Subgroups — #34442974

Order 20224.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772277	C₂	1	1	yes	maximal, characteristic
2	#23765561	C₂	1	2	yes	maximal, complemented
3	#23765554	C₂	1	2	yes	maximal, complemented
4	#23765549	C₂	1	2	yes	maximal, complemented
5	#23765534	C₂	1	2	yes	maximal, complemented
6	#23765189	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19098388	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765159	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765110	C₂	1	1	yes	maximal, characteristic
10	#23765105	C₂	1	1	yes	maximal, characteristic
11	C₇₉×#3164	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579183	C₂²	1	1	yes	characteristic
13	#19099043	—	2	1	no	—
14	C₂×#14579101	C₂²	1	1	yes	characteristic
15	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#19098665	—	2	2	no	—
17	C₄×Dic₃₁₆	C₂²	1	2	yes	—
18	C₄×D₆₃₂	C₂²	1	2	yes	—
19	C₂×#14579095	C₂²	1	1	yes	characteristic
20	#19098565	—	2	1	no	—
21	#19098390	C₂²	1	2	yes	—
22	#19098388	C₂²	1	2	yes	—
23	#19098388	C₂²	1	2	yes	—
24	#19098388	C₂²	1	2	yes	—
25	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
26	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
27	#19098366	C₂²	1	1	yes	characteristic
28	#19098366	C₂²	1	2	yes	—
29	#19098366	C₂²	1	1	yes	characteristic
30	#19098363	C₂²	1	1	yes	characteristic
31	#19098363	C₂²	1	2	yes	—
32	#19098363	C₂²	1	1	yes	characteristic
33	#544×C₇₉	—	2	1	no	—
34	#501×C₇₉	—	2	2	no	—
35	#500×C₇₉	C₂²	1	2	yes	—
36	#499×C₇₉	C₂²	1	2	yes	—
37	#428×C₇₉	—	2	1	no	—
38	#427×C₇₉	C₂²	1	1	yes	characteristic
39	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
40	#405×C₇₉	C₂²	1	1	yes	characteristic
41	C₂×#11929624	—	2	1	no	—
42	#14579183	—	2	2	no	—
43	#14579183	Q₈	1	4	yes	complemented
44	C₂×Dic₃₁₆	C₂³	1	1	yes	characteristic
45	C₂×#11929613	—	2	1	no	—
46	C₂×D₆₃₂	C₂³	1	1	yes	characteristic
47	C₄×Dic₁₅₈	—	2	2	no	—
48	C₄×D₃₁₆	—	2	2	no	—
49	(C₂²⋊D₇₉)⋊C₄	—	4	2	no	—
50	#14579119	—	4	1	no	—
51	C₂×C₄⋊Dic₇₉	—	2	1	no	—
52	C₄⋊C₄×D₇₉	—	4	1	no	—
53	#14579101	—	2	2	no	—
54	C₂×C₇₉⋊C₁₆	—	2	2	no	—
55	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
58	#14579095	—	2	1	no	—
59	#14579095	C₂³	1	2	yes	—
60	#14579086	—	2	2	no	—
61	#14579083	—	2	2	no	—
62	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₇₉	—	2	2	no	—
65	C₄⋊Q₈×C₇₉	—	4	1	no	—
66	#125×C₇₉	—	2	2	no	—
67	#125×C₇₉	—	4	1	no	—
68	C₂²⋊Q₈×C₇₉	—	4	2	no	—
69	#117×C₇₉	—	2	1	no	—
70	#117×C₇₉	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
72	#107×C₇₉	—	2	1	no	—
73	#107×C₇₉	—	2	2	no	—
74	#107×C₇₉	C₂³	1	2	yes	—
75	#107×C₇₉	C₂³	1	1	yes	characteristic
76	#107×C₇₉	C₂³	1	1	yes	characteristic
77	#106×C₇₉	—	2	1	no	—
78	#106×C₇₉	—	2	2	no	—
79	#97×C₇₉	D₄	1	2	yes	—
80	C₄×C₈×C₇₉	C₂³	1	2	yes	—
81	#11929624	—	2	4	no	—
82	#11929624	—	4	2	no	—
83	C₂×Dic₁₅₈	—	2	1	no	—
84	C₂×D₃₁₆	—	2	1	no	—
85	C₂×C₂²⋊D₇₉	—	4	1	no	—
86	C₂²×Dic₇₉	—	4	1	no	—
87	C₂×C₄×D₇₉	—	4	1	no	—
88	C₂×C₄×D₇₉	—	4	2	no	—
89	Dic₃₁₆	—	2	1	no	—
90	Dic₃₁₆	C₂×Q₈	1	2	yes	—
91	#11929613	—	2	4	no	—
92	D₆₃₂	—	2	1	no	—
93	D₆₃₂	C₂×Q₈	1	2	yes	—
94	C₄⋊Dic₇₉	—	2	2	no	—
95	C₄⋊Dic₇₉	—	4	1	no	—
96	Dic₇₉⋊C₄	—	4	2	no	—
97	(C₂×D₇₉)⋊C₄	—	4	2	no	—
98	C₄×Dic₇₉	—	4	2	no	—
99	C₇₉⋊C₁₆	—	2	1	no	—
100	C₇₉⋊C₁₆	—	2	1	no	—
101	C₇₉⋊C₁₆	—	2	2	no	—
102	C₂×Q₈×C₇₉	—	4	2	no	—
103	C₂²×C₄×C₇₉	—	4	1	no	—
104	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
105	M₁₆×C₇₉	—	2	2	no	—
106	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
109	C₂×C₈×C₇₉	#41	1	2	yes	—
110	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₇₉	—	2	1	no	—
112	C₄⋊C₄×C₇₉	—	2	2	no	—
113	C₄⋊C₄×C₇₉	—	2	1	no	—
114	C₄⋊C₄×C₇₉	—	4	1	no	—
115	C₄⋊C₄×C₇₉	—	4	2	no	—
116	C₄⋊C₄×C₇₉	—	4	2	no	—
117	C₄⋊C₄×C₇₉	—	4	2	no	—
118	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
119	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
120	C₂²⋊C₄×C₇₉	—	4	1	no	—
121	C₂²⋊C₄×C₇₉	D₈	1	2	yes	—
122	C₄²×C₇₉	—	4	1	no	—
123	C₄²×C₇₉	C₂×D₄	1	2	yes	—
124	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
125	C₂²×D₇₉	—	4	1	no	—
126	Q₈×C₇₉	—	16	2	no	—
127	C₂×C₄×C₇₉	—	4	1	no	—
128	C₂×C₄×C₇₉	—	4	1	no	—
129	C₂×C₄×C₇₉	—	4	2	no	—
130	C₂×C₄×C₇₉	—	16	1	no	—
131	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₇₉	#137	1	2	yes	—
136	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₇₉	—	4	4	no	—
138	C₂×Dic₇₉	—	4	1	no	—
139	C₂×Dic₇₉	—	4	2	no	—
140	C₂×Dic₇₉	—	16	1	no	—
141	D₃₁₆	—	2	2	no	—
142	D₃₁₆	—	4	1	no	—
143	C₄×D₇₉	—	4	4	no	—
144	C₄×D₇₉	—	16	2	no	—
145	Dic₁₅₈	—	2	2	no	—
146	Dic₁₅₈	—	4	1	no	—
147	C₈×C₇₉	—	2	2	no	—
148	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
149	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
150	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
151	C₂²×C₇₉	—	2	1	no	—
152	C₂²×C₇₉	—	2	1	no	—
153	C₂²×C₇₉	#474	1	2	yes	—
154	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₇₉	—	4	2	no	—
156	C₂×D₇₉	—	8	1	no	—
157	C₂×D₇₉	—	8	1	no	—
158	C₄×C₇₉	—	2	2	no	—
159	C₄×C₇₉	—	2	2	no	—
160	C₄×C₇₉	—	8	2	no	—
161	C₄×C₇₉	—	8	1	no	—
162	C₄×C₇₉	—	8	1	no	—
163	C₄×C₇₉	#474	1	2	yes	—
164	C₄×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
165	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
166	Dic₇₉	—	4	2	no	—
167	Dic₇₉	—	8	2	no	—
168	#33633	—	79	1	no	maximal, Sylow
169	C₂×C₇₉	—	2	2	no	—
170	C₂×C₇₉	C₂×#474	1	1	yes	characteristic
171	C₂×C₇₉	#3415	1	1	yes	characteristic
172	C₂×C₇₉	#3417	1	1	yes	characteristic
173	D₇₉	—	8	2	no	—
174	#3164	D₇₉	1	1	yes	characteristic, complemented
175	#2951	—	79	1	no	—
176	#2229	—	79	2	no	—
177	#2228	—	79	2	no	—
178	#2227	—	79	2	no	—
179	#2224	—	79	2	no	—
180	#2159	—	79	1	no	—
181	C₂×#366	—	79	1	no	—
182	#2151	—	79	1	no	—
183	#2139	—	79	1	no	—
184	#2138	—	79	1	no	—
185	C₇₉	#33633	1	1	yes	characteristic, complemented, Sylow
186	C₂×#135	—	79	1	no	—
187	#544	—	2	1	no	—
188	#520	—	158	1	no	—
189	C₂×M₃₂	—	79	1	no	—
190	C₂²×C₁₆	—	79	1	no	—
191	#501	—	2	2	no	—
192	#500	C₂×D₇₉	1	2	yes	—
193	#499	C₂×D₇₉	1	2	yes	—
194	#440	—	158	2	no	—
195	C₄×Q₁₆	—	79	2	no	—
196	C₄×D₈	—	79	2	no	—
197	#428	—	2	1	no	—
198	#427	C₂×D₇₉	1	1	yes	characteristic
199	C₂×#107	—	79	1	no	—
200	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
201	#417	—	158	1	no	—
202	#405	C₂×D₇₉	1	1	yes	characteristic
203	#367	—	79	2	no	—
204	#366	—	79	2	no	—
205	#366	—	79	2	no	—
206	#366	—	79	2	no	—
207	C₄⋊C₁₆	—	79	2	no	—
208	C₄⋊C₁₆	—	79	2	no	—
209	#358	—	79	1	no	—
210	#358	—	79	1	no	—
211	#358	—	79	2	no	—
212	#357	—	79	1	no	—
213	#357	—	79	1	no	—
214	#357	—	79	2	no	—
215	C₂×#41	—	158	1	no	—
216	#135	—	79	4	no	—
217	#135	—	158	2	no	—
218	C₂×Q₁₆	—	79	1	no	—
219	C₂×SD₁₆	—	158	1	no	—
220	C₂×D₈	—	79	1	no	—
221	C₂×M₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
222	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
223	C₄⋊Q₈	—	2	2	no	—
224	C₄⋊Q₈	—	4	1	no	—
225	#125	—	2	2	no	—
226	#125	—	4	1	no	—
227	C₂²⋊Q₈	—	4	2	no	—
228	C₄×Q₈	—	158	2	no	—
229	C₄×D₄	—	316	2	no	—
230	C₄×D₄	—	158	2	no	—
231	#117	—	2	1	no	—
232	#117	—	316	1	no	—
233	#117	C₂²⋊D₇₉	1	1	yes	characteristic
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	316	1	no	—
236	C₂×C₄⋊C₄	—	158	1	no	—
237	M₃₂	—	158	2	no	—
238	C₂×C₁₆	—	79	1	no	—
239	C₂×C₁₆	—	79	1	no	—
240	C₂×C₁₆	—	79	2	no	—
241	C₂×C₁₆	—	158	2	no	—
242	#107	—	2	1	no	—
243	#107	—	2	2	no	—
244	#107	—	79	2	no	—
245	#107	—	158	1	no	—
246	#107	C₂²×D₇₉	1	2	yes	—
247	#107	C₂²×D₇₉	1	1	yes	characteristic
248	#107	C₂²×D₇₉	1	1	yes	characteristic
249	#106	—	2	1	no	—
250	#106	—	2	2	no	—
251	#103	—	158	2	no	—
252	#102	—	158	2	no	—
253	#97	C₂²⋊D₇₉	1	2	yes	—
254	C₄×C₈	C₂²×D₇₉	1	2	yes	—
255	#41	—	316	2	no	—
256	#41	—	158	4	no	—
257	C₂×Q₈	—	4	2	no	—
258	C₂×Q₈	—	158	1	no	—
259	C₂×D₄	—	316	1	no	—
260	C₂×D₄	—	158	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	316	1	no	—
263	C₂²×C₄	—	316	1	no	—
264	C₂²×C₄	—	316	2	no	—
265	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
266	Q₁₆	—	79	2	no	—
267	Q₁₆	—	158	1	no	—
268	SD₁₆	—	158	4	no	—
269	D₈	—	79	2	no	—
270	D₈	—	158	1	no	—
271	M₁₆	—	2	2	no	—
272	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
273	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
274	C₂×C₈	Q₈×D₇₉	1	2	yes	—
275	C₂×C₈	#11929625	1	2	yes	—
276	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	316	2	no	—
285	C₄⋊C₄	—	316	1	no	—
286	C₄⋊C₄	—	158	2	no	—
287	C₄⋊C₄	#11929609	1	1	yes	characteristic
288	C₄⋊C₄	#11929609	1	1	yes	characteristic
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	316	2	no	—
291	C₂²⋊C₄	#11929609	1	2	yes	—
292	C₄²	—	4	1	no	—
293	C₄²	—	316	2	no	—
294	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
295	C₁₆	—	158	1	no	—
296	C₁₆	—	158	1	no	—
297	C₁₆	—	158	2	no	—
298	C₂³	—	316	1	no	—
299	C₂³	C₂×#11929609	1	1	yes	characteristic
300	Q₈	—	16	2	no	—
301	Q₈	—	316	1	no	—
302	Q₈	—	158	2	no	—
303	D₄	—	316	4	no	—
304	D₄	—	316	1	no	—
305	D₄	—	158	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	16	1	no	—
310	C₂×C₄	—	316	1	no	—
311	C₂×C₄	—	316	4	no	—
312	C₂×C₄	—	316	2	no	—
313	C₂×C₄	—	1264	2	no	—
314	C₂×C₄	—	1264	1	no	—
315	C₂×C₄	#14579137	1	2	yes	—
316	C₂×C₄	C₂×#11929609	1	2	yes	—
317	C₂×C₄	C₂×#11929609	1	2	yes	—
318	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
319	C₂×C₄	#14579192	1	2	yes	—
320	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	#14579137	1	2	yes	—
323	C₈	C₂×Q₈×D₇₉	1	1	yes	characteristic
324	C₈	C₂×#11929625	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	316	2	no	—
328	C₂²	—	632	1	no	—
329	C₂²	—	632	1	no	—
330	C₂²	#19098840	1	2	yes	—
331	C₂²	C₂²×#11929609	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	1	no	—
337	C₄	—	316	2	no	—
338	C₄	—	632	2	no	—
339	C₄	#19098840	1	2	yes	—
340	C₄	C₂×#14579137	1	1	yes	characteristic
341	C₄	C₂×#14579192	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	632	2	no	—
344	C₂	C₂×#19098840	1	1	yes	characteristic
345	C₂	#23776539	1	1	yes	characteristic
346	C₂	#23776551	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#3164	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×#14579183	C₂²
hypercenter	#3164	D₇₉
hypocenter	C₇₉	#33633
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3164

Complements

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

N	H ≅ G/N	type
2. #23765561	C₂	semidirect
3. #23765554	C₂	semidirect
4. #23765549	C₂	semidirect
5. #23765534	C₂	semidirect
6. #23765189	C₂	semidirect
7. C₂×#19098388	C₂	semidirect
8. #23765159	C₂	semidirect
11. C₇₉×#3164	C₂	semidirect
43. #14579183	Q₈	semidirect
174. #3164	D₇₉	semidirect
185. C₇₉	#33633	semidirect

Maximal subgroups

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

subgroup	index
#23772277	2
#23765561	2
#23765554	2
#23765549	2
#23765534	2
#23765189	2
C₂×#19098388	2
#23765159	2
#23765110	2
#23765105	2
C₇₉×#3164	2
#33633	79