Subgroups — #34279582

Order 20224.

690 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23775749	C₂	1	1	yes	maximal, characteristic
2	#23775046	C₂	1	2	yes	maximal, complemented
3	#23763017	C₂	1	1	yes	maximal, characteristic, complemented
4	#23762887	C₂	1	2	yes	maximal, complemented
5	#23762794	C₂	1	1	yes	maximal, characteristic, complemented
6	#23762786	C₂	1	1	yes	maximal, characteristic, complemented
7	#23762649	C₂	1	2	yes	maximal, complemented
8	#23762634	C₂	1	2	yes	maximal, complemented
9	#23762285	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11929611	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₈×D₈×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579192	—	2	2	no	—
13	#19099218	C₂²	1	2	yes	—
14	#19098968	C₂²	1	1	yes	characteristic
15	#19098966	C₂²	1	2	yes	—
16	#19098942	C₂²	1	2	yes	—
17	#19098875	—	2	2	no	—
18	#19098843	C₂²	1	2	yes	—
19	#19098819	—	2	2	no	—
20	#19098796	—	2	1	no	—
21	#19098773	C₂²	1	2	yes	—
22	#19098765	C₂²	1	2	yes	—
23	#19098723	—	2	2	no	—
24	#19098719	—	2	2	no	—
25	C₄×#11929611	C₂²	1	1	yes	characteristic
26	#19098641	C₂²	1	2	yes	—
27	C₈×D₃₁₆	C₂²	1	1	yes	characteristic
28	#107×D₇₉	—	2	1	no	—
29	#19098316	C₂²	1	1	yes	characteristic
30	#19098313	C₂²	1	1	yes	characteristic
31	#19098309	C₂²	1	2	yes	—
32	#19098308	C₂²	1	2	yes	—
33	Q₈⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
34	D₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
35	#19098290	C₂²	1	1	yes	characteristic
36	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
37	D₄×Q₈×C₇₉	C₂²	1	2	yes	—
38	#500×C₇₉	—	2	1	no	—
39	#500×C₇₉	C₂²	1	1	yes	characteristic
40	#474×C₇₉	—	2	2	no	—
41	#474×C₇₉	C₂²	1	2	yes	—
42	C₈×Q₈×C₇₉	C₂²	1	1	yes	characteristic
43	C₄×D₈×C₇₉	—	2	1	no	—
44	C₄×D₈×C₇₉	C₂²	1	1	yes	characteristic
45	C₂×#11929624	—	2	2	no	—
46	#14579192	—	4	8	no	—
47	C₂×#11929614	—	2	2	no	—
48	C₂×#11929614	—	2	2	no	—
49	C₂×#11929614	—	2	2	no	—
50	C₂×#11929611	—	2	2	no	—
51	C₂×#11929611	D₄	1	1	yes	characteristic
52	C₂×#11929608	—	2	2	no	—
53	C₂×C₈×D₇₉	—	2	1	no	—
54	#14579161	C₂³	1	2	yes	—
55	#14579143	—	4	2	no	—
56	#14579140	—	2	2	no	—
57	#14579132	—	2	2	no	—
58	C₄×D₃₁₆	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×D₇₉	—	2	2	no	—
60	#14579095	—	2	1	no	—
61	#14579095	—	2	1	no	—
62	#14579094	—	2	2	no	—
63	#14579094	—	4	2	no	—
64	#14579092	D₄	1	1	yes	characteristic
65	C₄⋊(C₇₉⋊C₈)	C₂³	1	1	yes	characteristic
66	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
67	#14579085	—	2	2	no	—
68	#14579085	D₄	1	1	yes	characteristic
69	#14579084	—	2	2	no	—
70	#14579082	—	2	2	no	—
71	#14579081	D₄	1	1	yes	characteristic
72	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
73	(C₂×D₇₉)⋊C₈	—	2	1	no	—
74	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
76	C₂²×Q₈×C₇₉	—	2	2	no	—
77	C₂×D₈×C₇₉	D₄	1	1	yes	characteristic
78	C₄⋊Q₈×C₇₉	—	2	2	no	—
79	C₄⋊Q₈×C₇₉	C₂³	1	2	yes	—
80	C₂²⋊Q₈×C₇₉	—	2	2	no	—
81	C₂²⋊Q₈×C₇₉	—	4	2	no	—
82	C₄×Q₈×C₇₉	C₂³	1	1	yes	characteristic
83	C₄×D₄×C₇₉	—	2	2	no	—
84	C₄×D₄×C₇₉	C₂³	1	2	yes	—
85	#107×C₇₉	—	2	1	no	—
86	#107×C₇₉	—	2	1	no	—
87	#107×C₇₉	—	2	2	no	—
88	#107×C₇₉	D₄	1	1	yes	characteristic
89	C₄⋊C₈×C₇₉	—	2	1	no	—
90	C₄⋊C₈×C₇₉	C₂³	1	1	yes	characteristic
91	#102×C₇₉	—	2	2	no	—
92	#102×C₇₉	D₄	1	2	yes	—
93	C₄×C₈×C₇₉	—	2	1	no	—
94	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
95	#11929624	—	4	4	no	—
96	#11929624	—	8	2	no	—
97	C₂×Dic₁₅₈	—	2	2	no	—
98	C₂×D₃₁₆	C₂×D₄	1	1	yes	characteristic
99	C₂×C₂²⋊D₇₉	—	4	2	no	—
100	C₂×C₄×D₇₉	—	2	1	no	—
101	C₂×C₄×D₇₉	—	4	2	no	—
102	#11929614	—	4	4	no	—
103	#11929614	—	4	4	no	—
104	#11929614	—	8	2	no	—
105	#11929611	—	4	4	no	—
106	#11929611	D₈	1	4	yes	complemented
107	#11929608	—	4	4	no	—
108	C₂×C₇₉⋊C₈	—	2	1	no	—
109	C₂×C₇₉⋊C₈	—	2	2	no	—
110	C₂×C₇₉⋊C₈	—	2	1	no	—
111	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
113	C₈×D₇₉	—	4	2	no	—
114	C₄⋊Dic₇₉	—	2	2	no	—
115	C₄⋊Dic₇₉	—	2	2	no	—
116	C₄⋊Dic₇₉	C₂×D₄	1	1	yes	characteristic
117	Dic₇₉⋊C₄	—	4	2	no	—
118	C₂²⋊Dic₇₉	—	4	2	no	—
119	(C₂×D₇₉)⋊C₄	—	2	1	no	—
120	C₂×Q₈×C₇₉	—	2	2	no	—
121	C₂×Q₈×C₇₉	—	4	2	no	—
122	C₂×Q₈×C₇₉	—	4	4	no	—
123	C₂×Q₈×C₇₉	—	8	2	no	—
124	C₂×Q₈×C₇₉	C₂×D₄	1	1	yes	characteristic
125	C₂×D₄×C₇₉	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₇₉	—	2	2	no	—
127	C₂²×C₄×C₇₉	—	4	2	no	—
128	D₈×C₇₉	—	2	2	no	—
129	C₂×C₈×C₇₉	—	2	1	no	—
130	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₇₉	—	2	1	no	—
133	C₄⋊C₄×C₇₉	—	2	2	no	—
134	C₄⋊C₄×C₇₉	—	2	2	no	—
135	C₄⋊C₄×C₇₉	—	4	2	no	—
136	C₄⋊C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄×C₇₉	C₂×D₄	1	2	yes	—
138	C₂²⋊C₄×C₇₉	—	2	2	no	—
139	C₂²⋊C₄×C₇₉	—	4	2	no	—
140	C₄²×C₇₉	—	2	1	no	—
141	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
142	C₂³×C₇₉	—	2	2	no	—
143	C₂²×D₇₉	—	2	1	no	—
144	Q₈×C₇₉	—	2	1	no	—
145	Q₈×C₇₉	—	4	2	no	—
146	Q₈×C₇₉	—	4	2	no	—
147	Q₈×C₇₉	—	8	4	no	—
148	Q₈×C₇₉	C₂×D₈	1	2	yes	—
149	D₄×C₇₉	—	2	2	no	—
150	D₄×C₇₉	—	2	2	no	—
151	C₂×C₄×C₇₉	—	2	1	no	—
152	C₂×C₄×C₇₉	—	2	2	no	—
153	C₂×C₄×C₇₉	—	2	1	no	—
154	C₂×C₄×C₇₉	—	4	2	no	—
155	C₂×C₄×C₇₉	—	4	2	no	—
156	C₂×C₄×C₇₉	—	4	2	no	—
157	C₂×C₄×C₇₉	—	8	4	no	—
158	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂²⋊D₇₉	—	8	4	no	—
162	C₂×Dic₇₉	—	2	1	no	—
163	C₂×Dic₇₉	—	4	2	no	—
164	D₃₁₆	—	2	1	no	—
165	D₃₁₆	C₂×D₈	1	2	yes	—
166	C₄×D₇₉	—	4	1	no	—
167	C₄×D₇₉	—	4	1	no	—
168	C₄×D₇₉	—	8	4	no	—
169	Dic₁₅₈	—	4	2	no	—
170	Dic₁₅₈	—	4	2	no	—
171	C₈×C₇₉	—	2	1	no	—
172	C₈×C₇₉	—	4	1	no	—
173	C₈×C₇₉	#137	1	2	yes	complemented
174	C₇₉⋊C₈	—	2	2	no	—
175	C₇₉⋊C₈	—	2	1	no	—
176	C₇₉⋊C₈	—	4	2	no	—
177	C₇₉⋊C₈	—	4	1	no	—
178	C₇₉⋊C₈	C₂×D₈	1	2	yes	—
179	C₂²×C₇₉	—	4	2	no	—
180	C₂²×C₇₉	—	4	2	no	—
181	C₂²×C₇₉	—	4	2	no	—
182	C₂²×C₇₉	D₄²	1	1	yes	characteristic
183	C₂×D₇₉	—	2	2	no	—
184	C₂×D₇₉	—	4	1	no	—
185	C₂×D₇₉	—	4	1	no	—
186	C₄×C₇₉	—	2	2	no	—
187	C₄×C₇₉	—	2	1	no	—
188	C₄×C₇₉	—	4	2	no	—
189	C₄×C₇₉	—	4	1	no	—
190	C₄×C₇₉	—	8	2	no	—
191	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
192	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
193	C₄×C₇₉	#576	1	1	yes	characteristic
194	C₄×C₇₉	#578	1	1	yes	characteristic
195	Dic₇₉	—	4	1	no	—
196	Dic₇₉	—	8	2	no	—
197	#19887	—	79	1	no	maximal, Sylow
198	C₂×C₇₉	—	8	2	no	—
199	C₂×C₇₉	#3276	1	1	yes	characteristic
200	C₂×C₇₉	D₄×D₈	1	1	yes	characteristic
201	C₂×C₇₉	#3293	1	1	yes	characteristic
202	D₇₉	—	4	2	no	—
203	Q₈×D₈	D₇₉	1	1	yes	characteristic, complemented
204	#3334	—	79	1	no	—
205	#3276	—	79	2	no	—
206	#1713	—	79	1	no	—
207	#1689	—	79	2	no	—
208	#1674	—	79	1	no	—
209	#1673	—	79	1	no	—
210	#1649	—	79	2	no	—
211	#1647	—	79	2	no	—
212	#1589	—	79	1	no	—
213	C₈×SD₁₆	—	79	1	no	—
214	C₇₉	#19887	1	1	yes	characteristic, complemented, Sylow
215	C₂×#137	—	158	2	no	—
216	D₄×Q₈	C₂×D₇₉	1	2	yes	—
217	#547	—	79	2	no	—
218	#500	—	2	1	no	—
219	#500	—	79	2	no	—
220	#500	—	79	1	no	—
221	#500	C₂×D₇₉	1	1	yes	characteristic
222	#494	—	79	2	no	—
223	#480	—	158	2	no	—
224	#474	—	2	2	no	—
225	#474	—	79	2	no	—
226	#474	C₂×D₇₉	1	2	yes	—
227	#470	—	158	2	no	—
228	#466	—	158	1	no	—
229	#462	—	79	2	no	—
230	#461	—	79	2	no	—
231	#452	—	158	2	no	—
232	#451	—	158	2	no	—
233	C₈×Q₈	C₂×D₇₉	1	1	yes	characteristic
234	C₄×SD₁₆	—	79	1	no	—
235	C₄×D₈	—	2	1	no	—
236	C₄×D₈	C₂×D₇₉	1	1	yes	characteristic
237	#436	—	79	2	no	—
238	C₈×D₄	—	79	1	no	—
239	C₂×#107	—	158	1	no	—
240	#335	—	79	1	no	—
241	#334	—	79	1	no	—
242	#332	—	79	2	no	—
243	#332	—	79	2	no	—
244	Q₈⋊C₈	—	79	1	no	—
245	D₄⋊C₈	—	79	2	no	—
246	D₄⋊C₈	—	79	1	no	—
247	C₈²	—	79	1	no	—
248	C₂×#41	—	158	2	no	—
249	C₂²×Q₈	—	2	2	no	—
250	#137	—	316	8	no	—
251	C₂×Q₁₆	—	158	2	no	—
252	C₂×Q₁₆	—	158	2	no	—
253	C₂×Q₁₆	—	158	2	no	—
254	C₂×SD₁₆	—	79	1	no	—
255	C₂×SD₁₆	—	158	2	no	—
256	C₂×D₈	C₂²⋊D₇₉	1	1	yes	characteristic
257	C₂×M₁₆	—	158	2	no	—
258	C₂²×C₈	—	158	1	no	—
259	C₄⋊Q₈	—	2	2	no	—
260	C₄⋊Q₈	—	79	2	no	—
261	C₄⋊Q₈	C₂²×D₇₉	1	2	yes	—
262	#123	—	316	2	no	—
263	C₂²⋊Q₈	—	2	2	no	—
264	C₂²⋊Q₈	—	4	2	no	—
265	C₂²⋊Q₈	—	158	2	no	—
266	#121	—	158	2	no	—
267	C₄×Q₈	C₂²×D₇₉	1	1	yes	characteristic
268	C₄×D₄	—	2	2	no	—
269	C₄×D₄	—	79	1	no	—
270	C₄×D₄	C₂²×D₇₉	1	2	yes	—
271	C₂×C₄⋊C₄	—	158	2	no	—
272	#107	—	2	1	no	—
273	#107	—	2	1	no	—
274	#107	—	2	2	no	—
275	#107	—	316	2	no	—
276	#107	—	158	1	no	—
277	#107	—	158	1	no	—
278	#107	—	158	2	no	—
279	#107	C₂²⋊D₇₉	1	1	yes	characteristic
280	#106	—	79	1	no	—
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	79	2	no	—
283	C₄⋊C₈	—	79	1	no	—
284	C₄⋊C₈	C₂²×D₇₉	1	1	yes	characteristic
285	#103	—	79	1	no	—
286	#103	—	158	2	no	—
287	#103	—	158	2	no	—
288	#102	—	2	2	no	—
289	#102	—	79	1	no	—
290	#102	—	158	2	no	—
291	#102	C₂²⋊D₇₉	1	2	yes	—
292	C₂²⋊C₈	—	158	2	no	—
293	C₂²⋊C₈	—	158	1	no	—
294	C₄×C₈	—	2	1	no	—
295	C₄×C₈	—	79	1	no	—
296	C₄×C₈	—	79	1	no	—
297	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
298	#41	—	316	4	no	—
299	#41	—	632	2	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	4	2	no	—
302	C₂×Q₈	—	4	4	no	—
303	C₂×Q₈	—	8	2	no	—
304	C₂×Q₈	—	158	2	no	—
305	C₂×Q₈	D₄×D₇₉	1	1	yes	characteristic
306	C₂×D₄	—	316	2	no	—
307	C₂×D₄	—	79	1	no	—
308	C₂×D₄	C₂×C₂²⋊D₇₉	1	2	yes	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	4	2	no	—
311	C₂²×C₄	—	316	2	no	—
312	C₂²×C₄	—	158	1	no	—
313	Q₁₆	—	316	4	no	—
314	Q₁₆	—	316	4	no	—
315	Q₁₆	—	632	2	no	—
316	SD₁₆	—	316	4	no	—
317	SD₁₆	—	79	4	no	—
318	D₈	—	2	2	no	—
319	M₁₆	—	316	4	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	316	2	no	—
322	C₂×C₈	—	79	1	no	—
323	C₂×C₈	—	79	1	no	—
324	C₂×C₈	—	158	2	no	—
325	C₂×C₈	—	158	1	no	—
326	C₂×C₈	—	158	1	no	—
327	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
328	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	4	2	no	—
333	C₄⋊C₄	—	316	2	no	—
334	C₄⋊C₄	—	79	1	no	—
335	C₄⋊C₄	—	158	2	no	—
336	C₄⋊C₄	—	158	2	no	—
337	C₄⋊C₄	D₄×D₇₉	1	1	yes	characteristic
338	C₄⋊C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
339	C₂²⋊C₄	—	2	2	no	—
340	C₂²⋊C₄	—	4	2	no	—
341	C₂²⋊C₄	—	316	2	no	—
342	C₂²⋊C₄	—	158	1	no	—
343	C₄²	—	2	1	no	—
344	C₄²	C₂³×D₇₉	1	1	yes	characteristic
345	C₂³	—	2	2	no	—
346	C₂³	—	158	1	no	—
347	Q₈	—	2	1	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	8	4	no	—
351	Q₈	—	316	2	no	—
352	Q₈	—	316	2	no	—
353	Q₈	D₈×D₇₉	1	2	yes	—
354	D₄	—	2	2	no	—
355	D₄	—	2	2	no	—
356	D₄	—	79	2	no	—
357	D₄	—	632	4	no	—
358	D₄	—	158	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	8	4	no	—
366	C₂×C₄	—	316	1	no	—
367	C₂×C₄	—	316	1	no	—
368	C₂×C₄	—	316	2	no	—
369	C₂×C₄	—	632	4	no	—
370	C₂×C₄	—	158	1	no	—
371	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
372	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
373	C₂×C₄	#14579206	1	1	yes	characteristic
374	C₈	—	2	1	no	—
375	C₈	—	4	1	no	—
376	C₈	—	316	2	no	—
377	C₈	—	316	1	no	—
378	C₈	—	79	2	no	—
379	C₈	—	158	2	no	—
380	C₈	—	158	1	no	—
381	C₈	#14579192	1	2	yes	complemented
382	C₂²	—	4	2	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	316	1	no	—
386	C₂²	—	316	1	no	—
387	C₂²	—	158	2	no	—
388	C₂²	D₄×C₂²⋊D₇₉	1	1	yes	characteristic
389	C₄	—	2	2	no	—
390	C₄	—	2	1	no	—
391	C₄	—	4	2	no	—
392	C₄	—	4	1	no	—
393	C₄	—	8	2	no	—
394	C₄	—	316	1	no	—
395	C₄	—	632	2	no	—
396	C₄	C₂×D₈×D₇₉	1	1	yes	characteristic
397	C₄	C₂×#14579192	1	1	yes	characteristic
398	C₄	#19099364	1	1	yes	characteristic
399	C₄	#19099378	1	1	yes	characteristic
400	C₂	—	8	2	no	—
401	C₂	—	316	2	no	—
402	C₂	#23775046	1	1	yes	characteristic
403	C₂	D₈×C₂²⋊D₇₉	1	1	yes	characteristic
404	C₂	#23775202	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₇₉
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	Q₈×D₈×C₇₉	C₂
socle	C₂²×C₇₉	D₄²
prime radical	#23775749	C₂
hypercenter	Q₈×D₈	D₇₉
hypocenter	C₇₉	#19887
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈×D₈

Complements

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

N	H ≅ G/N	type
2. #23775046	C₂	semidirect
3. #23763017	C₂	semidirect
4. #23762887	C₂	semidirect
5. #23762794	C₂	semidirect
6. #23762786	C₂	semidirect
7. #23762649	C₂	semidirect
8. #23762634	C₂	semidirect
9. #23762285	C₂	semidirect
10. C₈×#11929611	C₂	semidirect
11. Q₈×D₈×C₇₉	C₂	semidirect
106. #11929611	D₈	semidirect
173. C₈×C₇₉	#137	semidirect
203. Q₈×D₈	D₇₉	semidirect
214. C₇₉	#19887	semidirect
381. C₈	#14579192	semidirect

Maximal subgroups

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

subgroup	index
#23775749	2
#23775046	2
#23763017	2
#23762887	2
#23762794	2
#23762786	2
#23762649	2
#23762634	2
#23762285	2
C₈×#11929611	2
Q₈×D₈×C₇₉	2
#19887	79