Subgroups — #34205217

Order 20224.

574 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098804	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098366	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#23761491	C₂	1	2	yes	maximal, complemented
5	C₂²×#107×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14579140	—	2	1	no	—
8	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
9	#19098804	—	2	2	no	—
10	#19098804	C₄	1	4	yes	complemented
11	C₂×#14579095	C₂²	1	1	yes	characteristic
12	C₂×#14579086	—	2	1	no	—
13	#19098366	—	2	4	no	—
14	C₂²⋊(C₇₉⋊C₁₆)	—	2	2	no	—
15	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₇₉	—	2	1	no	—
17	C₂×#107×C₇₉	—	2	1	no	—
18	C₂×#107×C₇₉	—	2	4	no	—
19	C₂×#107×C₇₉	C₂²	1	2	yes	—
20	C₂²×Dic₁₅₈	—	2	1	no	—
21	C₂×Dic₃₁₆	—	2	2	no	—
22	C₂×Dic₃₁₆	—	2	2	no	—
23	C₂×Dic₃₁₆	—	2	1	no	—
24	C₂×Dic₃₁₆	C₂×C₄	1	2	yes	complemented
25	#14579140	—	2	4	no	—
26	#14579140	—	4	2	no	—
27	C₂×C₄⋊Dic₇₉	—	2	1	no	—
28	C₂×Dic₇₉⋊C₄	—	4	1	no	—
29	C₂×C₂²⋊Dic₇₉	—	4	1	no	—
30	C₂×C₇₉⋊C₁₆	—	2	2	no	—
31	C₂×C₇₉⋊C₁₆	—	2	2	no	—
32	C₂×C₇₉⋊C₁₆	—	2	2	no	—
33	#14579095	—	2	1	no	—
34	#14579095	C₂×C₄	1	2	yes	complemented
35	#14579086	—	2	4	no	—
36	C₂³×C₄×C₇₉	—	4	1	no	—
37	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₉	—	2	2	no	—
39	C₂²×C₈×C₇₉	—	2	2	no	—
40	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₇₉	D₄	1	2	yes	—
43	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₇₉	—	2	4	no	—
45	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₇₉	—	4	2	no	—
47	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
48	#107×C₇₉	—	2	4	no	—
49	#107×C₇₉	—	2	2	no	—
50	#107×C₇₉	D₄	1	4	yes	—
51	C₂×Dic₁₅₈	—	2	2	no	—
52	C₂×Dic₁₅₈	—	4	1	no	—
53	C₂×Dic₁₅₈	—	4	1	no	—
54	C₂×Dic₁₅₈	—	4	1	no	—
55	C₂×Dic₁₅₈	—	4	1	no	—
56	C₂×Dic₁₅₈	—	4	1	no	—
57	C₂²×Dic₇₉	—	4	1	no	—
58	C₂²×Dic₇₉	—	4	1	no	—
59	Dic₃₁₆	—	2	4	no	—
60	Dic₃₁₆	—	4	2	no	—
61	C₄⋊Dic₇₉	—	2	2	no	—
62	C₄⋊Dic₇₉	—	4	1	no	—
63	Dic₇₉⋊C₄	—	4	4	no	—
64	C₂²⋊Dic₇₉	—	4	4	no	—
65	C₇₉⋊C₁₆	—	4	2	no	—
66	C₂⁴×C₇₉	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	—	2	1	no	—
68	C₂²×C₄×C₇₉	—	2	1	no	—
69	C₂²×C₄×C₇₉	—	2	1	no	—
70	C₂²×C₄×C₇₉	—	2	1	no	—
71	C₂²×C₄×C₇₉	—	4	4	no	—
72	C₂²×C₄×C₇₉	—	4	2	no	—
73	C₂²×C₄×C₇₉	—	8	2	no	—
74	C₂²×C₄×C₇₉	—	8	2	no	—
75	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
76	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
78	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₇₉	—	2	4	no	—
80	C₂×C₈×C₇₉	—	2	4	no	—
81	C₂×C₈×C₇₉	—	2	1	no	—
82	C₂×C₈×C₇₉	—	2	1	no	—
83	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₇₉	#41	1	2	yes	—
88	C₄⋊C₄×C₇₉	—	2	4	no	—
89	C₄⋊C₄×C₇₉	—	4	4	no	—
90	C₄⋊C₄×C₇₉	—	4	2	no	—
91	C₂³×C₇₉	—	2	1	no	—
92	C₂³×C₇₉	—	2	1	no	—
93	C₂³×C₇₉	—	2	1	no	—
94	C₂³×C₇₉	—	2	1	no	—
95	C₂³×C₇₉	#102	1	2	yes	—
96	C₂³×C₇₉	SD₃₂	1	2	yes	—
97	C₂³×C₇₉	Q₃₂	1	2	yes	—
98	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₇₉	—	2	1	no	—
100	C₂×C₄×C₇₉	—	2	1	no	—
101	C₂×C₄×C₇₉	—	2	2	no	—
102	C₂×C₄×C₇₉	—	2	2	no	—
103	C₂×C₄×C₇₉	—	2	2	no	—
104	C₂×C₄×C₇₉	—	2	2	no	—
105	C₂×C₄×C₇₉	—	4	4	no	—
106	C₂×C₄×C₇₉	—	8	8	no	—
107	C₂×C₄×C₇₉	—	8	2	no	—
108	C₂×C₄×C₇₉	—	8	2	no	—
109	C₂×C₄×C₇₉	#102	1	2	yes	—
110	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
112	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
116	C₂×Dic₇₉	—	4	2	no	—
117	C₂×Dic₇₉	—	4	2	no	—
118	C₂×Dic₇₉	—	8	1	no	—
119	C₂×Dic₇₉	—	8	1	no	—
120	C₂×Dic₇₉	—	8	1	no	—
121	C₂×Dic₇₉	—	8	1	no	—
122	Dic₁₅₈	—	4	2	no	—
123	Dic₁₅₈	—	4	2	no	—
124	Dic₁₅₈	—	8	1	no	—
125	Dic₁₅₈	—	8	1	no	—
126	C₈×C₇₉	—	2	2	no	—
127	C₈×C₇₉	C₄×D₄	1	2	yes	—
128	C₈×C₇₉	#121	1	2	yes	—
129	C₂²×C₇₉	—	2	2	no	—
130	C₂²×C₇₉	—	2	2	no	—
131	C₂²×C₇₉	—	2	2	no	—
132	C₂²×C₇₉	—	2	2	no	—
133	C₂²×C₇₉	—	2	2	no	—
134	C₂²×C₇₉	—	2	2	no	—
135	C₂²×C₇₉	#358	1	4	yes	complemented
136	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
137	C₂²×C₇₉	#447	1	1	yes	characteristic
138	C₂²×C₇₉	#480	1	1	yes	characteristic
139	C₂²×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
140	C₂²×C₇₉	C₂×Q₃₂	1	1	yes	characteristic
141	C₂²×C₇₉	#509	1	1	yes	characteristic
142	C₂²×C₇₉	#510	1	1	yes	characteristic
143	C₄×C₇₉	—	2	2	no	—
144	C₄×C₇₉	—	8	4	no	—
145	C₄×C₇₉	#386	1	1	yes	characteristic
146	C₄×C₇₉	#417	1	1	yes	characteristic
147	C₄×C₇₉	#448	1	1	yes	characteristic
148	C₄×C₇₉	#481	1	1	yes	characteristic
149	Dic₇₉	—	8	2	no	—
150	Dic₇₉	—	8	2	no	—
151	#12993	—	79	1	no	maximal, Sylow
152	C₂×C₇₉	—	2	4	no	—
153	C₂×C₇₉	#1874	1	1	yes	characteristic
154	C₂×C₇₉	C₂×#358	1	1	yes	characteristic
155	C₂×C₇₉	#2139	1	1	yes	characteristic
156	C₂×C₇₉	#2187	1	1	yes	characteristic
157	C₂×C₇₉	#2188	1	1	yes	characteristic
158	C₂×C₇₉	#2235	1	1	yes	characteristic
159	C₂×C₇₉	#2236	1	1	yes	characteristic
160	C₂×#467	—	79	1	no	—
161	C₂²×#107	D₇₉	1	1	yes	characteristic
162	C₂×#358	—	79	2	no	—
163	C₂×C₂²⋊C₁₆	—	79	1	no	—
164	#1380	—	79	2	no	—
165	C₇₉	#12993	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₁₆	—	79	1	no	—
167	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
168	C₂×C₂²⋊Q₈	—	158	1	no	—
169	C₂²×C₄⋊C₄	—	2	1	no	—
170	C₂²×C₁₆	—	79	2	no	—
171	#467	—	79	4	no	—
172	#467	—	158	2	no	—
173	C₂×#107	—	2	1	no	—
174	C₂×#107	—	2	4	no	—
175	C₂×#107	—	79	1	no	—
176	C₂×#107	C₂×D₇₉	1	2	yes	—
177	C₂×#103	—	158	1	no	—
178	#358	—	158	4	no	—
179	C₂²⋊C₁₆	—	158	2	no	—
180	C₂²×Q₈	—	158	1	no	—
181	C₂³×C₄	—	4	1	no	—
182	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
183	C₂×Q₁₆	—	79	2	no	—
184	C₂×Q₁₆	—	158	2	no	—
185	C₂×Q₁₆	—	158	1	no	—
186	C₂×Q₁₆	—	158	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	C₄×D₇₉	1	2	yes	—
190	C₂²×C₈	D₃₁₆	1	2	yes	—
191	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
193	C₂²⋊Q₈	—	316	2	no	—
194	C₂²⋊Q₈	—	158	4	no	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	4	2	no	—
198	C₂×C₄⋊C₄	—	4	1	no	—
199	C₂×C₄⋊C₄	—	316	1	no	—
200	C₂×C₄⋊C₄	—	158	1	no	—
201	C₂×C₂²⋊C₄	—	316	1	no	—
202	C₂×C₁₆	—	158	2	no	—
203	C₂×C₁₆	—	158	2	no	—
204	C₂×C₁₆	—	158	2	no	—
205	#107	—	2	4	no	—
206	#107	—	2	2	no	—
207	#107	—	79	2	no	—
208	#107	—	158	1	no	—
209	#107	C₂²⋊D₇₉	1	4	yes	—
210	#103	—	158	4	no	—
211	C₂⁴	#11929609	1	1	yes	characteristic
212	C₂×Q₈	—	316	1	no	—
213	C₂×Q₈	—	316	1	no	—
214	C₂×Q₈	—	316	1	no	—
215	C₂×Q₈	—	316	1	no	—
216	C₂×Q₈	—	316	1	no	—
217	C₂×Q₈	—	158	2	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	8	2	no	—
225	C₂²×C₄	—	8	2	no	—
226	C₂²×C₄	—	316	1	no	—
227	C₂²×C₄	—	316	1	no	—
228	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
229	C₂²×C₄	#11929609	1	1	yes	characteristic
230	C₂²×C₄	#11929611	1	2	yes	—
231	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
232	Q₁₆	—	316	2	no	—
233	Q₁₆	—	158	4	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
239	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
240	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
241	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
242	C₂×C₈	#11929624	1	2	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	316	4	no	—
247	C₄⋊C₄	—	316	1	no	—
248	C₄⋊C₄	—	158	2	no	—
249	C₂²⋊C₄	—	316	4	no	—
250	C₁₆	—	316	2	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	#14579081	1	2	yes	—
256	C₂³	#14579105	1	2	yes	—
257	C₂³	#14579107	1	2	yes	—
258	C₂³	C₂×#11929609	1	1	yes	characteristic
259	Q₈	—	316	2	no	—
260	Q₈	—	316	2	no	—
261	Q₈	—	632	1	no	—
262	Q₈	—	632	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	8	8	no	—
271	C₂×C₄	—	8	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	316	2	no	—
274	C₂×C₄	—	316	2	no	—
275	C₂×C₄	—	632	1	no	—
276	C₂×C₄	—	632	1	no	—
277	C₂×C₄	—	632	1	no	—
278	C₂×C₄	—	632	1	no	—
279	C₂×C₄	#14579081	1	2	yes	—
280	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
281	C₂×C₄	#14579130	1	1	yes	characteristic
282	C₂×C₄	#14579144	1	1	yes	characteristic
283	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
284	C₂×C₄	#14579187	1	1	yes	characteristic
285	C₂×C₄	#14579192	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	C₄×C₂²⋊D₇₉	1	2	yes	—
288	C₈	#14579135	1	2	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	#19098366	1	4	yes	complemented
296	C₂²	C₂×#14579081	1	1	yes	characteristic
297	C₂²	#19098697	1	1	yes	characteristic
298	C₂²	#19098876	1	1	yes	characteristic
299	C₂²	C₂×#14579105	1	1	yes	characteristic
300	C₂²	C₂×#14579107	1	1	yes	characteristic
301	C₂²	#19098999	1	1	yes	characteristic
302	C₂²	#19099004	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	8	4	no	—
305	C₄	—	632	2	no	—
306	C₄	—	632	2	no	—
307	C₄	#19098442	1	1	yes	characteristic
308	C₄	#19098562	1	1	yes	characteristic
309	C₄	#19098702	1	1	yes	characteristic
310	C₄	#19098881	1	1	yes	characteristic
311	C₂	—	2	4	no	—
312	C₂	#23763717	1	1	yes	characteristic
313	C₂	C₂×#19098366	1	1	yes	characteristic
314	C₂	#23765110	1	1	yes	characteristic
315	C₂	#23765329	1	1	yes	characteristic
316	C₂	#23765334	1	1	yes	characteristic
317	C₂	#23765597	1	1	yes	characteristic
318	C₂	#23765602	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂²×#107×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂⁴×C₇₉	D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂²×#107	D₇₉
hypocenter	C₇₉	#12993
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
2. C₂×#19098366	C₂	semidirect
4. #23761491	C₂	semidirect
10. #19098804	C₄	semidirect
24. C₂×Dic₃₁₆	C₂×C₄	semidirect
34. #14579095	C₂×C₄	semidirect
135. C₂²×C₇₉	#358	semidirect
165. C₇₉	#12993	semidirect
295. C₂²	#19098366	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098804	2
C₂×#19098366	2
C₂×C₂²⋊(C₇₉⋊C₁₆)	2
#23761491	2
C₂²×#107×C₇₉	2
#12993	79