Subgroups — #34307057

Order 20224.

790 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#11929608	C₂	1	2	yes	maximal
2	#23764176	C₂	1	2	yes	maximal, complemented
3	#23763607	C₂	1	4	yes	maximal, complemented
4	#23763538	C₂	1	1	yes	maximal, characteristic, complemented
5	#23763442	C₂	1	2	yes	maximal, complemented
6	C₄×#14579097	C₂	1	1	yes	maximal, characteristic, complemented
7	#23763127	C₂	1	2	yes	maximal, complemented
8	C₂×#436×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11929608	C₂²	1	4	yes	—
10	#19098641	—	2	2	no	—
11	#19098635	—	2	4	no	—
12	D₄×C₇₉⋊C₈	—	2	2	no	—
13	C₂×#14579097	C₂²	1	2	yes	—
14	C₂×#14579097	C₂²	1	4	yes	—
15	#19098587	C₂²	1	2	yes	—
16	#19098527	C₂²	1	4	yes	—
17	C₄×#11929608	C₂²	1	2	yes	—
18	#19098328	C₂²	1	2	yes	—
19	#19098328	C₂²	1	2	yes	—
20	#19098328	C₂²	1	1	yes	characteristic
21	#19098328	C₂²	1	1	yes	characteristic
22	#19098328	C₂²	1	4	yes	—
23	C₂²×M₁₆×C₇₉	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₇₉	C₂²	1	1	yes	characteristic
25	#436×C₇₉	C₄	1	4	yes	—
26	#436×C₇₉	C₄	1	4	yes	—
27	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₈×C₇₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
30	C₂×#11929608	—	2	4	no	—
31	C₂×#11929608	—	2	8	no	—
32	C₂×#11929608	—	4	4	no	—
33	C₂×#11929608	C₂³	1	2	yes	—
34	C₂×#11929608	C₂³	1	4	yes	—
35	C₂×#11929608	C₂³	1	2	yes	—
36	C₂²×C₇₉⋊C₈	—	2	4	no	—
37	#14579097	—	2	4	no	—
38	#14579097	—	4	4	no	—
39	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
40	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
41	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
42	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
43	#14579072	—	2	2	no	—
44	C₄×C₇₉⋊C₈	—	2	2	no	—
45	C₂²×D₄×C₇₉	Q₈	1	1	yes	characteristic
46	C₂³×C₄×C₇₉	C₂³	1	2	yes	—
47	C₂×M₁₆×C₇₉	—	4	4	no	—
48	C₂×M₁₆×C₇₉	C₂×C₄	1	8	yes	—
49	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
50	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
52	C₄×D₄×C₇₉	C₂×C₄	1	2	yes	—
53	C₄×D₄×C₇₉	C₂×C₄	1	2	yes	—
54	C₄×D₄×C₇₉	D₄	1	2	yes	—
55	C₄×D₄×C₇₉	D₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₇₉	Q₈	1	1	yes	characteristic
57	C₂×C₂²⋊C₄×C₇₉	Q₈	1	2	yes	—
58	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
60	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
61	C₂²⋊C₈×C₇₉	C₂×C₄	1	8	yes	—
62	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
64	#11929608	—	2	2	no	—
65	#11929608	—	2	4	no	—
66	#11929608	—	2	2	no	—
67	#11929608	—	2	2	no	—
68	#11929608	—	2	4	no	—
69	#11929608	—	4	8	no	—
70	#11929608	#41	1	4	yes	—
71	C₂×C₇₉⋊C₈	—	2	2	no	—
72	C₂×C₇₉⋊C₈	—	2	2	no	—
73	C₂×C₇₉⋊C₈	—	2	4	no	—
74	C₂×C₇₉⋊C₈	—	4	8	no	—
75	C₂⁴×C₇₉	C₂×Q₈	1	2	yes	—
76	C₂×D₄×C₇₉	—	4	2	no	—
77	C₂×D₄×C₇₉	C₄⋊C₄	1	2	yes	—
78	C₂×D₄×C₇₉	C₄⋊C₄	1	2	yes	—
79	C₂²×C₄×C₇₉	—	2	4	no	—
80	C₂²×C₄×C₇₉	—	2	4	no	—
81	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₇₉	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
87	M₁₆×C₇₉	—	4	8	no	—
88	C₂×C₈×C₇₉	—	2	2	no	—
89	C₂×C₈×C₇₉	—	2	2	no	—
90	C₂×C₈×C₇₉	—	2	4	no	—
91	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
94	C₄⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
95	C₄⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
96	C₂²⋊C₄×C₇₉	C₄⋊C₄	1	4	yes	—
97	C₂²⋊C₄×C₇₉	C₄⋊C₄	1	4	yes	—
98	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
102	C₂³×C₇₉	—	2	4	no	—
103	C₂³×C₇₉	—	4	2	no	—
104	C₂³×C₇₉	—	4	2	no	—
105	C₂³×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
106	C₂³×C₇₉	C₂²×Q₈	1	1	yes	characteristic
107	D₄×C₇₉	—	2	4	no	—
108	D₄×C₇₉	—	4	2	no	—
109	C₂×C₄×C₇₉	—	2	4	no	—
110	C₂×C₄×C₇₉	—	2	4	no	—
111	C₂×C₄×C₇₉	—	2	4	no	—
112	C₂×C₄×C₇₉	—	2	4	no	—
113	C₂×C₄×C₇₉	—	2	4	no	—
114	C₂×C₄×C₇₉	—	2	2	no	—
115	C₂×C₄×C₇₉	—	4	1	no	—
116	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
124	C₂×C₄×C₇₉	H(5,𝔽₂)	1	2	yes	—
125	C₈×C₇₉	—	2	2	no	—
126	C₈×C₇₉	—	2	4	no	—
127	C₈×C₇₉	—	2	2	no	—
128	C₇₉⋊C₈	—	2	4	no	—
129	C₇₉⋊C₈	—	4	4	no	—
130	C₇₉⋊C₈	—	4	2	no	—
131	C₂²×C₇₉	—	2	1	no	—
132	C₂²×C₇₉	—	2	1	no	—
133	C₂²×C₇₉	—	2	4	no	—
134	C₂²×C₇₉	—	2	4	no	—
135	C₂²×C₇₉	—	2	4	no	—
136	C₂²×C₇₉	—	8	4	no	—
137	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₇₉	#554	1	2	yes	—
139	C₄×C₇₉	—	2	4	no	—
140	C₄×C₇₉	—	2	1	no	—
141	C₄×C₇₉	—	2	1	no	—
142	C₄×C₇₉	—	2	1	no	—
143	C₄×C₇₉	#430	1	2	yes	—
144	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
145	C₄×C₇₉	#518	1	1	yes	characteristic
146	C₄×C₇₉	#546	1	2	yes	—
147	#22719	—	79	1	no	maximal, Sylow
148	C₂×C₇₉	—	2	2	no	—
149	C₂×C₇₉	—	4	4	no	—
150	C₂×C₇₉	#2349	1	1	yes	characteristic
151	C₂×C₇₉	C₂×#430	1	1	yes	characteristic
152	C₂×C₇₉	#2916	1	1	yes	characteristic
153	D₄×M₁₆	—	79	2	no	—
154	C₂×#436	D₇₉	1	1	yes	characteristic
155	#1950	—	79	2	no	—
156	#1855	—	79	4	no	—
157	#1837	—	79	1	no	—
158	#1811	—	79	2	no	—
159	C₄×#108	—	79	1	no	—
160	#1738	—	79	2	no	—
161	C₇₉	#22719	1	1	yes	characteristic, complemented, Sylow
162	C₂²×M₁₆	—	79	4	no	—
163	C₂²×M₁₆	C₂×D₇₉	1	2	yes	—
164	C₂×C₄×D₄	C₂×D₇₉	1	1	yes	characteristic
165	#436	—	158	2	no	—
166	#436	Dic₇₉	1	4	yes	—
167	#436	Dic₇₉	1	4	yes	—
168	#435	—	158	4	no	—
169	C₈×D₄	—	158	2	no	—
170	C₂×#108	—	79	4	no	—
171	C₂×#108	—	79	2	no	—
172	#423	—	79	2	no	—
173	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
174	#407	—	79	4	no	—
175	C₂×C₂²⋊C₈	C₂×D₇₉	1	2	yes	—
176	C₄×M₁₆	—	79	2	no	—
177	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
178	#341	—	79	1	no	—
179	#341	—	79	2	no	—
180	#341	—	79	2	no	—
181	#341	—	79	4	no	—
182	#341	—	79	1	no	—
183	C₂²×D₄	Dic₁₅₈	1	1	yes	characteristic
184	C₂³×C₄	C₂²×D₇₉	1	2	yes	—
185	C₂×M₁₆	—	4	4	no	—
186	C₂×M₁₆	—	316	4	no	—
187	C₂×M₁₆	—	79	2	no	—
188	C₂×M₁₆	—	79	4	no	—
189	C₂×M₁₆	—	79	2	no	—
190	C₂×M₁₆	—	158	8	no	—
191	C₂×M₁₆	—	158	4	no	—
192	C₂×M₁₆	C₂×Dic₇₉	1	8	yes	—
193	C₂²×C₈	—	158	4	no	—
194	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
196	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
197	C₄×D₄	D₃₁₆	1	2	yes	—
198	C₄×D₄	D₃₁₆	1	2	yes	—
199	C₄×D₄	C₂×Dic₇₉	1	2	yes	—
200	C₄×D₄	C₂×Dic₇₉	1	2	yes	—
201	C₂×C₄⋊C₄	Dic₁₅₈	1	1	yes	characteristic
202	C₂×C₂²⋊C₄	Dic₁₅₈	1	2	yes	—
203	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
204	#108	—	316	4	no	—
205	#108	—	158	4	no	—
206	C₄⋊C₈	—	158	2	no	—
207	C₄⋊C₈	—	158	2	no	—
208	C₄⋊C₈	C₂×Dic₇₉	1	2	yes	—
209	C₄⋊C₈	C₂×Dic₇₉	1	2	yes	—
210	C₂²⋊C₈	—	158	4	no	—
211	C₂²⋊C₈	—	158	4	no	—
212	C₂²⋊C₈	C₂×Dic₇₉	1	8	yes	—
213	#97	—	158	2	no	—
214	C₄×C₈	—	158	2	no	—
215	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
216	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
217	C₂⁴	C₂×Dic₁₅₈	1	2	yes	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	C₄⋊Dic₇₉	1	2	yes	—
220	C₂×D₄	C₄⋊Dic₇₉	1	2	yes	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	C₂²×Dic₇₉	1	4	yes	—
224	C₂²×C₄	C₂×D₃₁₆	1	4	yes	—
225	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
226	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
227	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
228	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
229	M₁₆	—	4	8	no	—
230	M₁₆	—	316	8	no	—
231	M₁₆	—	79	4	no	—
232	M₁₆	—	158	2	no	—
233	M₁₆	—	158	4	no	—
234	M₁₆	—	158	2	no	—
235	M₁₆	—	158	4	no	—
236	M₁₆	—	158	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	316	8	no	—
241	C₂×C₈	—	158	2	no	—
242	C₂×C₈	—	158	4	no	—
243	C₂×C₈	—	158	2	no	—
244	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
245	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
246	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
247	C₄⋊C₄	C₄⋊Dic₇₉	1	2	yes	—
248	C₄⋊C₄	C₄⋊Dic₇₉	1	2	yes	—
249	C₂²⋊C₄	C₄⋊Dic₇₉	1	4	yes	—
250	C₂²⋊C₄	C₄⋊Dic₇₉	1	4	yes	—
251	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
252	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
253	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
254	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
255	C₂³	—	2	4	no	—
256	C₂³	—	4	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	C₂×C₄⋊Dic₇₉	1	4	yes	—
259	C₂³	C₂²×Dic₁₅₈	1	1	yes	characteristic
260	D₄	—	2	4	no	—
261	D₄	—	4	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
273	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
274	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
276	C₂×C₄	#41×D₇₉	1	2	yes	—
277	C₂×C₄	#14579206	1	2	yes	—
278	C₈	—	2	2	no	—
279	C₈	—	2	2	no	—
280	C₈	—	2	4	no	—
281	C₈	—	316	4	no	—
282	C₈	—	316	2	no	—
283	C₈	—	158	4	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	8	4	no	—
290	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
291	C₂²	#19099272	1	2	yes	—
292	C₄	—	2	4	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	#19098612	1	2	yes	—
297	C₄	#41×Dic₇₉	1	1	yes	characteristic
298	C₄	#19099038	1	1	yes	characteristic
299	C₄	#19099208	1	2	yes	—
300	C₂	—	2	2	no	—
301	C₂	—	4	4	no	—
302	C₂	#23766310	1	1	yes	characteristic
303	C₂	C₂×#19098612	1	1	yes	characteristic
304	C₂	#23772011	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×#436×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×C₄⋊C₄
prime radical	C₂²×D₄×C₇₉	Q₈
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×#436	D₇₉
hypocenter	C₇₉	#22719
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄ ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
2. #23764176	C₂	semidirect
3. #23763607	C₂	semidirect
4. #23763538	C₂	semidirect
5. #23763442	C₂	semidirect
6. C₄×#14579097	C₂	semidirect
7. #23763127	C₂	semidirect
161. C₇₉	#22719	semidirect

Maximal subgroups

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

subgroup	index
D₄×#11929608	2
#23764176	2
#23763607	2
#23763538	2
#23763442	2
C₄×#14579097	2
#23763127	2
C₂×#436×C₇₉	2
#22719	79