Subgroups — #34279818

Order 20224.

604 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23775824	C₂	1	1	yes	maximal, characteristic
2	#23775694	C₂	1	2	yes	maximal, complemented
3	#23763009	C₂	1	1	yes	maximal, characteristic, complemented
4	#23762973	C₂	1	2	yes	maximal, complemented
5	#23762804	C₂	1	1	yes	maximal, characteristic, complemented
6	#23762786	C₂	1	1	yes	maximal, characteristic, complemented
7	#23762627	C₂	1	2	yes	maximal, complemented
8	#23762565	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	#19099278	C₂²	1	2	yes	—
13	#19098968	C₂²	1	1	yes	characteristic
14	#19098915	C₂²	1	2	yes	—
15	#19098913	C₂²	1	2	yes	—
16	#19098901	—	2	2	no	—
17	#19098885	—	2	2	no	—
18	#19098858	C₂²	1	2	yes	—
19	#19098853	C₂²	1	2	yes	—
20	#19098811	—	2	2	no	—
21	#19098796	—	2	1	no	—
22	#19098765	C₂²	1	2	yes	—
23	#19098702	—	2	2	no	—
24	C₄×#11929611	C₂²	1	1	yes	characteristic
25	#19098641	C₂²	1	2	yes	—
26	C₈×D₃₁₆	C₂²	1	1	yes	characteristic
27	#19098600	—	2	1	no	—
28	#19098572	—	2	2	no	—
29	#19098316	C₂²	1	1	yes	characteristic
30	#19098313	C₂²	1	1	yes	characteristic
31	#19098311	C₂²	1	2	yes	—
32	#19098309	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×#11929611	—	2	2	no	—
46	C₂×#11929611	D₄	1	1	yes	characteristic
47	C₂×#11929608	—	2	2	no	—
48	C₂×C₈×D₇₉	—	2	1	no	—
49	#14579161	C₂³	1	2	yes	—
50	#14579147	—	4	2	no	—
51	#14579135	—	2	2	no	—
52	#14579132	—	2	2	no	—
53	C₄×D₃₁₆	C₂³	1	1	yes	characteristic
54	#14579120	—	2	2	no	—
55	#14579119	—	2	2	no	—
56	#14579095	—	2	1	no	—
57	#14579095	—	2	1	no	—
58	#14579094	—	2	2	no	—
59	#14579092	—	2	2	no	—
60	#14579092	D₄	1	1	yes	characteristic
61	C₄⋊(C₇₉⋊C₈)	C₂³	1	1	yes	characteristic
62	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
63	#14579085	—	2	2	no	—
64	#14579085	—	2	2	no	—
65	#14579085	—	2	2	no	—
66	#14579085	—	2	2	no	—
67	#14579085	D₄	1	1	yes	characteristic
68	#14579084	—	4	2	no	—
69	#14579082	—	2	2	no	—
70	#14579081	D₄	1	1	yes	characteristic
71	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
72	(C₂×D₇₉)⋊C₈	—	2	1	no	—
73	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
75	C₂²×Q₈×C₇₉	—	2	2	no	—
76	C₂×D₈×C₇₉	D₄	1	1	yes	characteristic
77	C₄⋊Q₈×C₇₉	—	2	2	no	—
78	C₄⋊Q₈×C₇₉	C₂³	1	2	yes	—
79	C₂²⋊Q₈×C₇₉	—	2	2	no	—
80	C₂²⋊Q₈×C₇₉	—	4	2	no	—
81	C₄×Q₈×C₇₉	C₂³	1	1	yes	characteristic
82	C₄×D₄×C₇₉	—	2	2	no	—
83	C₄×D₄×C₇₉	C₂³	1	2	yes	—
84	#107×C₇₉	—	2	1	no	—
85	#107×C₇₉	—	2	1	no	—
86	#107×C₇₉	—	2	2	no	—
87	#107×C₇₉	D₄	1	1	yes	characteristic
88	C₄⋊C₈×C₇₉	—	2	1	no	—
89	C₄⋊C₈×C₇₉	C₂³	1	1	yes	characteristic
90	#102×C₇₉	—	2	2	no	—
91	#102×C₇₉	D₄	1	2	yes	—
92	C₄×C₈×C₇₉	—	2	1	no	—
93	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₅₈	—	2	2	no	—
95	C₂×D₃₁₆	C₂×D₄	1	1	yes	characteristic
96	C₂×C₂²⋊D₇₉	—	4	2	no	—
97	C₂×C₄×D₇₉	—	2	1	no	—
98	#11929611	—	2	2	no	—
99	#11929611	—	8	2	no	—
100	#11929608	—	8	2	no	—
101	C₂×C₇₉⋊C₈	—	2	1	no	—
102	C₂×C₇₉⋊C₈	—	2	2	no	—
103	C₂×C₇₉⋊C₈	—	2	1	no	—
104	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
106	C₈×D₇₉	—	4	2	no	—
107	C₄⋊Dic₇₉	—	2	2	no	—
108	C₄⋊Dic₇₉	—	2	2	no	—
109	C₄⋊Dic₇₉	C₂×D₄	1	1	yes	characteristic
110	C₂²⋊Dic₇₉	—	4	2	no	—
111	(C₂×D₇₉)⋊C₄	—	2	1	no	—
112	(C₂×D₇₉)⋊C₄	—	4	2	no	—
113	C₄×Dic₇₉	—	4	2	no	—
114	C₂×Q₈×C₇₉	—	2	2	no	—
115	C₂×Q₈×C₇₉	—	4	2	no	—
116	C₂×Q₈×C₇₉	—	4	4	no	—
117	C₂×Q₈×C₇₉	—	8	2	no	—
118	C₂×Q₈×C₇₉	C₂×D₄	1	1	yes	characteristic
119	C₂×D₄×C₇₉	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₇₉	—	2	2	no	—
121	C₂²×C₄×C₇₉	—	4	2	no	—
122	D₈×C₇₉	—	2	2	no	—
123	C₂×C₈×C₇₉	—	2	1	no	—
124	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
126	C₄⋊C₄×C₇₉	—	2	1	no	—
127	C₄⋊C₄×C₇₉	—	2	2	no	—
128	C₄⋊C₄×C₇₉	—	2	2	no	—
129	C₄⋊C₄×C₇₉	—	4	2	no	—
130	C₄⋊C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
131	C₄⋊C₄×C₇₉	C₂×D₄	1	2	yes	—
132	C₂²⋊C₄×C₇₉	—	2	2	no	—
133	C₂²⋊C₄×C₇₉	—	4	2	no	—
134	C₄²×C₇₉	—	2	1	no	—
135	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
136	C₂³×C₇₉	—	2	2	no	—
137	C₂²×D₇₉	—	2	1	no	—
138	Q₈×C₇₉	—	2	1	no	—
139	Q₈×C₇₉	—	4	2	no	—
140	Q₈×C₇₉	—	4	2	no	—
141	Q₈×C₇₉	—	8	4	no	—
142	Q₈×C₇₉	#135	1	2	yes	—
143	D₄×C₇₉	—	2	2	no	—
144	D₄×C₇₉	—	2	2	no	—
145	C₂×C₄×C₇₉	—	2	1	no	—
146	C₂×C₄×C₇₉	—	2	2	no	—
147	C₂×C₄×C₇₉	—	2	1	no	—
148	C₂×C₄×C₇₉	—	4	2	no	—
149	C₂×C₄×C₇₉	—	4	2	no	—
150	C₂×C₄×C₇₉	—	4	2	no	—
151	C₂×C₄×C₇₉	—	8	4	no	—
152	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²⋊D₇₉	—	16	2	no	—
156	C₂×Dic₇₉	—	2	1	no	—
157	C₂×Dic₇₉	—	4	2	no	—
158	D₃₁₆	—	2	1	no	—
159	D₃₁₆	—	2	1	no	—
160	C₄×D₇₉	—	4	1	no	—
161	C₄×D₇₉	—	4	1	no	—
162	Dic₁₅₈	—	8	2	no	—
163	C₈×C₇₉	—	2	1	no	—
164	C₈×C₇₉	—	4	1	no	—
165	C₈×C₇₉	#137	1	2	yes	—
166	C₇₉⋊C₈	—	2	1	no	—
167	C₇₉⋊C₈	—	2	1	no	—
168	C₇₉⋊C₈	—	4	1	no	—
169	C₇₉⋊C₈	—	4	1	no	—
170	C₇₉⋊C₈	—	4	2	no	—
171	C₂²×C₇₉	—	4	2	no	—
172	C₂²×C₇₉	—	4	2	no	—
173	C₂²×C₇₉	—	4	2	no	—
174	C₂²×C₇₉	D₄²	1	1	yes	characteristic
175	C₂×D₇₉	—	4	1	no	—
176	C₂×D₇₉	—	4	1	no	—
177	C₂×D₇₉	—	4	1	no	—
178	C₄×C₇₉	—	2	2	no	—
179	C₄×C₇₉	—	2	1	no	—
180	C₄×C₇₉	—	4	2	no	—
181	C₄×C₇₉	—	4	1	no	—
182	C₄×C₇₉	—	8	2	no	—
183	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
184	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
185	C₄×C₇₉	#576	1	1	yes	characteristic
186	C₄×C₇₉	#578	1	1	yes	characteristic
187	Dic₇₉	—	4	1	no	—
188	Dic₇₉	—	8	2	no	—
189	#19905	—	79	1	no	maximal, Sylow
190	C₂×C₇₉	—	8	2	no	—
191	C₂×C₇₉	#3276	1	1	yes	characteristic
192	C₂×C₇₉	#3280	1	1	yes	characteristic
193	C₂×C₇₉	#3290	1	1	yes	characteristic
194	D₇₉	—	8	1	no	—
195	Q₈×D₈	D₇₉	1	1	yes	characteristic, complemented
196	#3341	—	79	1	no	—
197	#3329	—	79	2	no	—
198	#1710	—	79	1	no	—
199	#1703	—	79	2	no	—
200	#1676	—	79	1	no	—
201	#1673	—	79	1	no	—
202	#1646	—	79	2	no	—
203	#1637	—	79	2	no	—
204	#1589	—	79	1	no	—
205	C₈×SD₁₆	—	79	1	no	—
206	C₇₉	#19905	1	1	yes	characteristic, complemented, Sylow
207	#555	—	79	2	no	—
208	D₄×Q₈	C₂×D₇₉	1	2	yes	—
209	#500	—	2	1	no	—
210	#500	—	79	1	no	—
211	#500	C₂×D₇₉	1	1	yes	characteristic
212	#487	—	79	2	no	—
213	#487	—	79	2	no	—
214	#485	—	158	2	no	—
215	#482	—	158	2	no	—
216	#477	—	79	2	no	—
217	#476	—	79	2	no	—
218	#474	—	2	2	no	—
219	#474	C₂×D₇₉	1	2	yes	—
220	#468	—	158	2	no	—
221	#466	—	158	1	no	—
222	#461	—	79	2	no	—
223	#448	—	158	2	no	—
224	C₈×Q₈	C₂×D₇₉	1	1	yes	characteristic
225	C₄×SD₁₆	—	79	1	no	—
226	C₄×D₈	—	2	1	no	—
227	C₄×D₈	C₂×D₇₉	1	1	yes	characteristic
228	#436	—	79	2	no	—
229	C₈×D₄	—	79	1	no	—
230	#427	—	158	1	no	—
231	#419	—	158	2	no	—
232	#335	—	79	1	no	—
233	#334	—	79	1	no	—
234	#333	—	79	2	no	—
235	#332	—	79	2	no	—
236	Q₈⋊C₈	—	79	1	no	—
237	D₄⋊C₈	—	79	2	no	—
238	D₄⋊C₈	—	79	1	no	—
239	C₈²	—	79	1	no	—
240	C₂²×Q₈	—	2	2	no	—
241	C₂×SD₁₆	—	79	1	no	—
242	C₂×SD₁₆	—	158	2	no	—
243	C₂×D₈	C₂²⋊D₇₉	1	1	yes	characteristic
244	C₂×M₁₆	—	158	2	no	—
245	C₂²×C₈	—	158	1	no	—
246	C₄⋊Q₈	—	2	2	no	—
247	C₄⋊Q₈	—	79	2	no	—
248	C₄⋊Q₈	C₂²×D₇₉	1	2	yes	—
249	#124	—	316	2	no	—
250	C₂²⋊Q₈	—	2	2	no	—
251	C₂²⋊Q₈	—	4	2	no	—
252	#121	—	158	2	no	—
253	#121	—	158	2	no	—
254	C₄×Q₈	C₂²×D₇₉	1	1	yes	characteristic
255	C₄×D₄	—	2	2	no	—
256	C₄×D₄	—	79	1	no	—
257	C₄×D₄	C₂²×D₇₉	1	2	yes	—
258	#117	—	158	2	no	—
259	#117	—	158	2	no	—
260	#107	—	2	1	no	—
261	#107	—	2	1	no	—
262	#107	—	2	2	no	—
263	#107	—	158	1	no	—
264	#107	—	158	1	no	—
265	#107	—	158	2	no	—
266	#107	C₂²⋊D₇₉	1	1	yes	characteristic
267	#106	—	79	1	no	—
268	#106	—	158	2	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	79	2	no	—
271	C₄⋊C₈	—	79	1	no	—
272	C₄⋊C₈	C₂²×D₇₉	1	1	yes	characteristic
273	#103	—	316	2	no	—
274	#103	—	79	1	no	—
275	#103	—	158	2	no	—
276	#103	—	158	2	no	—
277	#103	—	158	2	no	—
278	#103	—	158	2	no	—
279	#102	—	2	2	no	—
280	#102	—	79	1	no	—
281	#102	—	158	2	no	—
282	#102	C₂²⋊D₇₉	1	2	yes	—
283	C₂²⋊C₈	—	158	2	no	—
284	C₂²⋊C₈	—	158	1	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	—	79	1	no	—
287	C₄×C₈	—	79	1	no	—
288	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	4	2	no	—
291	C₂×Q₈	—	4	4	no	—
292	C₂×Q₈	—	8	2	no	—
293	C₂×Q₈	—	158	2	no	—
294	C₂×Q₈	D₄×D₇₉	1	1	yes	characteristic
295	C₂×D₄	—	316	2	no	—
296	C₂×D₄	—	79	1	no	—
297	C₂×D₄	C₂×C₂²⋊D₇₉	1	2	yes	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	4	2	no	—
300	C₂²×C₄	—	158	1	no	—
301	SD₁₆	—	632	2	no	—
302	SD₁₆	—	158	2	no	—
303	D₈	—	2	2	no	—
304	M₁₆	—	632	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	316	2	no	—
307	C₂×C₈	—	79	1	no	—
308	C₂×C₈	—	79	1	no	—
309	C₂×C₈	—	158	1	no	—
310	C₂×C₈	—	158	2	no	—
311	C₂×C₈	—	158	1	no	—
312	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
313	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	4	2	no	—
318	C₄⋊C₄	—	79	1	no	—
319	C₄⋊C₄	—	158	2	no	—
320	C₄⋊C₄	—	158	2	no	—
321	C₄⋊C₄	D₄×D₇₉	1	1	yes	characteristic
322	C₄⋊C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
323	C₂²⋊C₄	—	2	2	no	—
324	C₂²⋊C₄	—	4	2	no	—
325	C₂²⋊C₄	—	316	2	no	—
326	C₂²⋊C₄	—	316	2	no	—
327	C₂²⋊C₄	—	158	1	no	—
328	C₄²	—	2	1	no	—
329	C₄²	—	316	2	no	—
330	C₄²	C₂³×D₇₉	1	1	yes	characteristic
331	C₂³	—	2	2	no	—
332	C₂³	—	158	1	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	4	2	no	—
335	Q₈	—	4	2	no	—
336	Q₈	—	8	4	no	—
337	Q₈	—	632	2	no	—
338	Q₈	#14579180	1	2	yes	—
339	D₄	—	2	2	no	—
340	D₄	—	2	2	no	—
341	D₄	—	158	1	no	—
342	D₄	—	158	1	no	—
343	D₄	—	1264	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	8	4	no	—
351	C₂×C₄	—	316	1	no	—
352	C₂×C₄	—	316	1	no	—
353	C₂×C₄	—	316	2	no	—
354	C₂×C₄	—	158	1	no	—
355	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
356	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
357	C₂×C₄	#14579206	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	4	1	no	—
360	C₈	—	316	1	no	—
361	C₈	—	316	1	no	—
362	C₈	—	316	2	no	—
363	C₈	—	158	1	no	—
364	C₈	—	158	1	no	—
365	C₈	#14579192	1	2	yes	—
366	C₂²	—	4	2	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	316	1	no	—
370	C₂²	—	316	1	no	—
371	C₂²	—	316	1	no	—
372	C₂²	D₄×C₂²⋊D₇₉	1	1	yes	characteristic
373	C₄	—	2	2	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	4	1	no	—
377	C₄	—	8	2	no	—
378	C₄	—	316	1	no	—
379	C₄	—	632	2	no	—
380	C₄	C₂×#14579180	1	1	yes	characteristic
381	C₄	C₂×#14579192	1	1	yes	characteristic
382	C₄	#19099364	1	1	yes	characteristic
383	C₄	#19099378	1	1	yes	characteristic
384	C₂	—	8	2	no	—
385	C₂	—	632	1	no	—
386	C₂	#23775046	1	1	yes	characteristic
387	C₂	#23775085	1	1	yes	characteristic
388	C₂	#23775186	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	#23775824	C₂
hypercenter	Q₈×D₈	D₇₉
hypocenter	C₇₉	#19905
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. #23775694	C₂	semidirect
3. #23763009	C₂	semidirect
4. #23762973	C₂	semidirect
5. #23762804	C₂	semidirect
6. #23762786	C₂	semidirect
7. #23762627	C₂	semidirect
8. #23762565	C₂	semidirect
9. #23762285	C₂	semidirect
10. C₈×#11929611	C₂	semidirect
11. Q₈×D₈×C₇₉	C₂	semidirect
195. Q₈×D₈	D₇₉	semidirect
206. C₇₉	#19905	semidirect

Maximal subgroups

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

subgroup	index
#23775824	2
#23775694	2
#23763009	2
#23762973	2
#23762804	2
#23762786	2
#23762627	2
#23762565	2
#23762285	2
C₈×#11929611	2
Q₈×D₈×C₇₉	2
#19905	79