Subgroups — #34204317

Order 20224.

494 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₆₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098363	C₂	1	2	yes	maximal
4	#23761491	C₂	1	2	yes	maximal, complemented
5	C₇₉×#1846	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₃₁₆	—	2	1	no	—
8	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
9	C₄×D₆₃₂	—	2	2	no	—
10	C₄×D₆₃₂	C₄	1	4	yes	complemented
11	C₂×#14579095	C₂²	1	1	yes	characteristic
12	C₂×#14579083	—	2	1	no	—
13	C₄⋊(C₇₉⋊C₁₆)	—	2	2	no	—
14	#19098363	—	2	4	no	—
15	C₂×#107×C₇₉	C₂²	1	2	yes	—
16	C₂×#106×C₇₉	—	2	1	no	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	#382×C₇₉	—	2	1	no	—
19	C₂²×D₃₁₆	—	2	1	no	—
20	C₂²×C₄×D₇₉	—	4	1	no	—
21	C₂×D₆₃₂	—	2	1	no	—
22	C₂×D₆₃₂	—	2	1	no	—
23	C₂×D₆₃₂	—	2	1	no	—
24	C₂×D₆₃₂	C₂×C₄	1	2	yes	—
25	C₂×D₆₃₂	D₄	1	2	yes	—
26	C₂×D₆₃₂	Q₈	1	2	yes	—
27	C₄×D₃₁₆	—	2	4	no	—
28	C₄×D₃₁₆	—	4	2	no	—
29	C₂×C₄⋊Dic₇₉	—	2	1	no	—
30	C₂×(C₂×D₇₉)⋊C₄	—	4	1	no	—
31	C₂×C₇₉⋊C₁₆	—	2	2	no	—
32	C₂×C₇₉⋊C₁₆	—	2	2	no	—
33	C₂×C₇₉⋊C₁₆	—	2	2	no	—
34	#14579095	—	2	1	no	—
35	#14579095	C₂×C₄	1	2	yes	—
36	#14579083	—	2	4	no	—
37	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
40	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
41	#107×C₇₉	—	2	2	no	—
42	#107×C₇₉	—	2	2	no	—
43	#106×C₇₉	—	4	2	no	—
44	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₇₉	D₄	1	2	yes	—
46	#95×C₇₉	—	4	1	no	—
47	#95×C₇₉	—	4	1	no	—
48	C₂³×D₇₉	—	4	1	no	—
49	C₂×D₃₁₆	—	2	2	no	—
50	C₂×D₃₁₆	—	2	2	no	—
51	C₂×D₃₁₆	—	2	2	no	—
52	C₂×D₃₁₆	—	4	1	no	—
53	C₂×D₃₁₆	—	4	1	no	—
54	C₂×D₃₁₆	—	4	1	no	—
55	C₂²×Dic₇₉	—	4	1	no	—
56	C₂×C₄×D₇₉	—	4	4	no	—
57	C₂×C₄×D₇₉	—	8	4	no	—
58	D₆₃₂	—	2	2	no	—
59	D₆₃₂	—	2	2	no	—
60	D₆₃₂	—	2	2	no	—
61	D₆₃₂	C₄⋊C₄	1	4	yes	—
62	C₄⋊Dic₇₉	—	2	2	no	—
63	C₄⋊Dic₇₉	—	4	1	no	—
64	(C₂×D₇₉)⋊C₄	—	4	4	no	—
65	C₇₉⋊C₁₆	—	4	2	no	—
66	C₂²×C₄×C₇₉	—	4	2	no	—
67	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₇₉	—	2	1	no	—
71	C₂×C₈×C₇₉	—	2	1	no	—
72	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
78	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
79	C₄⋊C₄×C₇₉	—	4	2	no	—
80	C₄⋊C₄×C₇₉	—	4	2	no	—
81	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₇₉	SD₁₆	1	2	yes	—
83	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
84	C₂²×D₇₉	—	4	2	no	—
85	C₂²×D₇₉	—	4	2	no	—
86	C₂²×D₇₉	—	4	2	no	—
87	C₂²×D₇₉	—	8	2	no	—
88	C₂²×D₇₉	—	8	2	no	—
89	C₂×C₄×C₇₉	—	2	2	no	—
90	C₂×C₄×C₇₉	—	2	2	no	—
91	C₂×C₄×C₇₉	—	8	2	no	—
92	C₂×C₄×C₇₉	—	8	2	no	—
93	C₂×C₄×C₇₉	—	8	2	no	—
94	C₂×C₄×C₇₉	#102	1	2	yes	—
95	C₂×C₄×C₇₉	#102	1	2	yes	—
96	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
98	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
102	C₂×Dic₇₉	—	4	2	no	—
103	C₂×Dic₇₉	—	8	2	no	—
104	D₃₁₆	—	2	4	no	—
105	D₃₁₆	—	4	2	no	—
106	D₃₁₆	—	4	2	no	—
107	D₃₁₆	—	4	2	no	—
108	C₄×D₇₉	—	8	8	no	—
109	C₈×C₇₉	—	2	2	no	—
110	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₈×C₇₉	#117	1	1	yes	characteristic
112	C₈×C₇₉	#121	1	1	yes	characteristic
113	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
115	C₂²×C₇₉	#459	1	1	yes	characteristic
116	C₂²×C₇₉	#474	1	1	yes	characteristic
117	C₂²×C₇₉	#508	1	2	yes	—
118	C₂²×C₇₉	#509	1	2	yes	—
119	C₂×D₇₉	—	4	4	no	—
120	C₂×D₇₉	—	8	2	no	—
121	C₂×D₇₉	—	8	2	no	—
122	C₂×D₇₉	—	8	4	no	—
123	C₂×D₇₉	—	8	4	no	—
124	C₄×C₇₉	—	2	2	no	—
125	C₄×C₇₉	—	2	2	no	—
126	C₄×C₇₉	—	16	2	no	—
127	C₄×C₇₉	#380	1	1	yes	characteristic
128	C₄×C₇₉	#417	1	1	yes	characteristic
129	C₄×C₇₉	#461	1	1	yes	characteristic
130	C₄×C₇₉	#476	1	1	yes	characteristic
131	Dic₇₉	—	8	2	no	—
132	#12813	—	79	1	no	maximal, Sylow
133	C₂×C₇₉	#1802	1	1	yes	characteristic
134	C₂×C₇₉	#2138	1	1	yes	characteristic
135	C₂×C₇₉	#2140	1	1	yes	characteristic
136	C₂×C₇₉	#2210	1	2	yes	—
137	C₂×C₇₉	#2228	1	2	yes	—
138	D₇₉	—	8	4	no	—
139	C₂×C₄×D₈	—	79	1	no	—
140	C₂×C₄⋊C₁₆	—	79	1	no	—
141	C₂×#357	—	79	2	no	—
142	#1846	D₇₉	1	1	yes	characteristic, complemented
143	#1380	—	79	2	no	—
144	C₇₉	#12813	1	1	yes	characteristic, complemented, Sylow
145	C₂²×D₈	—	79	1	no	—
146	C₂×C₄×D₄	—	158	1	no	—
147	C₂²×C₁₆	—	79	2	no	—
148	C₄×D₈	—	79	4	no	—
149	C₄×D₈	—	158	2	no	—
150	C₂×#107	—	79	1	no	—
151	C₂×#107	C₂×D₇₉	1	2	yes	—
152	C₂×#106	—	2	1	no	—
153	C₂×#102	—	158	1	no	—
154	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
155	#382	—	2	1	no	—
156	C₄⋊C₁₆	—	158	2	no	—
157	#357	—	158	4	no	—
158	C₂²×D₄	—	158	1	no	—
159	C₂³×C₄	—	316	1	no	—
160	C₂×D₈	—	79	2	no	—
161	C₂×D₈	—	79	2	no	—
162	C₂×D₈	—	79	2	no	—
163	C₂×D₈	—	158	1	no	—
164	C₂×D₈	—	158	1	no	—
165	C₂×D₈	—	158	1	no	—
166	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
168	C₄×D₄	—	316	2	no	—
169	C₄×D₄	—	158	4	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	158	1	no	—
172	C₂×C₂²⋊C₄	—	316	1	no	—
173	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
174	C₂×C₁₆	—	158	2	no	—
175	C₂×C₁₆	—	158	2	no	—
176	C₂×C₁₆	—	158	2	no	—
177	#107	—	2	2	no	—
178	#107	—	2	2	no	—
179	#107	—	79	2	no	—
180	#107	—	158	1	no	—
181	#106	—	4	2	no	—
182	#102	—	158	4	no	—
183	C₄×C₈	C₄×D₇₉	1	2	yes	—
184	C₄×C₈	D₃₁₆	1	2	yes	—
185	#95	—	4	1	no	—
186	#95	—	4	1	no	—
187	C₂⁴	—	316	1	no	—
188	C₂×D₄	—	316	1	no	—
189	C₂×D₄	—	316	1	no	—
190	C₂×D₄	—	316	1	no	—
191	C₂×D₄	—	158	2	no	—
192	C₂×D₄	—	158	2	no	—
193	C₂×D₄	—	158	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	316	1	no	—
196	C₂²×C₄	—	316	4	no	—
197	C₂²×C₄	—	632	4	no	—
198	C₂²×C₄	#11929609	1	1	yes	characteristic
199	C₂²×C₄	#11929609	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
201	D₈	—	79	4	no	—
202	D₈	—	158	2	no	—
203	D₈	—	158	2	no	—
204	D₈	—	158	2	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
208	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
209	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
210	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
211	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
212	C₂×C₈	#11929623	1	1	yes	characteristic
213	C₂×C₈	#11929625	1	1	yes	characteristic
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	316	1	no	—
217	C₄⋊C₄	—	158	2	no	—
218	C₂²⋊C₄	—	316	4	no	—
219	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
220	C₄²	#11929611	1	2	yes	—
221	C₁₆	—	316	2	no	—
222	C₂³	—	316	2	no	—
223	C₂³	—	316	2	no	—
224	C₂³	—	316	2	no	—
225	C₂³	—	632	2	no	—
226	C₂³	—	632	2	no	—
227	C₂³	C₂×#11929609	1	1	yes	characteristic
228	D₄	—	316	2	no	—
229	D₄	—	316	2	no	—
230	D₄	—	316	2	no	—
231	D₄	—	158	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	316	2	no	—
238	C₂×C₄	—	632	8	no	—
239	C₂×C₄	—	632	2	no	—
240	C₂×C₄	#14579081	1	2	yes	—
241	C₂×C₄	#14579081	1	2	yes	—
242	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
243	C₂×C₄	#14579132	1	1	yes	characteristic
244	C₂×C₄	#14579137	1	1	yes	characteristic
245	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
246	C₂×C₄	#14579187	1	1	yes	characteristic
247	C₂×C₄	#14579192	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	C₄⋊C₄×D₇₉	1	1	yes	characteristic
250	C₈	#14579119	1	1	yes	characteristic
251	C₈	#14579136	1	1	yes	characteristic
252	C₈	#14579141	1	1	yes	characteristic
253	C₂²	—	316	4	no	—
254	C₂²	—	632	4	no	—
255	C₂²	—	632	2	no	—
256	C₂²	—	632	2	no	—
257	C₂²	—	632	4	no	—
258	C₂²	C₂×#14579081	1	1	yes	characteristic
259	C₂²	#19098755	1	1	yes	characteristic
260	C₂²	#19098840	1	1	yes	characteristic
261	C₂²	#19098993	1	2	yes	—
262	C₂²	#19098998	1	2	yes	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	16	2	no	—
266	C₄	—	632	2	no	—
267	C₄	#19098420	1	1	yes	characteristic
268	C₄	#19098562	1	1	yes	characteristic
269	C₄	#19098765	1	1	yes	characteristic
270	C₄	#19098850	1	1	yes	characteristic
271	C₂	—	632	4	no	—
272	C₂	#23763397	1	1	yes	characteristic
273	C₂	#23765105	1	1	yes	characteristic
274	C₂	#23765115	1	1	yes	characteristic
275	C₂	#23765452	1	2	yes	—
276	C₂	#23765554	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1846	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂²×D₆₃₂	C₂²
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1846	D₇₉
hypocenter	C₇₉	#12813
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	semidirect
4. #23761491	C₂	semidirect
5. C₇₉×#1846	C₂	semidirect
10. C₄×D₆₃₂	C₄	semidirect
142. #1846	D₇₉	semidirect
144. C₇₉	#12813	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₆₃₂	2
C₂×C₄⋊(C₇₉⋊C₁₆)	2
C₂×#19098363	2
#23761491	2
C₇₉×#1846	2
#12813	79