Subgroups — #34279851

Order 20224.

602 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	#23776390	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₆₃₂	C₂	1	1	yes	maximal, characteristic
3	#23775680	C₂	1	2	yes	maximal, complemented
4	#23762994	C₂	1	1	yes	maximal, characteristic, complemented
5	#23762950	C₂	1	2	yes	maximal, complemented
6	#23762795	C₂	1	1	yes	maximal, characteristic, complemented
7	#23762671	C₂	1	2	yes	maximal, complemented
8	#23762662	C₂	1	2	yes	maximal, complemented
9	#23762284	C₂	1	1	yes	maximal, characteristic
10	C₈×#11929614	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#1670	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099287	C₂²	1	2	yes	—
13	Q₈×D₃₁₆	C₂²	1	2	yes	—
14	#19098968	—	2	1	no	—
15	#19098968	C₂²	1	1	yes	characteristic
16	#19098959	C₂²	1	2	yes	—
17	#19098948	—	2	1	no	—
18	#19098915	C₂²	1	2	yes	—
19	#19098907	—	2	2	no	—
20	#19098844	—	2	2	no	—
21	#19098844	C₂²	1	2	yes	—
22	#19098822	—	2	2	no	—
23	#19098787	—	2	2	no	—
24	#19098774	C₂²	1	2	yes	—
25	#19098765	C₂²	1	2	yes	—
26	#19098720	—	2	2	no	—
27	C₈×Dic₁₅₈	C₂²	1	1	yes	characteristic
28	Q₁₆×Dic₇₉	—	2	1	no	—
29	C₄×#11929614	C₂²	1	1	yes	characteristic
30	C₄×D₆₃₂	—	2	1	no	—
31	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
32	#19098640	C₂²	1	2	yes	—
33	#19098570	—	2	2	no	—
34	#19098315	C₂²	1	1	yes	characteristic
35	#19098310	C₂²	1	2	yes	—
36	Q₈⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
37	#19098293	C₂²	1	1	yes	characteristic
38	#19098290	C₂²	1	2	yes	—
39	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
40	#494×C₇₉	C₂²	1	1	yes	characteristic
41	#475×C₇₉	C₂²	1	2	yes	—
42	C₈×Q₈×C₇₉	C₂²	1	1	yes	characteristic
43	#335×C₇₉	C₂²	1	1	yes	characteristic
44	#332×C₇₉	C₂²	1	2	yes	—
45	C₂×Q₈×D₇₉	—	2	2	no	—
46	C₂×Dic₃₁₆	—	2	2	no	—
47	C₂×#11929614	D₄	1	1	yes	characteristic
48	C₂×#11929611	—	2	2	no	—
49	C₂×D₆₃₂	D₄	1	1	yes	characteristic
50	C₂×#11929607	—	2	2	no	—
51	#14579161	—	2	2	no	—
52	#14579161	C₂³	1	2	yes	—
53	#14579156	—	4	2	no	—
54	#14579150	—	2	2	no	—
55	#14579141	—	2	2	no	—
56	#14579141	—	2	2	no	—
57	#14579141	—	4	2	no	—
58	Q₈×Dic₇₉	—	2	2	no	—
59	C₄×Dic₁₅₈	C₂³	1	1	yes	characteristic
60	C₄×D₃₁₆	—	2	2	no	—
61	C₄×D₃₁₆	C₂³	1	2	yes	—
62	#14579119	—	2	2	no	—
63	#14579095	—	2	1	no	—
64	#14579095	—	2	1	no	—
65	#14579095	—	2	2	no	—
66	#14579095	D₄	1	1	yes	characteristic
67	#14579094	D₄	1	1	yes	characteristic
68	#14579092	—	4	2	no	—
69	C₄⋊(C₇₉⋊C₈)	C₂³	1	1	yes	characteristic
70	Dic₇₉⋊C₈	—	2	1	no	—
71	#14579086	—	2	2	no	—
72	#14579085	—	2	2	no	—
73	#14579085	D₄	1	1	yes	characteristic
74	#14579084	—	2	2	no	—
75	#14579084	—	2	2	no	—
76	#14579084	D₄	1	1	yes	characteristic
77	#14579083	—	2	2	no	—
78	#14579083	D₄	1	2	yes	—
79	(C₂×D₇₉)⋊C₈	—	2	2	no	—
80	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
81	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
82	C₈×Dic₇₉	—	2	1	no	—
83	C₂×Q₁₆×C₇₉	—	2	1	no	—
84	C₂×Q₁₆×C₇₉	—	2	1	no	—
85	C₄⋊Q₈×C₇₉	C₂³	1	2	yes	—
86	C₄×Q₈×C₇₉	C₂³	1	1	yes	characteristic
87	#106×C₇₉	—	2	2	no	—
88	C₄⋊C₈×C₇₉	—	2	1	no	—
89	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
90	C₄⋊C₈×C₇₉	C₂³	1	1	yes	characteristic
91	#103×C₇₉	—	2	2	no	—
92	C₄×C₈×C₇₉	—	2	1	no	—
93	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₅₈	—	2	2	no	—
95	C₂×Dic₁₅₈	—	4	2	no	—
96	C₂×Dic₁₅₈	C₂×D₄	1	1	yes	characteristic
97	Q₈×D₇₉	—	4	4	no	—
98	Q₈×D₇₉	—	8	2	no	—
99	C₂×D₃₁₆	C₂×D₄	1	2	yes	—
100	C₂×C₄×D₇₉	—	2	2	no	—
101	C₂×C₄×D₇₉	—	4	2	no	—
102	Dic₃₁₆	—	8	2	no	—
103	#11929614	—	2	2	no	—
104	#11929611	—	8	2	no	—
105	D₆₃₂	—	2	2	no	—
106	#11929607	—	8	2	no	—
107	C₂×C₇₉⋊C₈	—	2	1	no	—
108	C₂×C₇₉⋊C₈	—	2	1	no	—
109	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
111	C₄⋊Dic₇₉	—	2	2	no	—
112	C₄⋊Dic₇₉	—	2	2	no	—
113	C₄⋊Dic₇₉	—	4	2	no	—
114	C₄⋊Dic₇₉	—	4	2	no	—
115	C₄⋊Dic₇₉	C₂×D₄	1	2	yes	—
116	C₄⋊Dic₇₉	C₂×D₄	1	1	yes	characteristic
117	Dic₇₉⋊C₄	—	2	1	no	—
118	(C₂×D₇₉)⋊C₄	—	2	2	no	—
119	(C₂×D₇₉)⋊C₄	—	4	2	no	—
120	(C₂×D₇₉)⋊C₄	—	4	2	no	—
121	(C₂×D₇₉)⋊C₄	—	4	2	no	—
122	C₄×Dic₇₉	—	2	1	no	—
123	C₄×Dic₇₉	—	4	2	no	—
124	C₂×Q₈×C₇₉	—	2	2	no	—
125	C₂×Q₈×C₇₉	C₂×D₄	1	1	yes	characteristic
126	Q₁₆×C₇₉	—	4	2	no	—
127	Q₁₆×C₇₉	—	8	1	no	—
128	C₂×C₈×C₇₉	—	2	1	no	—
129	C₂×C₈×C₇₉	—	2	2	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	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₇₉	—	2	1	no	—
137	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
138	C₂²×D₇₉	—	2	2	no	—
139	Q₈×C₇₉	—	2	1	no	—
140	Q₈×C₇₉	—	4	2	no	—
141	Q₈×C₇₉	—	4	2	no	—
142	Q₈×C₇₉	#135	1	2	yes	—
143	C₂×C₄×C₇₉	—	2	1	no	—
144	C₂×C₄×C₇₉	—	2	1	no	—
145	C₂×C₄×C₇₉	—	4	2	no	—
146	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂×Dic₇₉	—	2	1	no	—
150	C₂×Dic₇₉	—	2	1	no	—
151	C₂×Dic₇₉	—	2	2	no	—
152	C₂×Dic₇₉	—	4	2	no	—
153	D₃₁₆	—	2	2	no	—
154	D₃₁₆	—	2	2	no	—
155	C₄×D₇₉	—	4	2	no	—
156	C₄×D₇₉	—	4	2	no	—
157	C₄×D₇₉	—	8	4	no	—
158	Dic₁₅₈	—	2	1	no	—
159	Dic₁₅₈	—	2	1	no	—
160	Dic₁₅₈	—	4	2	no	—
161	Dic₁₅₈	—	4	2	no	—
162	Dic₁₅₈	—	8	4	no	—
163	C₈×C₇₉	—	2	1	no	—
164	C₈×C₇₉	—	4	2	no	—
165	C₈×C₇₉	—	4	1	no	—
166	C₈×C₇₉	#137	1	2	yes	—
167	C₇₉⋊C₈	—	2	2	no	—
168	C₇₉⋊C₈	—	2	1	no	—
169	C₇₉⋊C₈	—	2	1	no	—
170	C₇₉⋊C₈	—	4	1	no	—
171	C₂²×C₇₉	D₄²	1	1	yes	characteristic
172	C₂×D₇₉	—	4	2	no	—
173	C₂×D₇₉	—	4	2	no	—
174	C₂×D₇₉	—	4	2	no	—
175	C₄×C₇₉	—	2	2	no	—
176	C₄×C₇₉	—	2	1	no	—
177	C₄×C₇₉	—	4	1	no	—
178	C₄×C₇₉	—	8	2	no	—
179	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
180	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
181	C₄×C₇₉	#576	1	1	yes	characteristic
182	C₄×C₇₉	#577	1	1	yes	characteristic
183	Dic₇₉	—	4	1	no	—
184	Dic₇₉	—	4	2	no	—
185	Dic₇₉	—	4	1	no	—
186	Dic₇₉	—	8	2	no	—
187	#19907	—	79	1	no	maximal, Sylow
188	C₂×C₇₉	#3278	1	1	yes	characteristic
189	C₂×C₇₉	#3280	1	1	yes	characteristic
190	C₂×C₇₉	#3293	1	1	yes	characteristic
191	D₇₉	—	8	2	no	—
192	#3393	—	79	1	no	—
193	Q₈×D₈	—	79	1	no	—
194	#3328	—	79	2	no	—
195	#1707	—	79	1	no	—
196	#1699	—	79	2	no	—
197	#1674	—	79	1	no	—
198	#1670	D₇₉	1	1	yes	characteristic, complemented
199	#1652	—	79	2	no	—
200	#1651	—	79	2	no	—
201	#1589	—	79	1	no	—
202	C₈×Q₁₆	—	79	1	no	—
203	C₇₉	#19907	1	1	yes	characteristic, complemented, Sylow
204	#556	—	79	2	no	—
205	D₄×Q₈	—	79	2	no	—
206	#500	—	79	1	no	—
207	#500	—	158	1	no	—
208	#498	—	79	2	no	—
209	#495	—	158	1	no	—
210	#494	C₂×D₇₉	1	1	yes	characteristic
211	#487	—	79	2	no	—
212	#485	—	158	2	no	—
213	#475	C₂×D₇₉	1	2	yes	—
214	#474	—	79	2	no	—
215	#474	—	158	2	no	—
216	#470	—	158	2	no	—
217	#464	—	158	2	no	—
218	#462	—	79	2	no	—
219	#461	—	79	2	no	—
220	#451	—	158	2	no	—
221	C₈×Q₈	—	79	1	no	—
222	C₈×Q₈	C₂×D₇₉	1	1	yes	characteristic
223	C₄×Q₁₆	—	79	1	no	—
224	C₄×Q₁₆	—	158	1	no	—
225	C₄×D₈	—	79	1	no	—
226	C₄×D₈	—	158	1	no	—
227	#436	—	79	2	no	—
228	#419	—	158	2	no	—
229	#335	—	79	1	no	—
230	#335	C₂×D₇₉	1	1	yes	characteristic
231	#332	—	79	2	no	—
232	#332	C₂×D₇₉	1	2	yes	—
233	Q₈⋊C₈	—	79	1	no	—
234	Q₈⋊C₈	—	79	1	no	—
235	D₄⋊C₈	—	79	2	no	—
236	C₈²	—	79	1	no	—
237	C₂²×Q₈	—	158	2	no	—
238	C₂×Q₁₆	—	2	1	no	—
239	C₂×Q₁₆	—	2	1	no	—
240	C₂×Q₁₆	—	79	1	no	—
241	C₂×Q₁₆	—	158	2	no	—
242	C₂×SD₁₆	—	158	2	no	—
243	C₂×D₈	—	79	1	no	—
244	C₂×M₁₆	—	158	2	no	—
245	C₄⋊Q₈	—	79	2	no	—
246	C₄⋊Q₈	—	158	2	no	—
247	C₄⋊Q₈	C₂²×D₇₉	1	2	yes	—
248	#126	—	316	2	no	—
249	#124	—	158	2	no	—
250	C₂²⋊Q₈	—	316	2	no	—
251	C₂²⋊Q₈	—	158	2	no	—
252	C₂²⋊Q₈	—	158	2	no	—
253	C₄×Q₈	—	79	1	no	—
254	C₄×Q₈	—	158	2	no	—
255	C₄×Q₈	C₂²×D₇₉	1	1	yes	characteristic
256	C₄×D₄	—	79	2	no	—
257	C₄×D₄	—	158	2	no	—
258	#117	—	158	2	no	—
259	#107	—	79	1	no	—
260	#107	—	79	1	no	—
261	#107	—	158	2	no	—
262	#107	—	158	1	no	—
263	#107	—	158	1	no	—
264	#106	—	2	2	no	—
265	#106	—	316	2	no	—
266	C₄⋊C₈	—	2	1	no	—
267	C₄⋊C₈	—	79	1	no	—
268	C₄⋊C₈	—	158	1	no	—
269	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
270	C₄⋊C₈	C₂²×D₇₉	1	1	yes	characteristic
271	#103	—	2	2	no	—
272	#103	—	79	1	no	—
273	#103	—	79	1	no	—
274	#103	—	158	2	no	—
275	#103	—	158	2	no	—
276	#103	—	158	2	no	—
277	#103	—	158	2	no	—
278	#102	—	79	2	no	—
279	#102	—	158	2	no	—
280	C₂²⋊C₈	—	158	2	no	—
281	C₄×C₈	—	2	1	no	—
282	C₄×C₈	—	79	1	no	—
283	C₄×C₈	—	79	1	no	—
284	C₄×C₈	—	158	1	no	—
285	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
286	C₂×Q₈	—	2	2	no	—
287	C₂×Q₈	—	316	2	no	—
288	C₂×Q₈	—	316	4	no	—
289	C₂×Q₈	—	79	1	no	—
290	C₂×Q₈	—	632	2	no	—
291	C₂×Q₈	—	158	2	no	—
292	C₂×Q₈	D₄×D₇₉	1	1	yes	characteristic
293	C₂×D₄	—	79	2	no	—
294	C₂²×C₄	—	316	2	no	—
295	C₂²×C₄	—	158	2	no	—
296	Q₁₆	—	4	2	no	—
297	Q₁₆	—	8	1	no	—
298	Q₁₆	—	632	2	no	—
299	Q₁₆	—	158	2	no	—
300	SD₁₆	—	632	2	no	—
301	D₈	—	158	2	no	—
302	M₁₆	—	632	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	79	1	no	—
306	C₂×C₈	—	79	1	no	—
307	C₂×C₈	—	158	1	no	—
308	C₂×C₈	—	158	1	no	—
309	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
310	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	316	2	no	—
315	C₄⋊C₄	—	316	2	no	—
316	C₄⋊C₄	—	79	2	no	—
317	C₄⋊C₄	—	79	1	no	—
318	C₄⋊C₄	—	158	2	no	—
319	C₄⋊C₄	—	158	2	no	—
320	C₄⋊C₄	—	158	1	no	—
321	C₄⋊C₄	D₄×D₇₉	1	1	yes	characteristic
322	C₂²⋊C₄	—	316	2	no	—
323	C₂²⋊C₄	—	316	2	no	—
324	C₂²⋊C₄	—	316	2	no	—
325	C₂²⋊C₄	—	158	2	no	—
326	C₄²	—	2	1	no	—
327	C₄²	—	316	2	no	—
328	C₄²	—	158	1	no	—
329	C₄²	C₂³×D₇₉	1	1	yes	characteristic
330	C₂³	—	158	2	no	—
331	Q₈	—	2	1	no	—
332	Q₈	—	4	2	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	316	2	no	—
335	Q₈	—	316	2	no	—
336	Q₈	—	632	4	no	—
337	Q₈	—	158	1	no	—
338	Q₈	—	158	1	no	—
339	Q₈	#14579184	1	2	yes	—
340	D₄	—	158	2	no	—
341	D₄	—	158	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	316	2	no	—
346	C₂×C₄	—	316	2	no	—
347	C₂×C₄	—	316	2	no	—
348	C₂×C₄	—	632	4	no	—
349	C₂×C₄	—	158	1	no	—
350	C₂×C₄	—	158	2	no	—
351	C₂×C₄	—	158	1	no	—
352	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
354	C₂×C₄	#14579207	1	1	yes	characteristic
355	C₈	—	2	1	no	—
356	C₈	—	4	2	no	—
357	C₈	—	4	1	no	—
358	C₈	—	316	1	no	—
359	C₈	—	158	2	no	—
360	C₈	—	158	1	no	—
361	C₈	—	158	1	no	—
362	C₈	#14579190	1	2	yes	—
363	C₂²	—	316	2	no	—
364	C₂²	—	316	2	no	—
365	C₂²	—	316	2	no	—
366	C₂²	#19099197	1	1	yes	characteristic
367	C₄	—	2	2	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	1	no	—
370	C₄	—	8	2	no	—
371	C₄	—	316	2	no	—
372	C₄	—	316	1	no	—
373	C₄	—	316	1	no	—
374	C₄	—	632	2	no	—
375	C₄	C₂×#14579184	1	1	yes	characteristic
376	C₄	C₂×#14579190	1	1	yes	characteristic
377	C₄	#19099365	1	1	yes	characteristic
378	C₄	#19099373	1	1	yes	characteristic
379	C₂	—	632	2	no	—
380	C₂	#23775069	1	1	yes	characteristic
381	C₂	#23775082	1	1	yes	characteristic
382	C₂	#23775203	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19099197
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1670	C₂
socle	C₂²×C₇₉	D₄²
prime radical	C₂×D₆₃₂	D₄
hypercenter	#1670	D₇₉
hypocenter	C₇₉	#19907
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1670

Complements

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

N	H ≅ G/N	type
3. #23775680	C₂	semidirect
4. #23762994	C₂	semidirect
5. #23762950	C₂	semidirect
6. #23762795	C₂	semidirect
7. #23762671	C₂	semidirect
8. #23762662	C₂	semidirect
10. C₈×#11929614	C₂	semidirect
11. C₇₉×#1670	C₂	semidirect
198. #1670	D₇₉	semidirect
203. C₇₉	#19907	semidirect

Maximal subgroups

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

subgroup	index
#23776390	2
Q₈×D₆₃₂	2
#23775680	2
#23762994	2
#23762950	2
#23762795	2
#23762671	2
#23762662	2
#23762284	2
C₈×#11929614	2
C₇₉×#1670	2
#19907	79