	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	79	4	no	—
188	C₂×M₁₆	—	158	2	no	—
189	C₂×M₁₆	—	158	2	no	—
190	C₂×M₁₆	—	158	2	no	—
191	C₂×M₁₆	—	158	2	no	—
192	C₂×M₁₆	—	158	2	no	—
193	C₂×M₁₆	C₄×D₇₉	1	4	yes	—
194	C₂×M₁₆	C₂²⋊D₇₉	1	2	yes	—
195	C₂×M₁₆	C₂²⋊D₇₉	1	2	yes	—
196	C₂²×C₈	—	158	2	no	—
197	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
198	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
199	#117	—	79	4	no	—
200	#117	—	158	2	no	—
201	#117	—	158	2	no	—
202	#117	—	158	2	no	—
203	C₂×C₄⋊C₄	—	158	2	no	—
204	C₂×C₂²⋊C₄	—	158	2	no	—
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	158	2	no	—
209	C₂×C₄²	Dic₁₅₈	1	2	yes	—
210	C₂×C₄²	D₃₁₆	1	2	yes	—
211	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
212	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
213	#97	—	2	4	no	—
214	C₄×C₈	—	2	4	no	—
215	C₂⁴	#11929624	1	1	yes	characteristic
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	316	2	no	—
222	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
223	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
227	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
228	C₂²×C₄	#11929624	1	1	yes	characteristic
229	C₂²×C₄	#11929624	1	2	yes	—
230	C₂²×C₄	#11929624	1	2	yes	—
231	C₂²×C₄	#11929624	1	2	yes	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	316	2	no	—
237	M₁₆	—	316	2	no	—
238	M₁₆	—	158	4	no	—
239	M₁₆	—	158	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	316	2	no	—
245	C₂×C₈	—	316	2	no	—
246	C₂×C₈	—	158	4	no	—
247	C₂×C₈	C₄×Dic₇₉	1	4	yes	—
248	C₂×C₈	C₂²⋊Dic₇₉	1	4	yes	—
249	C₄⋊C₄	—	316	2	no	—
250	C₄⋊C₄	—	158	4	no	—
251	C₂²⋊C₄	—	316	2	no	—
252	C₂²⋊C₄	—	158	4	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	316	2	no	—
258	C₄²	—	158	4	no	—
259	C₄²	C₄⋊Dic₇₉	1	2	yes	—
260	C₄²	C₄⋊Dic₇₉	1	2	yes	—
261	C₂³	—	4	2	no	—
262	C₂³	#14579118	1	2	yes	—
263	C₂³	#14579120	1	1	yes	characteristic
264	C₂³	#14579135	1	1	yes	characteristic
265	C₂³	#14579140	1	1	yes	characteristic
266	C₂³	#14579155	1	2	yes	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	316	4	no	—
274	C₂×C₄	—	632	4	no	—
275	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
276	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
278	C₂×C₄	#14579118	1	2	yes	—
279	C₂×C₄	#14579120	1	1	yes	characteristic
280	C₂×C₄	#14579120	1	1	yes	characteristic
281	C₂×C₄	#14579120	1	1	yes	characteristic
282	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
283	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
284	C₂×C₄	C₄×D₃₁₆	1	2	yes	—
285	C₂×C₄	C₄×Dic₁₅₈	1	2	yes	—
286	C₂×C₄	#14579135	1	1	yes	characteristic
287	C₂×C₄	#14579140	1	1	yes	characteristic
288	C₂×C₄	#14579144	1	2	yes	—
289	C₂×C₄	#14579144	1	2	yes	—
290	C₂×C₄	#14579149	1	2	yes	—
291	C₂×C₄	#14579153	1	2	yes	—
292	C₂×C₄	#14579155	1	2	yes	—
293	C₈	—	2	4	no	—
294	C₈	—	2	4	no	—
295	C₈	—	316	4	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	2	no	—
301	C₂²	#19098409	1	1	yes	characteristic
302	C₂²	#19098423	1	1	yes	characteristic
303	C₂²	#19098431	1	1	yes	characteristic
304	C₂²	#19098447	1	2	yes	—
305	C₂²	#19098454	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	—	632	4	no	—
309	C₄	C₄×C₂²⋊Dic₇₉	1	1	yes	characteristic
310	C₄	C₄×C₄⋊Dic₇₉	1	1	yes	characteristic
311	C₄	#19098426	1	1	yes	characteristic
312	C₄	#19098429	1	1	yes	characteristic
313	C₄	#19098447	1	2	yes	—
314	C₄	#19098454	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	#23761627	1	1	yes	characteristic
320	C₂	#23763549	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14579120
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₂×C₄×M₁₆×C₇₉	C₂
socle	C₂²×C₇₉	#383
prime radical	C₂⁴×C₇₉	#41
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₇₉
hypocenter	C₇₉	#10805
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
8. #19098334	C₄	semidirect
17. C₂×#97×C₇₉	C₄	semidirect
18. C₂×C₄×C₈×C₇₉	C₄	semidirect
166. C₇₉	#10805	semidirect
174. C₂×#97	Dic₇₉	semidirect
175. 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×#19098334	2
C₂×#19098323	2
C₂×C₇₉⋊#338	2
C₂×#19098320	2
C₂×C₄×M₁₆×C₇₉	2
#10805	79