Subgroups — #34205216

Order 20224.

574 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098805	C₂	1	1	yes	maximal, characteristic
2	C₂×#107×D₇₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098368	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19098366	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×(C₂×D₇₉)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#23761491	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₃₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₈×D₇₉	C₂²	1	1	yes	characteristic
10	C₂×#14579141	—	2	1	no	—
11	C₂×C₄⋊C₄×D₇₉	—	2	1	no	—
12	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	#19098805	—	2	2	no	—
14	#19098805	C₄	1	4	yes	complemented
15	C₂×#14579095	C₂²	1	1	yes	characteristic
16	C₂×#14579094	—	2	1	no	—
17	#107×D₇₉	—	2	2	no	—
18	#107×D₇₉	—	2	2	no	—
19	C₂×#14579084	—	2	1	no	—
20	#19098368	—	2	2	no	—
21	#19098366	—	2	2	no	—
22	(C₂×D₇₉)⋊C₁₆	—	2	2	no	—
23	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
24	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₅₈	—	2	1	no	—
27	C₂²×C₄×D₇₉	—	4	1	no	—
28	C₂²×C₄×D₇₉	D₄	1	1	yes	characteristic
29	C₂×Dic₃₁₆	—	2	1	no	—
30	C₂×Dic₃₁₆	—	2	1	no	—
31	C₂×Dic₃₁₆	—	2	1	no	—
32	C₂×Dic₃₁₆	—	2	1	no	—
33	C₂×Dic₃₁₆	—	2	1	no	—
34	C₂×Dic₃₁₆	C₂×C₄	1	2	yes	complemented
35	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
36	C₂×C₈×D₇₉	—	2	1	no	—
37	C₂×C₈×D₇₉	—	2	1	no	—
38	C₂×C₈×D₇₉	—	2	1	no	—
39	C₂×C₈×D₇₉	—	2	1	no	—
40	C₂×C₈×D₇₉	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₇₉	D₄	1	2	yes	—
42	#14579141	—	2	4	no	—
43	#14579141	—	4	2	no	—
44	C₂×C₄⋊Dic₇₉	—	2	1	no	—
45	C₂×C₄⋊Dic₇₉	—	4	1	no	—
46	C₂×Dic₇₉⋊C₄	—	4	1	no	—
47	C₄⋊C₄×D₇₉	—	2	4	no	—
48	C₄⋊C₄×D₇₉	—	4	2	no	—
49	C₂×(C₂×D₇₉)⋊C₄	—	4	1	no	—
50	C₂×C₇₉⋊C₁₆	—	2	1	no	—
51	C₂×C₇₉⋊C₁₆	—	2	1	no	—
52	C₂×C₇₉⋊C₁₆	—	2	1	no	—
53	#14579095	—	2	1	no	—
54	#14579095	D₄	1	2	yes	—
55	#14579094	—	2	4	no	—
56	#14579084	—	2	4	no	—
57	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
59	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
60	C₂×C₁₆×C₇₉	—	2	1	no	—
61	C₂×C₁₆×C₇₉	—	2	1	no	—
62	C₂×C₁₆×C₇₉	—	2	1	no	—
63	#107×C₇₉	—	2	1	no	—
64	#107×C₇₉	—	2	1	no	—
65	#107×C₇₉	C₂×C₄	1	2	yes	complemented
66	#107×C₇₉	D₄	1	2	yes	—
67	C₂³×D₇₉	D₈	1	1	yes	characteristic
68	C₂×Dic₁₅₈	—	2	2	no	—
69	C₂×Dic₁₅₈	—	4	1	no	—
70	C₂×Dic₁₅₈	—	4	1	no	—
71	C₂×Dic₁₅₈	—	4	1	no	—
72	C₂×Dic₁₅₈	—	4	1	no	—
73	C₂×Dic₁₅₈	—	4	1	no	—
74	C₂²×Dic₇₉	—	4	1	no	—
75	C₂²×Dic₇₉	—	4	1	no	—
76	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
77	C₂×C₄×D₇₉	—	2	1	no	—
78	C₂×C₄×D₇₉	—	2	1	no	—
79	C₂×C₄×D₇₉	—	2	1	no	—
80	C₂×C₄×D₇₉	—	2	1	no	—
81	C₂×C₄×D₇₉	—	4	4	no	—
82	C₂×C₄×D₇₉	—	8	2	no	—
83	C₂×C₄×D₇₉	—	8	2	no	—
84	C₂×C₄×D₇₉	C₂²⋊C₄	1	2	yes	—
85	C₂×C₄×D₇₉	SD₁₆	1	2	yes	—
86	Dic₃₁₆	—	2	2	no	—
87	Dic₃₁₆	—	2	2	no	—
88	Dic₃₁₆	—	4	1	no	—
89	Dic₃₁₆	—	4	1	no	—
90	C₂×C₇₉⋊C₈	—	2	1	no	—
91	C₂×C₇₉⋊C₈	—	2	1	no	—
92	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
93	C₈×D₇₉	—	2	2	no	—
94	C₈×D₇₉	—	2	2	no	—
95	C₈×D₇₉	—	2	2	no	—
96	C₈×D₇₉	—	2	2	no	—
97	C₄⋊Dic₇₉	—	2	2	no	—
98	C₄⋊Dic₇₉	—	4	4	no	—
99	C₄⋊Dic₇₉	—	4	1	no	—
100	Dic₇₉⋊C₄	—	4	4	no	—
101	(C₂×D₇₉)⋊C₄	—	4	4	no	—
102	C₇₉⋊C₁₆	—	4	1	no	—
103	C₂²×C₄×C₇₉	—	4	1	no	—
104	C₂²×C₄×C₇₉	—	4	1	no	—
105	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
111	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
112	C₄⋊C₄×C₇₉	—	2	2	no	—
113	C₄⋊C₄×C₇₉	—	2	2	no	—
114	C₄⋊C₄×C₇₉	—	4	1	no	—
115	C₄⋊C₄×C₇₉	—	4	1	no	—
116	C₁₆×C₇₉	—	4	1	no	—
117	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
118	C₂²×D₇₉	—	2	1	no	—
119	C₂²×D₇₉	—	2	1	no	—
120	C₂²×D₇₉	—	2	1	no	—
121	C₂²×D₇₉	—	2	1	no	—
122	C₂²×D₇₉	#102	1	2	yes	—
123	C₂²×D₇₉	SD₃₂	1	2	yes	—
124	C₂²×D₇₉	Q₃₂	1	2	yes	—
125	C₂×C₄×C₇₉	—	4	2	no	—
126	C₂×C₄×C₇₉	—	4	2	no	—
127	C₂×C₄×C₇₉	—	8	1	no	—
128	C₂×C₄×C₇₉	—	8	1	no	—
129	C₂×C₄×C₇₉	—	8	1	no	—
130	C₂×C₄×C₇₉	—	8	1	no	—
131	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
133	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
134	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
137	C₂×Dic₇₉	—	2	1	no	—
138	C₂×Dic₇₉	—	2	1	no	—
139	C₂×Dic₇₉	—	4	2	no	—
140	C₂×Dic₇₉	—	4	2	no	—
141	C₂×Dic₇₉	—	8	1	no	—
142	C₂×Dic₇₉	—	8	1	no	—
143	C₂×Dic₇₉	—	8	1	no	—
144	C₂×Dic₇₉	—	8	1	no	—
145	C₂×Dic₇₉	#102	1	2	yes	—
146	C₄×D₇₉	—	2	2	no	—
147	C₄×D₇₉	—	2	2	no	—
148	C₄×D₇₉	—	2	2	no	—
149	C₄×D₇₉	—	2	2	no	—
150	C₄×D₇₉	—	8	8	no	—
151	Dic₁₅₈	—	4	2	no	—
152	Dic₁₅₈	—	4	2	no	—
153	Dic₁₅₈	—	8	1	no	—
154	Dic₁₅₈	—	8	1	no	—
155	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
156	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
157	C₈×C₇₉	#121	1	1	yes	characteristic
158	C₈×C₇₉	#121	1	1	yes	characteristic
159	C₇₉⋊C₈	—	2	2	no	—
160	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
161	C₂²×C₇₉	#447	1	1	yes	characteristic
162	C₂²×C₇₉	#480	1	1	yes	characteristic
163	C₂²×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
164	C₂²×C₇₉	C₂×Q₃₂	1	1	yes	characteristic
165	C₂²×C₇₉	#509	1	1	yes	characteristic
166	C₂²×C₇₉	#510	1	1	yes	characteristic
167	C₂×D₇₉	—	2	2	no	—
168	C₂×D₇₉	—	2	2	no	—
169	C₂×D₇₉	—	2	2	no	—
170	C₂×D₇₉	—	2	2	no	—
171	C₂×D₇₉	—	2	2	no	—
172	C₂×D₇₉	—	2	2	no	—
173	C₂×D₇₉	#358	1	4	yes	complemented
174	C₄×C₇₉	—	8	2	no	—
175	C₄×C₇₉	—	8	2	no	—
176	C₄×C₇₉	#386	1	1	yes	characteristic
177	C₄×C₇₉	#417	1	1	yes	characteristic
178	C₄×C₇₉	#448	1	1	yes	characteristic
179	C₄×C₇₉	#481	1	1	yes	characteristic
180	Dic₇₉	—	2	2	no	—
181	Dic₇₉	—	8	2	no	—
182	Dic₇₉	—	8	2	no	—
183	#12993	—	79	1	no	maximal, Sylow
184	C₂×C₇₉	#1874	1	1	yes	characteristic
185	C₂×C₇₉	C₂×#358	1	1	yes	characteristic
186	C₂×C₇₉	#2139	1	1	yes	characteristic
187	C₂×C₇₉	#2187	1	1	yes	characteristic
188	C₂×C₇₉	#2188	1	1	yes	characteristic
189	C₂×C₇₉	#2235	1	1	yes	characteristic
190	C₂×C₇₉	#2236	1	1	yes	characteristic
191	D₇₉	—	2	4	no	—
192	C₂×#467	—	79	1	no	—
193	C₂²×#107	—	79	1	no	—
194	C₂×#358	—	79	1	no	—
195	C₂×#358	—	79	1	no	—
196	C₂×C₂²⋊C₁₆	—	79	1	no	—
197	#1380	—	79	1	no	—
198	#1380	D₇₉	1	1	yes	characteristic, complemented
199	C₇₉	#12993	1	1	yes	characteristic, complemented, Sylow
200	C₂²×Q₁₆	—	79	1	no	—
201	C₂³×C₈	—	79	1	no	—
202	C₂×C₂²⋊Q₈	—	158	1	no	—
203	C₂²×C₄⋊C₄	—	158	1	no	—
204	C₂²×C₁₆	—	79	1	no	—
205	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
206	#467	—	79	4	no	—
207	#467	—	158	2	no	—
208	C₂×#107	—	79	1	no	—
209	C₂×#107	—	158	2	no	—
210	C₂×#107	—	158	1	no	—
211	C₂×#107	—	158	2	no	—
212	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
213	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
214	C₂×#103	—	158	1	no	—
215	#358	—	158	2	no	—
216	#358	—	158	2	no	—
217	C₂²⋊C₁₆	—	158	2	no	—
218	C₂²×Q₈	—	158	1	no	—
219	C₂³×C₄	—	316	1	no	—
220	C₂³×C₄	—	79	1	no	—
221	C₂×Q₁₆	—	79	2	no	—
222	C₂×Q₁₆	—	158	1	no	—
223	C₂×Q₁₆	—	158	1	no	—
224	C₂×Q₁₆	—	158	1	no	—
225	C₂×Q₁₆	—	158	1	no	—
226	C₂×Q₁₆	—	158	1	no	—
227	C₂²×C₈	—	79	1	no	—
228	C₂²×C₈	—	79	2	no	—
229	C₂²×C₈	—	79	2	no	—
230	C₂²×C₈	—	158	1	no	—
231	C₂²×C₈	—	158	1	no	—
232	C₂²×C₈	—	158	1	no	—
233	C₂²×C₈	—	158	1	no	—
234	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
235	C₂²⋊Q₈	—	316	2	no	—
236	C₂²⋊Q₈	—	158	4	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	316	2	no	—
240	C₂×C₄⋊C₄	—	316	1	no	—
241	C₂×C₄⋊C₄	—	316	1	no	—
242	C₂×C₄⋊C₄	—	158	4	no	—
243	C₂×C₄⋊C₄	—	158	1	no	—
244	C₂×C₂²⋊C₄	—	316	1	no	—
245	C₂×C₁₆	—	2	1	no	—
246	C₂×C₁₆	—	2	1	no	—
247	C₂×C₁₆	—	2	1	no	—
248	C₂×C₁₆	—	158	1	no	—
249	C₂×C₁₆	—	158	1	no	—
250	C₂×C₁₆	—	158	1	no	—
251	#107	—	2	1	no	—
252	#107	—	2	1	no	—
253	#107	—	79	2	no	—
254	#107	—	158	4	no	—
255	#107	—	158	1	no	—
256	#107	C₄×D₇₉	1	2	yes	complemented
257	#107	D₃₁₆	1	2	yes	—
258	#103	—	158	4	no	—
259	C₂⁴	—	79	1	no	—
260	C₂×Q₈	—	316	1	no	—
261	C₂×Q₈	—	316	1	no	—
262	C₂×Q₈	—	316	1	no	—
263	C₂×Q₈	—	316	1	no	—
264	C₂×Q₈	—	316	1	no	—
265	C₂×Q₈	—	158	2	no	—
266	C₂²×C₄	—	4	1	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	316	4	no	—
269	C₂²×C₄	—	316	1	no	—
270	C₂²×C₄	—	316	1	no	—
271	C₂²×C₄	—	79	1	no	—
272	C₂²×C₄	—	79	2	no	—
273	C₂²×C₄	—	79	2	no	—
274	C₂²×C₄	—	632	2	no	—
275	C₂²×C₄	—	632	2	no	—
276	C₂²×C₄	—	158	1	no	—
277	C₂²×C₄	—	158	1	no	—
278	C₂²×C₄	—	158	1	no	—
279	C₂²×C₄	—	158	1	no	—
280	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
281	Q₁₆	—	316	1	no	—
282	Q₁₆	—	316	1	no	—
283	Q₁₆	—	158	2	no	—
284	Q₁₆	—	158	2	no	—
285	C₂×C₈	—	79	2	no	—
286	C₂×C₈	—	158	1	no	—
287	C₂×C₈	—	158	1	no	—
288	C₂×C₈	—	158	2	no	—
289	C₂×C₈	—	158	2	no	—
290	C₂×C₈	—	158	2	no	—
291	C₂×C₈	—	158	2	no	—
292	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
293	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
294	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
295	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
296	C₂×C₈	#11929624	1	1	yes	characteristic
297	C₂×C₈	#11929625	1	1	yes	characteristic
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	316	4	no	—
303	C₄⋊C₄	—	316	4	no	—
304	C₄⋊C₄	—	316	1	no	—
305	C₄⋊C₄	—	158	2	no	—
306	C₂²⋊C₄	—	316	4	no	—
307	C₁₆	—	4	1	no	—
308	C₁₆	—	316	1	no	—
309	C₂³	—	79	2	no	—
310	C₂³	—	79	2	no	—
311	C₂³	—	79	2	no	—
312	C₂³	—	158	1	no	—
313	C₂³	—	158	1	no	—
314	C₂³	—	158	1	no	—
315	C₂³	—	158	1	no	—
316	C₂³	D₈×D₇₉	1	1	yes	characteristic
317	Q₈	—	316	2	no	—
318	Q₈	—	316	2	no	—
319	Q₈	—	632	1	no	—
320	Q₈	—	632	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	8	1	no	—
324	C₂×C₄	—	8	1	no	—
325	C₂×C₄	—	8	1	no	—
326	C₂×C₄	—	8	1	no	—
327	C₂×C₄	—	316	2	no	—
328	C₂×C₄	—	316	2	no	—
329	C₂×C₄	—	79	2	no	—
330	C₂×C₄	—	632	8	no	—
331	C₂×C₄	—	632	1	no	—
332	C₂×C₄	—	632	1	no	—
333	C₂×C₄	—	632	1	no	—
334	C₂×C₄	—	632	1	no	—
335	C₂×C₄	—	158	1	no	—
336	C₂×C₄	—	158	1	no	—
337	C₂×C₄	—	158	2	no	—
338	C₂×C₄	—	158	2	no	—
339	C₂×C₄	—	158	2	no	—
340	C₂×C₄	—	158	2	no	—
341	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
342	C₂×C₄	#14579131	1	1	yes	characteristic
343	C₂×C₄	#14579146	1	1	yes	characteristic
344	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
345	C₂×C₄	#14579188	1	1	yes	characteristic
346	C₂×C₄	#14579193	1	1	yes	characteristic
347	C₈	—	158	2	no	—
348	C₈	#14579123	1	1	yes	characteristic
349	C₈	C₄×D₃₁₆	1	1	yes	characteristic
350	C₈	#14579134	1	1	yes	characteristic
351	C₈	#14579136	1	1	yes	characteristic
352	C₂²	—	79	4	no	—
353	C₂²	—	158	2	no	—
354	C₂²	—	158	2	no	—
355	C₂²	—	158	2	no	—
356	C₂²	—	158	2	no	—
357	C₂²	—	158	2	no	—
358	C₂²	—	158	2	no	—
359	C₂²	#102×D₇₉	1	1	yes	characteristic
360	C₂²	#19098698	1	1	yes	characteristic
361	C₂²	#19098878	1	1	yes	characteristic
362	C₂²	SD₃₂×D₇₉	1	1	yes	characteristic
363	C₂²	Q₃₂×D₇₉	1	1	yes	characteristic
364	C₂²	#19098997	1	1	yes	characteristic
365	C₂²	#19099002	1	1	yes	characteristic
366	C₄	—	8	2	no	—
367	C₄	—	8	2	no	—
368	C₄	—	632	2	no	—
369	C₄	—	632	2	no	—
370	C₄	—	158	2	no	—
371	C₄	#19098441	1	1	yes	characteristic
372	C₄	#19098563	1	1	yes	characteristic
373	C₄	#19098703	1	1	yes	characteristic
374	C₄	#19098883	1	1	yes	characteristic
375	C₂	—	158	4	no	—
376	C₂	#23763716	1	1	yes	characteristic
377	C₂	#358×D₇₉	1	1	yes	characteristic
378	C₂	#23765109	1	1	yes	characteristic
379	C₂	#23765330	1	1	yes	characteristic
380	C₂	#23765335	1	1	yes	characteristic
381	C₂	#23765599	1	1	yes	characteristic
382	C₂	#23765604	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₇₉
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1380	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂³×D₇₉	D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1380	D₇₉
hypocenter	C₇₉	#12993
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

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

N	H ≅ G/N	type
3. C₂×#19098368	C₂	semidirect
4. C₂×#19098366	C₂	semidirect
6. #23761491	C₂	semidirect
7. C₇₉×#1380	C₂	semidirect
14. #19098805	C₄	semidirect
34. C₂×Dic₃₁₆	C₂×C₄	semidirect
65. #107×C₇₉	C₂×C₄	semidirect
173. C₂×D₇₉	#358	semidirect
198. #1380	D₇₉	semidirect
199. C₇₉	#12993	semidirect
256. #107	C₄×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098805	2
C₂×#107×D₇₉	2
C₂×#19098368	2
C₂×#19098366	2
C₂×(C₂×D₇₉)⋊C₁₆	2
#23761491	2
C₇₉×#1380	2
#12993	79