Subgroups — #34275967

Order 20224.

922 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773979	C₂	1	1	yes	maximal, characteristic, complemented
2	#23773975	C₂	1	2	yes	maximal, complemented
3	#23773826	C₂	1	2	yes	maximal, complemented
4	#23763017	C₂	1	1	yes	maximal, characteristic, complemented
5	#23763014	C₂	1	1	yes	maximal, characteristic, complemented
6	#23763004	C₂	1	2	yes	maximal, complemented
7	#23762993	C₂	1	1	yes	maximal, characteristic, complemented
8	#23762991	C₂	1	2	yes	maximal, complemented
9	#23762989	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#3169	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579181	—	2	4	no	—
13	#19099113	C₂²	1	2	yes	—
14	#19098968	C₂²	1	1	yes	characteristic
15	#19098966	C₂²	1	2	yes	—
16	#19098945	—	2	4	no	—
17	#19098942	C₂²	1	2	yes	—
18	#19098940	C₂²	1	1	yes	characteristic
19	#19098939	C₂²	1	2	yes	—
20	#19098939	C₂²	1	2	yes	—
21	#19098937	C₂²	1	2	yes	—
22	#19098935	C₂²	1	2	yes	—
23	#19098911	C₂²	1	2	yes	—
24	#19098910	C₂²	1	2	yes	—
25	#19098909	C₂²	1	2	yes	—
26	#19098799	—	2	2	no	—
27	#19098797	—	2	4	no	—
28	C₄×D₆₃₂	—	2	2	no	—
29	C₄×#11929609	—	2	4	no	—
30	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
32	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
33	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
34	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
36	#532×C₇₉	C₂²	1	2	yes	—
37	#500×C₇₉	C₂²	1	1	yes	characteristic
38	#493×C₇₉	C₂²	1	1	yes	characteristic
39	#486×C₇₉	C₂²	1	2	yes	—
40	#466×C₇₉	—	2	2	no	—
41	C₄×D₈×C₇₉	—	2	2	no	—
42	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
43	C₂×#11929624	—	2	2	no	—
44	#14579181	—	4	16	no	—
45	C₂×#11929614	—	2	4	no	—
46	C₂×#11929612	—	2	4	no	—
47	C₂×#11929611	—	2	4	no	—
48	C₂×D₆₃₂	—	2	2	no	—
49	C₂×#11929609	—	2	4	no	—
50	C₂×#11929609	—	2	4	no	—
51	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
52	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
53	#14579161	C₂³	1	2	yes	—
54	#14579158	C₂³	1	2	yes	—
55	#14579149	—	2	2	no	—
56	#14579135	—	2	2	no	—
57	#14579135	—	2	2	no	—
58	C₄×D₃₁₆	—	2	2	no	—
59	C₄×D₃₁₆	—	2	2	no	—
60	#14579095	—	2	2	no	—
61	#14579094	—	2	4	no	—
62	#14579083	—	2	4	no	—
63	#14579082	—	2	4	no	—
64	#14579081	—	2	4	no	—
65	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
66	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
67	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
68	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
69	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
70	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×#41×C₇₉	—	2	2	no	—
72	C₂×D₈×C₇₉	—	2	2	no	—
73	C₂²×C₈×C₇₉	D₄	1	2	yes	—
74	C₄⋊Q₈×C₇₉	C₂³	1	2	yes	—
75	#127×C₇₉	C₂³	1	2	yes	—
76	#124×C₇₉	—	2	2	no	—
77	#121×C₇₉	—	2	2	no	—
78	#121×C₇₉	—	2	2	no	—
79	C₄×D₄×C₇₉	—	2	2	no	—
80	C₄×D₄×C₇₉	—	2	2	no	—
81	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
82	#107×C₇₉	—	2	2	no	—
83	#102×C₇₉	—	2	4	no	—
84	C₄×C₈×C₇₉	D₄	1	2	yes	—
85	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
87	#11929624	—	4	8	no	—
88	C₂×Dic₁₅₈	—	2	2	no	—
89	C₂×D₃₁₆	—	2	2	no	—
90	C₂×D₃₁₆	—	2	2	no	—
91	C₂×D₃₁₆	—	2	2	no	—
92	C₂×C₂²⋊D₇₉	—	4	2	no	—
93	C₂×C₄×D₇₉	—	4	2	no	—
94	#11929614	—	4	8	no	—
95	#11929612	—	4	8	no	—
96	#11929611	—	4	8	no	—
97	D₆₃₂	—	4	4	no	—
98	#11929609	—	4	8	no	—
99	#11929609	—	4	8	no	—
100	C₂×C₇₉⋊C₈	—	2	4	no	—
101	C₂×C₇₉⋊C₈	D₈	1	8	yes	complemented
102	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
103	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
104	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
105	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
106	C₄⋊Dic₇₉	—	2	2	no	—
107	C₄⋊Dic₇₉	—	2	2	no	—
108	(C₂×D₇₉)⋊C₄	—	4	2	no	—
109	(C₂×D₇₉)⋊C₄	—	4	2	no	—
110	#41×C₇₉	—	4	8	no	—
111	C₂×Q₈×C₇₉	—	2	2	no	—
112	C₂×D₄×C₇₉	—	2	2	no	—
113	C₂×D₄×C₇₉	—	2	2	no	—
114	C₂×D₄×C₇₉	—	2	2	no	—
115	C₂×D₄×C₇₉	—	4	2	no	—
116	C₂²×C₄×C₇₉	—	4	2	no	—
117	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
120	D₈×C₇₉	—	4	4	no	—
121	C₂×C₈×C₇₉	—	2	4	no	—
122	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
123	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
124	C₄⋊C₄×C₇₉	—	2	2	no	—
125	C₄⋊C₄×C₇₉	—	2	2	no	—
126	C₂²⋊C₄×C₇₉	—	4	2	no	—
127	C₂²⋊C₄×C₇₉	—	4	2	no	—
128	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
132	C₂³×C₇₉	—	4	2	no	—
133	C₂³×C₇₉	#127	1	1	yes	characteristic
134	C₂²×D₇₉	—	4	2	no	—
135	Q₈×C₇₉	—	4	4	no	—
136	D₄×C₇₉	—	4	4	no	—
137	D₄×C₇₉	—	4	4	no	—
138	D₄×C₇₉	—	4	4	no	—
139	D₄×C₇₉	—	8	4	no	—
140	C₂×C₄×C₇₉	—	2	2	no	—
141	C₂×C₄×C₇₉	—	2	2	no	—
142	C₂×C₄×C₇₉	—	4	2	no	—
143	C₂×C₄×C₇₉	—	8	4	no	—
144	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
145	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
146	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
147	C₂×C₄×C₇₉	C₂×D₈	1	4	yes	—
148	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
151	C₂²⋊D₇₉	—	8	4	no	—
152	C₂×Dic₇₉	—	4	2	no	—
153	D₃₁₆	—	4	4	no	—
154	D₃₁₆	—	4	4	no	—
155	D₃₁₆	—	4	4	no	—
156	C₄×D₇₉	—	8	4	no	—
157	Dic₁₅₈	—	4	4	no	—
158	C₈×C₇₉	—	2	2	no	—
159	C₈×C₇₉	#135	1	4	yes	complemented
160	C₇₉⋊C₈	—	2	2	no	—
161	C₇₉⋊C₈	C₂×D₈	1	4	yes	—
162	C₇₉⋊C₈	C₂×D₈	1	4	yes	—
163	C₇₉⋊C₈	#135	1	4	yes	complemented
164	C₂²×C₇₉	—	2	1	no	—
165	C₂²×C₇₉	—	2	1	no	—
166	C₂²×C₇₉	—	8	2	no	—
167	C₂²×C₇₉	—	8	2	no	—
168	C₂²×C₇₉	—	8	2	no	—
169	C₂²×C₇₉	#493	1	2	yes	—
170	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
171	C₂×D₇₉	—	8	2	no	—
172	C₂×D₇₉	—	8	2	no	—
173	C₂×D₇₉	—	8	2	no	—
174	C₄×C₇₉	—	2	1	no	—
175	C₄×C₇₉	—	2	1	no	—
176	C₄×C₇₉	—	8	2	no	—
177	C₄×C₇₉	#493	1	2	yes	—
178	C₄×C₇₉	C₂²×D₈	1	2	yes	—
179	C₄×C₇₉	C₂×#135	1	2	yes	—
180	C₄×C₇₉	C₂×#135	1	2	yes	—
181	Dic₇₉	—	8	2	no	—
182	#19603	—	79	1	no	maximal, Sylow
183	C₂×C₇₉	—	2	2	no	—
184	C₂×C₇₉	—	16	2	no	—
185	C₂×C₇₉	C₂×#493	1	1	yes	characteristic
186	C₂×C₇₉	#3147	1	1	yes	characteristic
187	C₂×C₇₉	#3147	1	1	yes	characteristic
188	D₇₉	—	16	2	no	—
189	#3169	—	79	1	no	—
190	#3169	—	79	2	no	—
191	#3169	D₇₉	1	1	yes	characteristic, complemented
192	#3147	—	79	2	no	—
193	#1713	—	79	1	no	—
194	#1712	—	79	1	no	—
195	#1709	—	79	2	no	—
196	#1706	—	79	1	no	—
197	#1706	—	79	1	no	—
198	#1706	—	79	2	no	—
199	C₂×C₈²	—	79	1	no	—
200	C₇₉	#19603	1	1	yes	characteristic, complemented, Sylow
201	C₂×#135	—	158	4	no	—
202	#532	—	79	2	no	—
203	#532	C₂×D₇₉	1	2	yes	—
204	#500	—	79	2	no	—
205	#500	—	79	1	no	—
206	#500	C₂×D₇₉	1	1	yes	characteristic
207	#495	—	158	4	no	—
208	#494	—	79	2	no	—
209	#493	—	79	2	no	—
210	#493	—	79	2	no	—
211	#493	—	79	1	no	—
212	#493	C₂×D₇₉	1	1	yes	characteristic
213	#492	—	79	2	no	—
214	#492	—	79	2	no	—
215	#486	—	79	2	no	—
216	#486	—	79	2	no	—
217	#486	—	79	2	no	—
218	#486	C₂×D₇₉	1	2	yes	—
219	#466	—	2	2	no	—
220	#466	—	158	4	no	—
221	#466	—	158	2	no	—
222	C₄×D₈	—	2	2	no	—
223	C₄×D₈	—	158	4	no	—
224	C₄×D₈	—	158	2	no	—
225	C₂×C₄×C₈	—	79	1	no	—
226	C₂×C₄×C₈	—	79	1	no	—
227	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
228	C₈²	—	79	1	no	—
229	C₈²	—	79	1	no	—
230	C₈²	—	79	1	no	—
231	C₈²	—	79	1	no	—
232	C₂×#41	—	2	2	no	—
233	C₂×#41	—	158	2	no	—
234	#135	—	316	16	no	—
235	C₂×Q₁₆	—	158	4	no	—
236	C₂×SD₁₆	—	158	4	no	—
237	C₂×SD₁₆	—	158	4	no	—
238	C₂×D₈	—	2	2	no	—
239	C₂×D₈	—	158	4	no	—
240	C₂×D₈	—	158	4	no	—
241	C₂×D₈	—	158	2	no	—
242	C₂²×C₈	—	79	2	no	—
243	C₂²×C₈	—	79	2	no	—
244	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
245	C₄⋊Q₈	—	79	2	no	—
246	C₄⋊Q₈	C₂²×D₇₉	1	2	yes	—
247	#127	—	79	2	no	—
248	#127	C₂²×D₇₉	1	2	yes	—
249	#124	—	2	2	no	—
250	#124	—	158	2	no	—
251	#121	—	2	2	no	—
252	#121	—	2	2	no	—
253	#121	—	158	2	no	—
254	#121	—	158	2	no	—
255	C₄×D₄	—	2	2	no	—
256	C₄×D₄	—	2	2	no	—
257	C₄×D₄	—	158	2	no	—
258	C₄×D₄	—	158	2	no	—
259	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
260	#107	—	2	2	no	—
261	#107	—	158	4	no	—
262	#107	—	158	2	no	—
263	#102	—	2	4	no	—
264	#102	—	158	4	no	—
265	#102	—	158	4	no	—
266	#102	—	158	4	no	—
267	C₄×C₈	—	79	2	no	—
268	C₄×C₈	—	79	2	no	—
269	C₄×C₈	—	79	1	no	—
270	C₄×C₈	—	79	1	no	—
271	C₄×C₈	—	79	1	no	—
272	C₄×C₈	—	79	1	no	—
273	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
274	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
276	#41	—	4	8	no	—
277	#41	—	316	8	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	—	158	2	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	4	2	no	—
284	C₂×D₄	—	316	2	no	—
285	C₂×D₄	—	158	2	no	—
286	C₂×D₄	—	158	2	no	—
287	C₂×D₄	—	158	2	no	—
288	C₂²×C₄	—	4	2	no	—
289	C₂²×C₄	—	316	2	no	—
290	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
291	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
292	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
293	Q₁₆	—	316	8	no	—
294	SD₁₆	—	316	8	no	—
295	SD₁₆	—	316	8	no	—
296	D₈	—	4	4	no	—
297	D₈	—	316	8	no	—
298	D₈	—	316	8	no	—
299	D₈	—	316	4	no	—
300	C₂×C₈	—	2	4	no	—
301	C₂×C₈	—	79	8	no	—
302	C₂×C₈	—	79	2	no	—
303	C₂×C₈	—	79	2	no	—
304	C₂×C₈	—	79	2	no	—
305	C₂×C₈	—	79	2	no	—
306	C₂×C₈	—	158	4	no	—
307	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
308	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	158	2	no	—
312	C₄⋊C₄	—	158	2	no	—
313	C₂²⋊C₄	—	4	2	no	—
314	C₂²⋊C₄	—	4	2	no	—
315	C₂²⋊C₄	—	316	2	no	—
316	C₂²⋊C₄	—	316	2	no	—
317	C₄²	D₄×D₇₉	1	1	yes	characteristic
318	C₄²	D₄×D₇₉	1	1	yes	characteristic
319	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
320	C₄²	C₂³×D₇₉	1	1	yes	characteristic
321	C₂³	—	4	2	no	—
322	C₂³	—	316	2	no	—
323	C₂³	#14579157	1	1	yes	characteristic
324	Q₈	—	4	4	no	—
325	Q₈	—	316	4	no	—
326	D₄	—	4	4	no	—
327	D₄	—	4	4	no	—
328	D₄	—	4	4	no	—
329	D₄	—	8	4	no	—
330	D₄	—	316	4	no	—
331	D₄	—	316	4	no	—
332	D₄	—	316	4	no	—
333	D₄	—	632	4	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	8	4	no	—
338	C₂×C₄	—	316	2	no	—
339	C₂×C₄	—	632	4	no	—
340	C₂×C₄	#14579157	1	1	yes	characteristic
341	C₂×C₄	#14579157	1	1	yes	characteristic
342	C₂×C₄	#14579157	1	1	yes	characteristic
343	C₂×C₄	D₈×D₇₉	1	4	yes	—
344	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
345	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
346	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
347	C₈	—	2	2	no	—
348	C₈	—	79	4	no	—
349	C₈	—	79	4	no	—
350	C₈	—	79	4	no	—
351	C₈	—	158	2	no	—
352	C₈	#14579181	1	4	yes	complemented
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	632	2	no	—
359	C₂²	—	632	2	no	—
360	C₂²	—	632	2	no	—
361	C₂²	#19098938	1	2	yes	—
362	C₂²	C₂×#14579157	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	8	2	no	—
366	C₄	—	632	2	no	—
367	C₄	#19098938	1	2	yes	—
368	C₄	C₂×D₈×D₇₉	1	2	yes	—
369	C₄	#135×D₇₉	1	2	yes	—
370	C₄	C₂×#14579181	1	2	yes	—
371	C₂	—	2	2	no	—
372	C₂	—	16	2	no	—
373	C₂	—	1264	2	no	—
374	C₂	C₂×#19098938	1	1	yes	characteristic
375	C₂	#23773824	1	1	yes	characteristic
376	C₂	#23773826	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579157
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#3169	C₂
socle	C₂²×C₇₉	C₂×#127
hypercenter	#3169	D₇₉
hypocenter	C₇₉	#19603
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #23773979	C₂	semidirect
2. #23773975	C₂	semidirect
3. #23773826	C₂	semidirect
4. #23763017	C₂	semidirect
5. #23763014	C₂	semidirect
6. #23763004	C₂	semidirect
7. #23762993	C₂	semidirect
8. #23762991	C₂	semidirect
9. #23762989	C₂	semidirect
10. C₂×C₈×C₇₉⋊C₈	C₂	semidirect
11. C₇₉×#3169	C₂	semidirect
101. C₂×C₇₉⋊C₈	D₈	semidirect
159. C₈×C₇₉	#135	semidirect
163. C₇₉⋊C₈	#135	semidirect
191. #3169	D₇₉	semidirect
200. C₇₉	#19603	semidirect
352. C₈	#14579181	semidirect

Maximal subgroups

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

subgroup	index
#23773979	2
#23773975	2
#23773826	2
#23763017	2
#23763014	2
#23763004	2
#23762993	2
#23762991	2
#23762989	2
C₂×C₈×C₇₉⋊C₈	2
C₇₉×#3169	2
#19603	79