Subgroups — #34445141

Order 20224.

580 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23775747	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765629	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765599	C₂	1	2	yes	maximal, complemented
5	#23765497	C₂	1	2	yes	maximal, complemented
6	#23765446	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765318	C₂	1	2	yes	maximal, complemented
8	C₄×#14579104	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765219	C₂	1	1	yes	maximal, characteristic, complemented
10	#358×D₇₉	C₂	1	2	yes	maximal, complemented
11	C₇₉×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₉	C₂²	1	2	yes	—
13	Q₈×D₃₁₆	—	2	1	no	—
14	#19099216	—	2	1	no	—
15	C₂×#14579106	C₂²	1	2	yes	—
16	C₂×#14579104	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₇₉	C₂²	1	2	yes	—
18	#19098943	C₂²	1	1	yes	characteristic
19	#19098878	—	2	2	no	—
20	#19098840	—	2	1	no	—
21	#19098805	C₂²	1	2	yes	—
22	#19098800	C₂²	1	2	yes	—
23	#19098773	—	2	1	no	—
24	#19098720	—	2	2	no	—
25	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
26	C₈×D₃₁₆	C₂²	1	1	yes	characteristic
27	#107×D₇₉	C₂²	1	2	yes	—
28	#19098390	C₂²	1	1	yes	characteristic
29	#19098368	C₂²	1	2	yes	—
30	#19098367	C₂²	1	1	yes	characteristic
31	#19098366	C₂²	1	2	yes	—
32	#19098366	C₂²	1	2	yes	—
33	#19098363	C₂²	1	1	yes	characteristic
34	(C₂×D₇₉)⋊C₁₆	C₂²	1	2	yes	—
35	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
36	#500×C₇₉	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
38	#367×C₇₉	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
40	#358×C₇₉	C₂²	1	2	yes	—
41	C₂×#11929625	—	4	1	no	—
42	C₂×Q₈×D₇₉	—	2	2	no	—
43	C₂×Dic₃₁₆	—	2	1	no	—
44	C₂×Dic₃₁₆	C₂³	1	2	yes	—
45	C₂×#11929614	—	2	2	no	—
46	Q₁₆×D₇₉	—	2	4	no	—
47	Q₁₆×D₇₉	—	4	2	no	—
48	C₂×D₆₃₂	C₂³	1	1	yes	characteristic
49	C₂×C₈×D₇₉	C₂³	1	2	yes	—
50	#14579161	—	2	1	no	—
51	#14579161	—	4	1	no	—
52	#14579146	—	4	2	no	—
53	#14579141	—	2	2	no	—
54	#14579141	—	4	2	no	—
55	#14579137	—	4	1	no	—
56	#14579136	—	2	2	no	—
57	C₄×D₃₁₆	—	2	1	no	—
58	C₄×D₃₁₆	—	4	1	no	—
59	C₄×D₃₁₆	D₄	1	1	yes	characteristic
60	C₄⋊C₄×D₇₉	—	2	2	no	—
61	#14579106	—	4	2	no	—
62	#14579104	—	2	2	no	—
63	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
65	C₁₆×D₇₉	—	4	2	no	—
66	#14579095	C₂³	1	1	yes	characteristic
67	#14579094	—	2	2	no	—
68	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
69	#14579084	—	2	2	no	—
70	#14579083	—	2	1	no	—
71	#14579081	—	2	2	no	—
72	(C₂×D₇₉)⋊C₈	D₄	1	2	yes	—
73	C₂×Q₁₆×C₇₉	C₂³	1	1	yes	characteristic
74	C₄⋊Q₈×C₇₉	—	2	1	no	—
75	C₄×Q₈×C₇₉	—	2	1	no	—
76	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
77	#107×C₇₉	—	2	1	no	—
78	#107×C₇₉	C₂³	1	1	yes	characteristic
79	#107×C₇₉	C₂³	1	2	yes	—
80	#103×C₇₉	—	2	1	no	—
81	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
82	#11929625	—	8	4	no	—
83	C₂×Dic₁₅₈	—	2	2	no	—
84	C₂×Dic₁₅₈	—	4	2	no	—
85	Q₈×D₇₉	—	4	4	no	—
86	Q₈×D₇₉	—	8	2	no	—
87	C₂×D₃₁₆	—	2	1	no	—
88	C₂×D₃₁₆	—	4	2	no	—
89	C₂×D₃₁₆	D₈	1	1	yes	characteristic
90	C₂×C₄×D₇₉	—	4	2	no	—
91	C₂×C₄×D₇₉	—	4	2	no	—
92	C₂×C₄×D₇₉	—	4	1	no	—
93	C₂×C₄×D₇₉	C₂×D₄	1	2	yes	—
94	Dic₃₁₆	—	2	2	no	—
95	Dic₃₁₆	—	2	2	no	—
96	Dic₃₁₆	—	8	1	no	—
97	#11929614	—	4	4	no	—
98	D₆₃₂	—	2	1	no	—
99	D₆₃₂	—	2	1	no	—
100	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
101	C₈×D₇₉	—	2	2	no	—
102	C₈×D₇₉	—	2	2	no	—
103	C₄⋊Dic₇₉	—	2	1	no	—
104	C₄⋊Dic₇₉	—	4	2	no	—
105	C₄⋊Dic₇₉	—	4	1	no	—
106	C₄⋊Dic₇₉	—	4	1	no	—
107	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
108	Dic₇₉⋊C₄	—	4	2	no	—
109	(C₂×D₇₉)⋊C₄	—	4	2	no	—
110	(C₂×D₇₉)⋊C₄	—	4	2	no	—
111	(C₂×D₇₉)⋊C₄	—	4	1	no	—
112	(C₂×D₇₉)⋊C₄	D₈	1	2	yes	—
113	C₇₉⋊C₁₆	—	2	1	no	—
114	C₇₉⋊C₁₆	—	2	1	no	—
115	C₂×Q₈×C₇₉	—	2	1	no	—
116	C₂×Q₈×C₇₉	—	4	1	no	—
117	Q₁₆×C₇₉	—	2	1	no	—
118	Q₁₆×C₇₉	#41	1	2	yes	complemented
119	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
121	C₄⋊C₄×C₇₉	—	2	1	no	—
122	C₄⋊C₄×C₇₉	—	2	2	no	—
123	C₄⋊C₄×C₇₉	—	4	1	no	—
124	C₄⋊C₄×C₇₉	—	4	1	no	—
125	C₄²×C₇₉	—	4	1	no	—
126	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
127	C₁₆×C₇₉	—	2	2	no	—
128	C₂²×D₇₉	—	4	1	no	—
129	C₂²×D₇₉	C₂×D₈	1	2	yes	—
130	Q₈×C₇₉	—	2	2	no	—
131	Q₈×C₇₉	—	4	1	no	—
132	Q₈×C₇₉	—	8	2	no	—
133	C₂×C₄×C₇₉	—	4	1	no	—
134	C₂×C₄×C₇₉	—	4	1	no	—
135	C₂×C₄×C₇₉	—	4	2	no	—
136	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₇₉	—	4	1	no	—
140	C₂×Dic₇₉	—	4	2	no	—
141	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
142	D₃₁₆	—	4	1	no	—
143	D₃₁₆	—	4	1	no	—
144	D₃₁₆	—	8	4	no	—
145	D₃₁₆	SD₃₂	1	4	yes	complemented
146	C₄×D₇₉	—	2	2	no	—
147	C₄×D₇₉	—	2	2	no	—
148	C₄×D₇₉	—	8	4	no	—
149	C₄×D₇₉	—	8	4	no	—
150	C₄×D₇₉	—	8	2	no	—
151	Dic₁₅₈	—	4	2	no	—
152	Dic₁₅₈	—	4	2	no	—
153	Dic₁₅₈	—	8	4	no	—
154	C₈×C₇₉	—	2	1	no	—
155	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
156	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
157	C₇₉⋊C₈	—	2	2	no	—
158	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
159	C₂×D₇₉	—	2	2	no	—
160	C₂×D₇₉	—	2	2	no	—
161	C₂×D₇₉	—	8	1	no	—
162	C₂×D₇₉	—	8	1	no	—
163	C₂×D₇₉	—	8	1	no	—
164	C₂×D₇₉	C₂×SD₃₂	1	4	yes	—
165	C₄×C₇₉	—	2	1	no	—
166	C₄×C₇₉	—	4	2	no	—
167	C₄×C₇₉	—	8	2	no	—
168	C₄×C₇₉	—	8	1	no	—
169	C₄×C₇₉	C₂×SD₃₂	1	2	yes	—
170	C₄×C₇₉	#546	1	1	yes	characteristic
171	C₄×C₇₉	#577	1	1	yes	characteristic
172	Dic₇₉	—	2	2	no	—
173	Dic₇₉	—	8	1	no	—
174	Dic₇₉	—	8	2	no	—
175	#33865	—	79	1	no	maximal, Sylow
176	C₂×C₇₉	#3294	1	1	yes	characteristic
177	C₂×C₇₉	C₂²×SD₃₂	1	1	yes	characteristic
178	C₂×C₇₉	#3417	1	1	yes	characteristic
179	D₇₉	—	2	4	no	—
180	D₇₉	—	16	1	no	—
181	#3334	—	79	1	no	—
182	D₄×Q₁₆	—	79	1	no	—
183	#2241	—	79	1	no	—
184	#2235	—	79	2	no	—
185	#2225	D₇₉	1	1	yes	characteristic, complemented
186	#2217	—	79	2	no	—
187	#2208	—	79	1	no	—
188	#2185	—	79	2	no	—
189	C₄×SD₃₂	—	79	1	no	—
190	D₄×C₁₆	—	79	1	no	—
191	C₂×#358	—	79	2	no	—
192	C₇₉	#33865	1	1	yes	characteristic, complemented, Sylow
193	C₂²×Q₁₆	—	79	2	no	—
194	D₄×Q₈	—	158	1	no	—
195	#547	—	158	1	no	—
196	C₂×SD₃₂	—	79	1	no	—
197	C₂×SD₃₂	—	79	2	no	—
198	C₂²×C₁₆	—	79	2	no	—
199	#500	C₂×D₇₉	1	1	yes	characteristic
200	#494	—	79	1	no	—
201	#480	—	158	2	no	—
202	#474	—	158	1	no	—
203	#467	—	79	2	no	—
204	#466	—	79	2	no	—
205	#462	—	158	1	no	—
206	#451	—	158	2	no	—
207	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
208	C₄×D₈	—	79	1	no	—
209	C₈×D₄	—	79	1	no	—
210	C₂×#107	—	79	2	no	—
211	#367	—	79	1	no	—
212	#367	C₂×D₇₉	1	2	yes	—
213	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
214	#358	—	79	1	no	—
215	#358	—	79	2	no	—
216	#358	—	79	2	no	—
217	#358	—	79	2	no	—
218	#358	C₂×D₇₉	1	2	yes	—
219	#357	—	79	1	no	—
220	C₂²⋊C₁₆	—	79	2	no	—
221	C₄×C₁₆	—	79	1	no	—
222	C₂×#41	—	316	1	no	—
223	C₂²×Q₈	—	158	2	no	—
224	C₂×Q₁₆	—	316	2	no	—
225	C₂×Q₁₆	—	79	2	no	—
226	C₂×Q₁₆	—	158	2	no	—
227	C₂×Q₁₆	—	158	4	no	—
228	C₂×Q₁₆	—	158	1	no	—
229	C₂×Q₁₆	C₂²×D₇₉	1	1	yes	characteristic
230	C₂×D₈	—	79	1	no	—
231	C₂²×C₈	—	79	2	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	316	1	no	—
234	C₄⋊Q₈	—	158	1	no	—
235	#123	—	316	2	no	—
236	C₂²⋊Q₈	—	316	1	no	—
237	C₂²⋊Q₈	—	316	2	no	—
238	C₂²⋊Q₈	—	158	2	no	—
239	#121	—	158	2	no	—
240	C₄×Q₈	—	2	1	no	—
241	C₄×D₄	—	316	1	no	—
242	C₄×D₄	—	79	1	no	—
243	C₄×D₄	—	158	1	no	—
244	C₂×C₄⋊C₄	—	158	2	no	—
245	SD₃₂	—	316	2	no	—
246	SD₃₂	—	158	2	no	—
247	C₂×C₁₆	—	316	2	no	—
248	C₂×C₁₆	—	79	1	no	—
249	C₂×C₁₆	—	79	1	no	—
250	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
251	#107	—	2	1	no	—
252	#107	—	79	1	no	—
253	#107	—	158	2	no	—
254	#107	C₂²×D₇₉	1	1	yes	characteristic
255	#107	C₂²×D₇₉	1	2	yes	—
256	C₄⋊C₈	—	79	1	no	—
257	#103	—	2	1	no	—
258	#103	—	158	2	no	—
259	#102	—	158	2	no	—
260	#102	—	158	1	no	—
261	C₂²⋊C₈	—	79	2	no	—
262	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
263	#41	—	632	4	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	316	2	no	—
267	C₂×Q₈	—	316	4	no	—
268	C₂×Q₈	—	632	2	no	—
269	C₂×Q₈	—	158	2	no	—
270	C₂×D₄	—	316	2	no	—
271	C₂×D₄	—	79	1	no	—
272	C₂×D₄	—	158	1	no	—
273	C₂²×C₄	—	316	2	no	—
274	C₂²×C₄	—	316	2	no	—
275	C₂²×C₄	—	316	1	no	—
276	C₂²×C₄	—	79	2	no	—
277	Q₁₆	—	2	1	no	—
278	Q₁₆	—	316	4	no	—
279	Q₁₆	—	632	1	no	—
280	Q₁₆	—	158	2	no	—
281	Q₁₆	—	158	2	no	—
282	Q₁₆	#11929625	1	2	yes	complemented
283	D₈	—	158	1	no	—
284	D₈	—	158	1	no	—
285	C₂×C₈	—	79	2	no	—
286	C₂×C₈	—	158	2	no	—
287	C₂×C₈	—	158	2	no	—
288	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
289	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	316	2	no	—
295	C₄⋊C₄	—	316	2	no	—
296	C₄⋊C₄	—	316	1	no	—
297	C₄⋊C₄	—	316	1	no	—
298	C₄⋊C₄	—	79	1	no	—
299	C₄⋊C₄	—	158	1	no	—
300	C₂²⋊C₄	—	316	2	no	—
301	C₂²⋊C₄	—	316	2	no	—
302	C₂²⋊C₄	—	316	1	no	—
303	C₂²⋊C₄	—	79	2	no	—
304	C₄²	—	4	1	no	—
305	C₄²	D₄×D₇₉	1	1	yes	characteristic
306	C₁₆	—	2	2	no	—
307	C₁₆	—	158	1	no	—
308	C₁₆	—	158	1	no	—
309	C₂³	—	316	1	no	—
310	C₂³	—	79	2	no	—
311	Q₈	—	2	2	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	8	2	no	—
314	Q₈	—	316	2	no	—
315	Q₈	—	316	2	no	—
316	Q₈	—	632	4	no	—
317	D₄	—	316	1	no	—
318	D₄	—	316	1	no	—
319	D₄	—	79	4	no	—
320	D₄	—	632	4	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	316	2	no	—
325	C₂×C₄	—	316	1	no	—
326	C₂×C₄	—	79	2	no	—
327	C₂×C₄	—	632	4	no	—
328	C₂×C₄	—	632	4	no	—
329	C₂×C₄	—	632	2	no	—
330	C₂×C₄	—	158	2	no	—
331	C₂×C₄	—	158	2	no	—
332	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
333	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	158	2	no	—
337	C₈	C₂×#11929625	1	1	yes	characteristic
338	C₈	#14579207	1	1	yes	characteristic
339	C₂²	—	79	4	no	—
340	C₂²	—	632	1	no	—
341	C₂²	—	632	1	no	—
342	C₂²	—	632	1	no	—
343	C₂²	—	158	2	no	—
344	C₂²	—	158	2	no	—
345	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	4	2	no	—
348	C₄	—	8	1	no	—
349	C₄	—	8	2	no	—
350	C₄	—	632	2	no	—
351	C₄	—	632	1	no	—
352	C₄	—	158	2	no	—
353	C₄	SD₃₂×D₇₉	1	2	yes	—
354	C₄	#19099204	1	1	yes	characteristic
355	C₄	#19099372	1	1	yes	characteristic
356	C₂	—	158	4	no	—
357	C₂	—	1264	1	no	—
358	C₂	#23775210	1	1	yes	characteristic
359	C₂	C₂×SD₃₂×D₇₉	1	1	yes	characteristic
360	C₂	#23776552	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2225	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	#23775747	C₂
hypercenter	#2225	D₇₉
hypocenter	C₇₉	#33865
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2225

Complements

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

N	H ≅ G/N	type
2. Q₁₆×D₃₁₆	C₂	semidirect
3. #23765629	C₂	semidirect
4. #23765599	C₂	semidirect
5. #23765497	C₂	semidirect
6. #23765446	C₂	semidirect
7. #23765318	C₂	semidirect
8. C₄×#14579104	C₂	semidirect
9. #23765219	C₂	semidirect
10. #358×D₇₉	C₂	semidirect
11. C₇₉×#2225	C₂	semidirect
118. Q₁₆×C₇₉	#41	semidirect
145. D₃₁₆	SD₃₂	semidirect
185. #2225	D₇₉	semidirect
192. C₇₉	#33865	semidirect
282. Q₁₆	#11929625	semidirect

Maximal subgroups

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

subgroup	index
#23775747	2
Q₁₆×D₃₁₆	2
#23765629	2
#23765599	2
#23765497	2
#23765446	2
#23765318	2
C₄×#14579104	2
#23765219	2
#358×D₇₉	2
C₇₉×#2225	2
#33865	79