Subgroups — #34442568

Order 20224.

702 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099003	C₂	1	2	yes	maximal
2	C₂×#19098943	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765450	C₂	1	4	yes	maximal, complemented
4	#23765445	C₂	1	4	yes	maximal, complemented
5	#23765156	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765119	C₂	1	2	yes	maximal, complemented
7	C₄×#135×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₃₁₆	—	2	1	no	—
9	C₂²×Dic₃₁₆	C₂²	1	2	yes	—
10	C₂×#14579161	—	2	1	no	—
11	#19099003	—	2	8	no	—
12	C₂×#14579107	C₂²	1	4	yes	—
13	C₂×#14579105	C₂²	1	4	yes	—
14	C₂×#14579101	C₂²	1	2	yes	—
15	#19098943	—	2	2	no	—
16	#19098943	C₂²	1	4	yes	—
17	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	4	yes	—
18	#19098366	C₂²	1	8	yes	—
19	C₂×#135×C₇₉	C₂²	1	1	yes	characteristic
20	C₄×#41×C₇₉	—	2	1	no	—
21	C₄×Q₁₆×C₇₉	C₂²	1	2	yes	—
22	C₄×SD₁₆×C₇₉	—	2	2	no	—
23	C₄×D₈×C₇₉	C₂²	1	2	yes	—
24	#427×C₇₉	C₂²	1	1	yes	characteristic
25	#416×C₇₉	—	2	1	no	—
26	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
27	C₂²×Dic₁₅₈	—	2	2	no	—
28	C₂×Dic₃₁₆	—	2	4	no	—
29	C₂×Dic₃₁₆	—	2	4	no	—
30	C₂×Dic₃₁₆	—	2	2	no	—
31	C₂×Dic₃₁₆	—	2	4	no	—
32	C₂×Dic₃₁₆	—	4	4	no	—
33	C₂×Dic₃₁₆	C₂³	1	4	yes	—
34	#14579161	—	2	4	no	—
35	#14579161	—	4	1	no	—
36	#14579161	—	4	1	no	—
37	C₂×C₄⋊Dic₇₉	—	4	2	no	—
38	#14579107	—	2	8	no	—
39	#14579105	—	2	8	no	—
40	#14579101	—	2	4	no	—
41	C₂×C₇₉⋊C₁₆	C₂³	1	4	yes	—
42	C₂×#41×C₇₉	—	2	1	no	—
43	#135×C₇₉	—	2	2	no	—
44	#135×C₇₉	D₄	1	4	yes	—
45	C₂×Q₁₆×C₇₉	C₂³	1	1	yes	characteristic
46	C₂×SD₁₆×C₇₉	—	2	1	no	—
47	C₂×D₈×C₇₉	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₇₉	—	2	2	no	—
51	C₄×D₄×C₇₉	—	2	2	no	—
52	C₄×D₄×C₇₉	—	4	2	no	—
53	#117×C₇₉	—	2	1	no	—
54	#117×C₇₉	—	4	1	no	—
55	C₂×C₄²×C₇₉	—	4	1	no	—
56	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
57	#107×C₇₉	C₂³	1	2	yes	—
58	#106×C₇₉	—	2	1	no	—
59	#103×C₇₉	—	2	2	no	—
60	#102×C₇₉	—	2	2	no	—
61	C₄×C₈×C₇₉	D₄	1	2	yes	—
62	C₄×C₈×C₇₉	C₂³	1	2	yes	—
63	C₂×Dic₁₅₈	—	2	4	no	—
64	C₂×Dic₁₅₈	—	4	4	no	—
65	C₂×Dic₁₅₈	—	4	2	no	—
66	C₂×Dic₁₅₈	—	4	4	no	—
67	C₂²×Dic₇₉	—	4	2	no	—
68	Dic₃₁₆	—	2	4	no	—
69	Dic₃₁₆	—	2	4	no	—
70	Dic₃₁₆	—	4	8	no	—
71	Dic₃₁₆	—	4	4	no	—
72	C₄⋊Dic₇₉	—	4	4	no	—
73	C₄⋊Dic₇₉	—	4	4	no	—
74	C₇₉⋊C₁₆	—	2	4	no	—
75	#41×C₇₉	—	2	4	no	—
76	#41×C₇₉	—	4	2	no	—
77	C₂×Q₈×C₇₉	—	2	1	no	—
78	C₂×D₄×C₇₉	—	2	1	no	—
79	C₂×D₄×C₇₉	—	4	1	no	—
80	C₂²×C₄×C₇₉	—	4	1	no	—
81	C₂²×C₄×C₇₉	—	4	2	no	—
82	C₂²×C₄×C₇₉	D₈	1	2	yes	—
83	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
84	Q₁₆×C₇₉	—	2	1	no	—
85	Q₁₆×C₇₉	C₂×D₄	1	2	yes	—
86	SD₁₆×C₇₉	—	2	4	no	—
87	D₈×C₇₉	—	2	1	no	—
88	D₈×C₇₉	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₇₉	—	2	2	no	—
90	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₇₉	#41	1	2	yes	—
94	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₇₉	—	2	2	no	—
96	C₄⋊C₄×C₇₉	—	4	2	no	—
97	C₂²⋊C₄×C₇₉	—	4	1	no	—
98	C₂²⋊C₄×C₇₉	—	4	2	no	—
99	C₄²×C₇₉	—	4	1	no	—
100	C₄²×C₇₉	—	4	2	no	—
101	C₄²×C₇₉	D₈	1	1	yes	characteristic
102	C₄²×C₇₉	D₈	1	1	yes	characteristic
103	C₄²×C₇₉	C₂×D₄	1	2	yes	—
104	C₂³×C₇₉	—	4	1	no	—
105	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
106	Q₈×C₇₉	—	2	2	no	—
107	Q₈×C₇₉	—	4	1	no	—
108	D₄×C₇₉	—	2	2	no	—
109	D₄×C₇₉	—	4	4	no	—
110	D₄×C₇₉	—	4	1	no	—
111	C₂×C₄×C₇₉	—	2	4	no	—
112	C₂×C₄×C₇₉	—	4	1	no	—
113	C₂×C₄×C₇₉	—	4	4	no	—
114	C₂×C₄×C₇₉	—	4	2	no	—
115	C₂×C₄×C₇₉	—	8	4	no	—
116	C₂×C₄×C₇₉	#121	1	2	yes	—
117	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
121	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×Dic₇₉	—	4	4	no	—
123	C₂×Dic₇₉	—	8	4	no	—
124	Dic₁₅₈	—	4	4	no	—
125	Dic₁₅₈	—	4	4	no	—
126	Dic₁₅₈	—	8	4	no	—
127	C₈×C₇₉	—	2	2	no	—
128	C₈×C₇₉	#121	1	2	yes	—
129	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
131	C₂²×C₇₉	—	2	2	no	—
132	C₂²×C₇₉	—	4	2	no	—
133	C₂²×C₇₉	—	8	2	no	—
134	C₂²×C₇₉	#459	1	2	yes	—
135	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
136	C₄×C₇₉	—	2	2	no	—
137	C₄×C₇₉	—	4	2	no	—
138	C₄×C₇₉	—	8	2	no	—
139	C₄×C₇₉	#459	1	2	yes	—
140	C₄×C₇₉	#510	1	4	yes	complemented
141	C₄×C₇₉	C₂×#121	1	1	yes	characteristic
142	C₄×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
143	Dic₇₉	—	8	4	no	—
144	#33583	—	79	1	no	maximal, Sylow
145	C₂×C₇₉	—	2	2	no	—
146	C₂×C₇₉	—	8	2	no	—
147	C₂×C₇₉	C₂×#459	1	1	yes	characteristic
148	C₂×C₇₉	C₂×#510	1	2	yes	—
149	C₂×#510	—	79	2	no	—
150	C₂×#494	—	79	1	no	—
151	C₄×#135	D₇₉	1	1	yes	characteristic
152	#2209	—	79	4	no	—
153	#2208	—	79	4	no	—
154	#2150	—	79	1	no	—
155	#2141	—	79	2	no	—
156	C₇₉	#33583	1	1	yes	characteristic, complemented, Sylow
157	C₂×#135	C₂×D₇₉	1	1	yes	characteristic
158	C₂²×Q₁₆	—	79	2	no	—
159	C₂²×Q₁₆	—	158	1	no	—
160	C₂×C₄⋊Q₈	—	158	1	no	—
161	C₄×#41	—	2	1	no	—
162	#510	—	158	8	no	—
163	C₂×Q₃₂	—	79	4	no	—
164	C₂×SD₃₂	—	79	4	no	—
165	C₂×M₃₂	—	79	2	no	—
166	#494	—	79	4	no	—
167	#494	—	158	2	no	—
168	C₄×Q₁₆	C₂×D₇₉	1	2	yes	—
169	C₄×SD₁₆	—	2	2	no	—
170	C₄×D₈	C₂×D₇₉	1	2	yes	—
171	#427	C₂×D₇₉	1	1	yes	characteristic
172	#416	—	2	1	no	—
173	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
174	C₄⋊C₁₆	—	79	4	no	—
175	#358	—	79	8	no	—
176	C₂×#41	—	2	1	no	—
177	C₂²×Q₈	—	158	2	no	—
178	#135	—	2	2	no	—
179	#135	D₃₁₆	1	4	yes	—
180	C₂×Q₁₆	—	316	4	no	—
181	C₂×Q₁₆	—	79	4	no	—
182	C₂×Q₁₆	—	158	2	no	—
183	C₂×Q₁₆	—	158	4	no	—
184	C₂×Q₁₆	—	158	4	no	—
185	C₂×Q₁₆	—	158	4	no	—
186	C₂×Q₁₆	C₂²×D₇₉	1	1	yes	characteristic
187	C₂×SD₁₆	—	2	1	no	—
188	C₂×D₈	C₂²×D₇₉	1	1	yes	characteristic
189	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
191	C₄⋊Q₈	—	316	1	no	—
192	C₄⋊Q₈	—	316	1	no	—
193	C₄⋊Q₈	—	158	4	no	—
194	C₄×Q₈	—	2	2	no	—
195	C₄×D₄	—	2	2	no	—
196	C₄×D₄	—	4	2	no	—
197	#117	—	2	1	no	—
198	#117	—	4	1	no	—
199	C₂×C₄⋊C₄	—	316	2	no	—
200	C₂×C₄²	—	4	1	no	—
201	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
202	Q₃₂	—	158	8	no	—
203	SD₃₂	—	158	8	no	—
204	M₃₂	—	158	4	no	—
205	C₂×C₁₆	—	79	4	no	—
206	#107	C₂²×D₇₉	1	2	yes	—
207	#106	—	2	1	no	—
208	#103	—	2	2	no	—
209	#102	—	2	2	no	—
210	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
211	C₄×C₈	C₂²×D₇₉	1	2	yes	—
212	#41	—	2	4	no	—
213	#41	—	4	2	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	316	2	no	—
216	C₂×Q₈	—	316	4	no	—
217	C₂×Q₈	—	316	4	no	—
218	C₂×Q₈	—	158	4	no	—
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	316	2	no	—
224	C₂²×C₄	#11929609	1	2	yes	—
225	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
226	Q₁₆	—	2	1	no	—
227	Q₁₆	—	316	8	no	—
228	Q₁₆	—	316	4	no	—
229	Q₁₆	—	158	4	no	—
230	Q₁₆	—	158	4	no	—
231	Q₁₆	C₂×D₃₁₆	1	2	yes	—
232	SD₁₆	—	2	4	no	—
233	D₈	—	2	1	no	—
234	D₈	C₂×D₃₁₆	1	2	yes	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
237	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
238	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
239	C₂×C₈	#11929624	1	2	yes	—
240	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	316	4	no	—
244	C₄⋊C₄	—	316	4	no	—
245	C₂²⋊C₄	—	4	1	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	2	no	—
249	C₄²	#11929609	1	1	yes	characteristic
250	C₄²	#11929609	1	1	yes	characteristic
251	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
252	C₁₆	—	158	4	no	—
253	C₂³	—	4	1	no	—
254	C₂³	C₂×#11929609	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	Q₈	—	4	1	no	—
257	Q₈	—	316	4	no	—
258	Q₈	—	316	4	no	—
259	Q₈	—	632	4	no	—
260	D₄	—	2	2	no	—
261	D₄	—	4	4	no	—
262	D₄	—	4	1	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	—	316	4	no	—
269	C₂×C₄	—	632	4	no	—
270	C₂×C₄	#14579135	1	2	yes	—
271	C₂×C₄	C₂×#11929609	1	2	yes	—
272	C₂×C₄	C₂×#11929609	1	2	yes	—
273	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
274	C₂×C₄	#14579186	1	2	yes	—
275	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	#14579135	1	2	yes	—
278	C₈	C₂²×D₃₁₆	1	1	yes	characteristic
279	C₈	C₂×#11929624	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	8	2	no	—
283	C₂²	#19098758	1	2	yes	—
284	C₂²	C₂²×#11929609	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	8	2	no	—
288	C₄	—	632	4	no	—
289	C₄	#19098758	1	2	yes	—
290	C₄	#19099003	1	4	yes	complemented
291	C₄	C₂×#14579135	1	1	yes	characteristic
292	C₄	C₂×#14579186	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	8	2	no	—
295	C₂	C₂×#19098758	1	1	yes	characteristic
296	C₂	C₂×#19099003	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₄×#135×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×#135×C₇₉	C₂²
hypercenter	C₄×#135	D₇₉
hypocenter	C₇₉	#33583
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. C₂×#19098943	C₂	semidirect
3. #23765450	C₂	semidirect
4. #23765445	C₂	semidirect
5. #23765156	C₂	semidirect
6. #23765119	C₂	semidirect
140. C₄×C₇₉	#510	semidirect
156. C₇₉	#33583	semidirect
290. C₄	#19099003	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099003	2
C₂×#19098943	2
#23765450	2
#23765445	2
#23765156	2
#23765119	2
C₄×#135×C₇₉	2
#33583	79