Subgroups — #34178363

Order 20224.

726 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098620	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098433	C₂	1	1	yes	maximal, characteristic
3	#23761284	C₂	1	4	yes	maximal, complemented
4	C₂×#432×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₇₉	—	2	1	no	—
6	C₂²×C₄⋊Dic₇₉	—	2	1	no	—
7	C₂²×C₄⋊Dic₇₉	C₄	1	2	yes	—
8	#19098620	—	2	2	no	—
9	#19098620	C₄	1	4	yes	complemented
10	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
12	C₂×#14579072	—	2	1	no	—
13	#19098433	—	2	1	no	—
14	#19098433	—	2	1	no	—
15	#19098433	C₄	1	2	yes	—
16	#19098433	C₂²	1	2	yes	—
17	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
18	#432×C₇₉	—	2	2	no	—
19	#432×C₇₉	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₇₉	—	2	1	no	—
22	C₂×#97×C₇₉	—	2	1	no	—
23	C₂³×Dic₇₉	—	2	2	no	—
24	C₂²×C₇₉⋊C₈	—	2	2	no	—
25	C₂×C₄⋊Dic₇₉	—	2	2	no	—
26	C₂×C₄⋊Dic₇₉	—	2	2	no	—
27	C₂×C₄⋊Dic₇₉	—	2	2	no	—
28	C₂×C₄⋊Dic₇₉	—	2	2	no	—
29	C₂×C₄⋊Dic₇₉	—	2	2	no	—
30	C₂×C₄⋊Dic₇₉	—	4	4	no	—
31	C₂×C₄⋊Dic₇₉	—	4	4	no	—
32	C₂×C₄⋊Dic₇₉	—	4	4	no	—
33	C₂×C₄⋊Dic₇₉	C₂×C₄	1	4	yes	—
34	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
35	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	complemented
36	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
37	#14579072	—	2	4	no	—
38	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₉	—	2	2	no	—
42	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₉	D₄	1	2	yes	—
47	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
49	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₇₉	—	2	2	no	—
51	C₄⋊C₈×C₇₉	C₂×C₄	1	4	yes	complemented
52	C₂²⋊C₈×C₇₉	—	2	4	no	—
53	#97×C₇₉	—	2	4	no	—
54	C₂²×Dic₇₉	—	2	4	no	—
55	C₂²×Dic₇₉	—	4	4	no	—
56	C₂²×Dic₇₉	—	4	4	no	—
57	C₂²×Dic₇₉	—	4	4	no	—
58	C₂×C₇₉⋊C₈	—	2	4	no	—
59	C₂×C₇₉⋊C₈	—	4	2	no	—
60	C₂×C₇₉⋊C₈	—	4	2	no	—
61	C₄⋊Dic₇₉	—	4	4	no	—
62	C₄⋊Dic₇₉	—	4	4	no	—
63	C₄⋊Dic₇₉	—	4	4	no	—
64	C₄⋊Dic₇₉	—	4	4	no	—
65	C₄⋊Dic₇₉	—	4	4	no	—
66	C₄⋊Dic₇₉	—	4	4	no	—
67	C₂⁴×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	—	2	2	no	—
69	C₂²×C₄×C₇₉	—	2	2	no	—
70	C₂²×C₄×C₇₉	—	2	2	no	—
71	C₂²×C₄×C₇₉	—	2	2	no	—
72	C₂²×C₄×C₇₉	—	2	2	no	—
73	C₂²×C₄×C₇₉	—	2	2	no	—
74	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
75	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
80	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
81	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
82	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
83	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₇₉	—	2	4	no	—
87	C₂×C₈×C₇₉	—	4	2	no	—
88	C₂×C₈×C₇₉	—	4	2	no	—
89	C₄²×C₇₉	C₄²	1	2	yes	—
90	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
91	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
93	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
94	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
95	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
99	C₂³×C₇₉	—	2	1	no	—
100	C₂³×C₇₉	—	2	1	no	—
101	C₂³×C₇₉	—	2	1	no	—
102	C₂³×C₇₉	—	2	1	no	—
103	C₂³×C₇₉	#95	1	2	yes	—
104	C₂³×C₇₉	#100	1	2	yes	—
105	C₂³×C₇₉	#101	1	2	yes	—
106	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₉	—	2	4	no	—
108	C₂×C₄×C₇₉	—	2	4	no	—
109	C₂×C₄×C₇₉	—	2	1	no	—
110	C₂×C₄×C₇₉	—	2	1	no	—
111	C₂×C₄×C₇₉	—	2	1	no	—
112	C₂×C₄×C₇₉	—	2	1	no	—
113	C₂×C₄×C₇₉	—	2	1	no	—
114	C₂×C₄×C₇₉	—	2	1	no	—
115	C₂×C₄×C₇₉	—	2	1	no	—
116	C₂×C₄×C₇₉	—	2	1	no	—
117	C₂×C₄×C₇₉	—	2	4	no	—
118	C₂×C₄×C₇₉	—	2	4	no	—
119	C₂×C₄×C₇₉	—	2	4	no	—
120	C₂×C₄×C₇₉	—	2	4	no	—
121	C₂×C₄×C₇₉	#95	1	2	yes	—
122	C₂×C₄×C₇₉	#95	1	2	yes	—
123	C₂×C₄×C₇₉	#95	1	2	yes	—
124	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
132	C₂×C₄×C₇₉	#137	1	2	yes	—
133	C₂×C₄×C₇₉	#137	1	2	yes	—
134	C₂×Dic₇₉	—	4	4	no	—
135	C₂×Dic₇₉	—	4	4	no	—
136	C₂×Dic₇₉	—	4	4	no	—
137	C₂×Dic₇₉	—	8	8	no	—
138	C₈×C₇₉	—	4	4	no	—
139	C₇₉⋊C₈	—	4	4	no	—
140	C₂²×C₇₉	—	2	2	no	—
141	C₂²×C₇₉	—	2	2	no	—
142	C₂²×C₇₉	—	2	2	no	—
143	C₂²×C₇₉	—	2	2	no	—
144	C₂²×C₇₉	—	2	2	no	—
145	C₂²×C₇₉	—	2	2	no	—
146	C₂²×C₇₉	#343	1	4	yes	—
147	C₂²×C₇₉	C₂×#95	1	1	yes	characteristic
148	C₂²×C₇₉	C₂×#100	1	1	yes	characteristic
149	C₂²×C₇₉	C₂×#101	1	1	yes	characteristic
150	C₂²×C₇₉	#418	1	1	yes	characteristic
151	C₂²×C₇₉	#418	1	1	yes	characteristic
152	C₂²×C₇₉	#419	1	1	yes	characteristic
153	C₂²×C₇₉	#419	1	1	yes	characteristic
154	C₄×C₇₉	—	2	2	no	—
155	C₄×C₇₉	—	2	2	no	—
156	C₄×C₇₉	—	2	2	no	—
157	C₄×C₇₉	—	2	1	no	—
158	C₄×C₇₉	—	2	1	no	—
159	C₄×C₇₉	#417	1	2	yes	—
160	C₄×C₇₉	#417	1	2	yes	—
161	C₄×C₇₉	#428	1	2	yes	—
162	C₄×C₇₉	#428	1	2	yes	—
163	Dic₇₉	—	8	4	no	—
164	#8040	—	79	1	no	maximal, Sylow
165	C₂×C₇₉	—	2	4	no	—
166	C₂×C₇₉	#1555	1	1	yes	characteristic
167	C₂×C₇₉	#1556	1	1	yes	characteristic
168	C₂×C₇₉	#1556	1	1	yes	characteristic
169	C₂×C₇₉	#1558	1	1	yes	characteristic
170	C₂×C₇₉	#1736	1	1	yes	characteristic
171	C₂×C₇₉	#1736	1	1	yes	characteristic
172	C₂×C₇₉	C₂×#343	1	1	yes	characteristic
173	C₂×#432	—	79	1	no	—
174	C₂×#432	D₇₉	1	1	yes	characteristic
175	C₂×#384	—	79	1	no	—
176	#1296	—	79	4	no	—
177	C₇₉	#8040	1	1	yes	characteristic, complemented, Sylow
178	C₂²×C₄⋊C₄	—	79	2	no	—
179	C₂²×C₄⋊C₄	—	158	1	no	—
180	C₂²×C₄⋊C₄	—	158	1	no	—
181	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
182	#432	—	2	2	no	—
183	#432	—	79	4	no	—
184	#432	—	158	2	no	—
185	#432	Dic₇₉	1	4	yes	complemented
186	C₂×C₄⋊C₈	—	79	2	no	—
187	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
188	C₂×C₂²⋊C₈	—	2	1	no	—
189	C₂×C₂²⋊C₈	—	158	1	no	—
190	C₂×#97	—	2	1	no	—
191	C₂×#97	—	158	1	no	—
192	#384	—	79	2	no	—
193	#384	—	79	2	no	—
194	#384	—	158	1	no	—
195	#384	—	158	1	no	—
196	C₂³×C₄	—	158	2	no	—
197	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
198	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
199	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	158	2	no	—
202	C₂×C₄⋊C₄	—	316	4	no	—
203	C₂×C₄⋊C₄	—	316	4	no	—
204	C₂×C₄⋊C₄	—	316	4	no	—
205	C₂×C₄⋊C₄	—	79	4	no	—
206	C₂×C₄⋊C₄	—	158	2	no	—
207	C₂×C₄⋊C₄	—	158	2	no	—
208	C₂×C₄⋊C₄	—	158	2	no	—
209	C₂×C₄⋊C₄	—	158	2	no	—
210	C₂×C₄⋊C₄	—	158	2	no	—
211	C₂×C₄²	Dic₁₅₈	1	2	yes	—
212	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₇₉	1	2	yes	—
214	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
215	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
216	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
217	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
218	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	—	79	4	no	—
221	C₄⋊C₈	—	158	2	no	—
222	C₄⋊C₈	C₂×Dic₇₉	1	4	yes	complemented
223	C₂²⋊C₈	—	2	4	no	—
224	C₂²⋊C₈	—	158	4	no	—
225	#97	—	2	4	no	—
226	#97	—	158	4	no	—
227	C₂⁴	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	316	4	no	—
235	C₂²×C₄	—	316	4	no	—
236	C₂²×C₄	—	316	4	no	—
237	C₂²×C₄	—	158	4	no	—
238	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
239	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
240	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
241	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
242	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
243	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
244	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
245	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
246	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
247	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	316	2	no	—
254	C₂×C₈	—	316	2	no	—
255	C₂×C₈	—	158	4	no	—
256	C₄⋊C₄	—	316	4	no	—
257	C₄⋊C₄	—	316	4	no	—
258	C₄⋊C₄	—	316	4	no	—
259	C₄⋊C₄	—	316	4	no	—
260	C₄⋊C₄	—	316	4	no	—
261	C₄⋊C₄	—	316	4	no	—
262	C₄²	C₄×Dic₇₉	1	2	yes	—
263	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
264	C₄²	C₂²⋊Dic₇₉	1	2	yes	—
265	C₄²	Dic₇₉⋊C₄	1	2	yes	—
266	C₄²	Dic₇₉⋊C₄	1	2	yes	—
267	C₄²	C₄⋊Dic₇₉	1	2	yes	—
268	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
269	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
270	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
271	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	C₇₉⋊#95	1	2	yes	—
277	C₂³	#14579077	1	2	yes	—
278	C₂³	#14579079	1	2	yes	—
279	C₂³	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	316	4	no	—
295	C₂×C₄	—	316	4	no	—
296	C₂×C₄	—	316	4	no	—
297	C₂×C₄	—	632	8	no	—
298	C₂×C₄	C₇₉⋊#95	1	2	yes	—
299	C₂×C₄	C₇₉⋊#95	1	2	yes	—
300	C₂×C₄	C₇₉⋊#95	1	2	yes	—
301	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
302	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
303	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
304	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
305	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
307	C₂×C₄	#14579186	1	2	yes	—
308	C₂×C₄	#14579189	1	2	yes	—
309	C₂×C₄	#14579191	1	2	yes	—
310	C₂×C₄	#14579194	1	2	yes	—
311	C₈	—	4	4	no	—
312	C₈	—	316	4	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	#19098331	1	4	yes	—
320	C₂²	C₂×C₇₉⋊#95	1	1	yes	characteristic
321	C₂²	C₂×#14579077	1	1	yes	characteristic
322	C₂²	C₂×#14579079	1	1	yes	characteristic
323	C₂²	#19098567	1	1	yes	characteristic
324	C₂²	#19098569	1	1	yes	characteristic
325	C₂²	#19098571	1	1	yes	characteristic
326	C₂²	#19098573	1	1	yes	characteristic
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	632	4	no	—
333	C₄	#19098564	1	2	yes	—
334	C₄	#19098565	1	2	yes	—
335	C₄	#19098605	1	2	yes	—
336	C₄	#19098606	1	2	yes	—
337	C₂	—	2	4	no	—
338	C₂	#23762113	1	1	yes	characteristic
339	C₂	#23762116	1	1	yes	characteristic
340	C₂	#23762118	1	1	yes	characteristic
341	C₂	#23762125	1	1	yes	characteristic
342	C₂	#23763120	1	1	yes	characteristic
343	C₂	#23763121	1	1	yes	characteristic
344	C₂	C₂×#19098331	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#432×C₇₉	C₂
socle	C₂³×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#432	D₇₉
hypocenter	C₇₉	#8040
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #23761284	C₂	semidirect
9. #19098620	C₄	semidirect
19. #432×C₇₉	C₄	semidirect
35. C₄⋊(C₇₉⋊C₈)	C₂×C₄	semidirect
51. C₄⋊C₈×C₇₉	C₂×C₄	semidirect
177. C₇₉	#8040	semidirect
185. #432	Dic₇₉	semidirect
222. C₄⋊C₈	C₂×Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098620	2
C₂×#19098433	2
#23761284	2
C₂×#432×C₇₉	2
#8040	79