Subgroups — #34307270

Order 20224.

642 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772351	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic
3	#23764174	C₂	1	4	yes	maximal
4	#23764173	C₂	1	2	yes	maximal
5	#23763536	C₂	1	2	yes	maximal
6	C₄×#14579097	C₂	1	1	yes	maximal, characteristic
7	#23763301	C₂	1	2	yes	maximal
8	C₇₉×#1837	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
10	#19098692	—	2	4	no	—
11	#19098690	—	2	2	no	—
12	C₈×Dic₁₅₈	C₄	1	4	yes	—
13	C₈×Dic₁₅₈	C₄	1	2	yes	—
14	C₈×Dic₁₅₈	C₄	1	2	yes	—
15	#19098687	—	2	2	no	—
16	C₂×#14579097	C₂²	1	2	yes	—
17	C₂×#14579096	C₂²	1	4	yes	—
18	#19098586	C₂²	1	4	yes	—
19	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×Dic₇₉⋊C₈	C₂²	1	2	yes	—
21	C₄×#11929608	C₂²	1	2	yes	—
22	M₁₆×Dic₇₉	C₂²	1	4	yes	—
23	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
24	#19098328	C₂²	1	1	yes	characteristic
25	#19098328	C₂²	1	1	yes	characteristic
26	#19098327	C₂²	1	4	yes	—
27	#423×C₇₉	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
29	#341×C₇₉	C₂²	1	2	yes	—
30	#341×C₇₉	C₂²	1	2	yes	—
31	C₂²×Dic₁₅₈	Q₈	1	1	yes	characteristic
32	C₂×#11929608	C₂³	1	2	yes	—
33	C₂×#11929608	C₂³	1	2	yes	—
34	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
35	C₄×Dic₁₅₈	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₁₅₈	C₂×C₄	1	2	yes	—
37	C₄×Dic₁₅₈	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₁₅₈	D₄	1	1	yes	characteristic
39	C₄×Dic₁₅₈	D₄	1	1	yes	characteristic
40	C₄×Dic₁₅₈	D₄	1	2	yes	—
41	C₂×C₄⋊Dic₇₉	Q₈	1	1	yes	characteristic
42	C₂×Dic₇₉⋊C₄	Q₈	1	2	yes	—
43	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
44	#14579097	—	2	4	no	—
45	#14579096	—	2	8	no	—
46	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
47	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
48	Dic₇₉⋊C₈	—	2	4	no	—
49	Dic₇₉⋊C₈	—	2	4	no	—
50	Dic₇₉⋊C₈	C₂×C₄	1	4	yes	—
51	Dic₇₉⋊C₈	C₂×C₄	1	4	yes	—
52	#14579072	—	2	2	no	—
53	#14579071	—	2	4	no	—
54	C₄×C₇₉⋊C₈	—	2	2	no	—
55	C₈×Dic₇₉	—	2	4	no	—
56	C₈×Dic₇₉	C₂×C₄	1	4	yes	—
57	C₈×Dic₇₉	C₂×C₄	1	4	yes	—
58	C₂×M₁₆×C₇₉	C₂³	1	4	yes	—
59	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₇₉	—	2	2	no	—
63	C₄⋊C₈×C₇₉	—	2	2	no	—
64	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
66	C₂×Dic₁₅₈	—	2	4	no	—
67	C₂×Dic₁₅₈	C₄⋊C₄	1	2	yes	—
68	C₂×Dic₁₅₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂×Dic₁₅₈	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×Dic₇₉	C₂×Q₈	1	2	yes	—
71	C₂²×Dic₇₉	C₂×Q₈	1	2	yes	—
72	#11929608	—	2	2	no	—
73	#11929608	—	2	2	no	—
74	#11929608	—	2	2	no	—
75	#11929608	#41	1	4	yes	—
76	C₂×C₇₉⋊C₈	—	2	2	no	—
77	C₂×C₇₉⋊C₈	—	2	2	no	—
78	C₂×C₇₉⋊C₈	—	2	4	no	—
79	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
80	C₄⋊Dic₇₉	C₄⋊C₄	1	2	yes	—
81	C₄⋊Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
82	C₄⋊Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
83	Dic₇₉⋊C₄	C₄⋊C₄	1	2	yes	—
84	Dic₇₉⋊C₄	C₄⋊C₄	1	2	yes	—
85	Dic₇₉⋊C₄	C₄⋊C₄	1	2	yes	—
86	Dic₇₉⋊C₄	C₄⋊C₄	1	2	yes	—
87	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
88	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
89	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
90	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
92	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
94	M₁₆×C₇₉	—	2	4	no	—
95	M₁₆×C₇₉	—	2	4	no	—
96	C₂×C₈×C₇₉	—	2	4	no	—
97	C₂×C₈×C₇₉	—	2	2	no	—
98	C₂×C₈×C₇₉	—	2	2	no	—
99	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
101	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
105	C₂³×C₇₉	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₇₉	—	2	2	no	—
107	C₂×C₄×C₇₉	—	4	1	no	—
108	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
115	C₂×C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
116	C₂×Dic₇₉	—	2	4	no	—
117	C₂×Dic₇₉	—	4	2	no	—
118	C₂×Dic₇₉	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×Dic₇₉	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×Dic₇₉	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×Dic₇₉	C₂×C₄⋊C₄	1	2	yes	—
122	Dic₁₅₈	—	2	2	no	—
123	Dic₁₅₈	—	2	4	no	—
124	Dic₁₅₈	#108	1	4	yes	complemented
125	C₈×C₇₉	—	2	2	no	—
126	C₈×C₇₉	—	2	2	no	—
127	C₈×C₇₉	—	4	4	no	—
128	C₇₉⋊C₈	—	2	4	no	—
129	C₇₉⋊C₈	—	2	4	no	—
130	C₇₉⋊C₈	—	4	2	no	—
131	C₂²×C₇₉	—	2	1	no	—
132	C₂²×C₇₉	—	2	1	no	—
133	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₇₉	#557	1	2	yes	—
135	C₄×C₇₉	—	2	1	no	—
136	C₄×C₇₉	—	2	1	no	—
137	C₄×C₇₉	—	2	1	no	—
138	C₄×C₇₉	C₂×#108	1	2	yes	—
139	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
140	C₄×C₇₉	#519	1	1	yes	characteristic
141	C₄×C₇₉	#548	1	2	yes	—
142	Dic₇₉	—	2	2	no	—
143	Dic₇₉	—	2	2	no	—
144	Dic₇₉	—	2	2	no	—
145	Dic₇₉	C₂×#108	1	4	yes	—
146	#22746	—	79	1	no	maximal, Sylow
147	C₂×C₇₉	—	2	2	no	—
148	C₂×C₇₉	#2351	1	1	yes	characteristic
149	C₂×C₇₉	C₂²×#108	1	1	yes	characteristic
150	C₂×C₇₉	#2916	1	1	yes	characteristic
151	Q₈×M₁₆	—	79	2	no	—
152	C₂×C₈×Q₈	—	79	1	no	—
153	#1950	—	79	2	no	—
154	#1950	—	79	4	no	—
155	#1837	—	79	2	no	—
156	#1837	D₇₉	1	1	yes	characteristic
157	C₄×#108	—	79	1	no	—
158	C₄×#108	—	79	2	no	—
159	C₇₉	#22746	1	1	yes	characteristic, complemented, Sylow
160	C₂×C₄×Q₈	—	79	1	no	—
161	#446	—	158	2	no	—
162	#446	—	158	4	no	—
163	C₈×Q₈	—	79	2	no	—
164	C₈×Q₈	—	79	4	no	—
165	C₈×Q₈	—	79	2	no	—
166	C₈×Q₈	—	158	2	no	—
167	C₂×#108	—	79	4	no	—
168	C₂×#108	—	79	2	no	—
169	#423	—	79	4	no	—
170	#423	C₂×D₇₉	1	2	yes	—
171	C₂×C₄⋊C₈	—	79	2	no	—
172	C₂×C₄⋊C₈	—	79	1	no	—
173	C₄×M₁₆	—	79	4	no	—
174	C₄×M₁₆	—	79	2	no	—
175	C₂×C₄×C₈	—	79	2	no	—
176	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
177	#341	—	79	4	no	—
178	#341	—	79	1	no	—
179	#341	—	79	1	no	—
180	#341	C₂×D₇₉	1	2	yes	—
181	#341	C₂×D₇₉	1	2	yes	—
182	C₂²×Q₈	—	79	1	no	—
183	C₂×M₁₆	—	79	2	no	—
184	C₂×M₁₆	—	79	2	no	—
185	C₂×M₁₆	C₂²×D₇₉	1	4	yes	—
186	C₂²×C₈	—	79	2	no	—
187	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
189	C₄×Q₈	—	79	1	no	—
190	C₄×Q₈	—	79	2	no	—
191	C₄×Q₈	—	79	1	no	—
192	C₄×Q₈	—	79	1	no	—
193	C₄×Q₈	—	79	2	no	—
194	C₄×Q₈	—	79	1	no	—
195	C₂×C₄⋊C₄	—	79	2	no	—
196	C₂×C₄⋊C₄	—	79	1	no	—
197	C₂×C₄²	—	79	2	no	—
198	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
199	#108	—	158	4	no	—
200	#108	—	158	8	no	—
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	—	79	4	no	—
204	C₄⋊C₈	—	79	4	no	—
205	C₄⋊C₈	—	79	2	no	—
206	C₄⋊C₈	—	79	2	no	—
207	C₄⋊C₈	—	158	4	no	—
208	C₄⋊C₈	—	158	4	no	—
209	#97	—	158	4	no	—
210	#97	—	158	2	no	—
211	C₄×C₈	—	79	4	no	—
212	C₄×C₈	—	79	4	no	—
213	C₄×C₈	—	158	4	no	—
214	C₄×C₈	—	158	2	no	—
215	C₄×C₈	C₄×D₇₉	1	2	yes	—
216	C₄×C₈	C₄×D₇₉	1	2	yes	—
217	C₂×Q₈	—	79	1	no	—
218	C₂×Q₈	—	79	2	no	—
219	C₂×Q₈	—	79	1	no	—
220	C₂×Q₈	—	158	4	no	—
221	C₂²×C₄	—	79	2	no	—
222	C₂²×C₄	—	79	2	no	—
223	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
224	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	79	4	no	—
229	M₁₆	—	158	2	no	—
230	M₁₆	—	158	2	no	—
231	M₁₆	—	158	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	79	4	no	—
236	C₂×C₈	—	158	2	no	—
237	C₂×C₈	—	158	4	no	—
238	C₂×C₈	—	158	2	no	—
239	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
240	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
241	C₄⋊C₄	—	79	2	no	—
242	C₄⋊C₄	—	79	2	no	—
243	C₄⋊C₄	—	79	2	no	—
244	C₄⋊C₄	—	79	2	no	—
245	C₄⋊C₄	—	79	2	no	—
246	C₄⋊C₄	—	79	1	no	—
247	C₄⋊C₄	—	79	1	no	—
248	C₄²	—	79	2	no	—
249	C₄²	—	79	2	no	—
250	C₄²	—	79	2	no	—
251	C₄²	—	79	2	no	—
252	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
253	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
254	C₄²	D₄×D₇₉	1	1	yes	characteristic
255	C₄²	D₄×D₇₉	1	1	yes	characteristic
256	C₂³	C₂×Q₈×D₇₉	1	1	yes	characteristic
257	Q₈	—	79	4	no	—
258	Q₈	—	158	4	no	—
259	Q₈	—	158	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	316	2	no	—
263	C₂×C₄	—	79	2	no	—
264	C₂×C₄	—	79	2	no	—
265	C₂×C₄	—	79	2	no	—
266	C₂×C₄	—	79	2	no	—
267	C₂×C₄	—	158	4	no	—
268	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
269	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
270	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
271	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
273	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
274	C₂×C₄	#41×D₇₉	1	2	yes	—
275	C₂×C₄	#14579208	1	2	yes	—
276	C₈	—	2	2	no	—
277	C₈	—	2	2	no	—
278	C₈	—	4	4	no	—
279	C₈	—	316	2	no	—
280	C₈	—	158	4	no	—
281	C₈	—	158	4	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	C₂×C₄⋊C₄×D₇₉	1	1	yes	characteristic
285	C₂²	#19099290	1	2	yes	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	79	4	no	—
290	C₄	—	158	2	no	—
291	C₄	—	158	2	no	—
292	C₄	—	158	2	no	—
293	C₄	#108×D₇₉	1	2	yes	—
294	C₄	#19099031	1	1	yes	characteristic
295	C₄	#19099040	1	1	yes	characteristic
296	C₄	#19099221	1	2	yes	—
297	C₂	—	2	2	no	—
298	C₂	#23766317	1	1	yes	characteristic
299	C₂	C₂×#108×D₇₉	1	1	yes	characteristic
300	C₂	#23772008	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#1837	C₂
socle	C₂²×C₇₉	C₂²×C₄⋊C₄
prime radical	C₂³×C₇₉	C₂²×Q₈
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1837	D₇₉
hypocenter	C₇₉	#22746
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

Complements

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

N	H ≅ G/N	type
124. Dic₁₅₈	#108	semidirect
159. C₇₉	#22746	semidirect

Maximal subgroups

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

subgroup	index
#23772351	2
C₂×C₈×Dic₁₅₈	2
#23764174	2
#23764173	2
#23763536	2
C₄×#14579097	2
#23763301	2
C₇₉×#1837	2
#22746	79