Subgroups — #34443802

Order 20224.

602 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	#23773979	C₂	1	1	yes	maximal, characteristic, complemented
2	#23765673	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765649	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765625	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765623	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765484	C₂	1	4	yes	maximal, complemented
7	C₄×#14579102	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₉×#3169	C₂	1	1	yes	maximal, characteristic, complemented
10	#19099113	—	2	1	no	—
11	C₂×#14579102	C₂²	1	4	yes	—
12	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
13	#19098968	C₂²	1	1	yes	characteristic
14	#19098940	C₂²	1	1	yes	characteristic
15	#19098910	—	2	1	no	—
16	#19098799	—	2	1	no	—
17	#19098799	C₂²	1	2	yes	—
18	C₄×D₆₃₂	—	2	1	no	—
19	C₄×D₆₃₂	C₂²	1	2	yes	—
20	#19098388	C₂²	1	4	yes	—
21	#19098364	C₂²	1	4	yes	—
22	#19098363	C₂²	1	4	yes	—
23	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₄×C₇₉⋊C₁₆	C₂²	1	2	yes	—
26	#532×C₇₉	—	2	1	no	—
27	#500×C₇₉	C₂²	1	1	yes	characteristic
28	#493×C₇₉	C₂²	1	1	yes	characteristic
29	#486×C₇₉	—	2	1	no	—
30	#466×C₇₉	—	2	1	no	—
31	#466×C₇₉	C₂²	1	2	yes	—
32	C₄×D₈×C₇₉	—	2	1	no	—
33	C₄×D₈×C₇₉	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
35	C₂×#11929624	—	4	1	no	—
36	C₂×D₆₃₂	—	2	1	no	—
37	C₂×D₆₃₂	C₂³	1	2	yes	—
38	#14579161	—	2	1	no	—
39	#14579158	—	2	1	no	—
40	#14579149	—	4	1	no	—
41	#14579135	—	2	2	no	—
42	#14579135	—	4	1	no	—
43	C₄×D₃₁₆	—	2	2	no	—
44	C₄×D₃₁₆	—	4	1	no	—
45	#14579102	—	2	8	no	—
46	C₂×C₇₉⋊C₁₆	—	2	4	no	—
47	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
48	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
49	#14579095	—	2	1	no	—
50	#14579095	C₂³	1	2	yes	—
51	#14579083	—	2	2	no	—
52	#14579083	—	2	2	no	—
53	C₂×#41×C₇₉	—	4	1	no	—
54	C₂×D₈×C₇₉	—	2	1	no	—
55	C₂×D₈×C₇₉	C₂³	1	2	yes	—
56	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
58	C₄⋊Q₈×C₇₉	—	2	1	no	—
59	#127×C₇₉	—	2	1	no	—
60	#124×C₇₉	—	4	1	no	—
61	#121×C₇₉	—	2	2	no	—
62	#121×C₇₉	—	4	1	no	—
63	C₄×D₄×C₇₉	—	2	2	no	—
64	C₄×D₄×C₇₉	—	4	1	no	—
65	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
66	#107×C₇₉	—	2	1	no	—
67	#107×C₇₉	C₂³	1	2	yes	—
68	#102×C₇₉	—	2	2	no	—
69	#102×C₇₉	—	2	2	no	—
70	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
71	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
72	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
74	#11929624	—	8	4	no	—
75	C₂×Dic₁₅₈	—	4	1	no	—
76	C₂×D₃₁₆	—	2	2	no	—
77	C₂×D₃₁₆	—	4	1	no	—
78	C₂×D₃₁₆	—	4	1	no	—
79	C₂×C₂²⋊D₇₉	—	4	2	no	—
80	C₂×C₄×D₇₉	—	4	2	no	—
81	D₆₃₂	—	2	4	no	—
82	D₆₃₂	—	4	2	no	—
83	C₄⋊Dic₇₉	—	2	2	no	—
84	C₄⋊Dic₇₉	—	4	1	no	—
85	(C₂×D₇₉)⋊C₄	—	4	2	no	—
86	(C₂×D₇₉)⋊C₄	—	4	2	no	—
87	C₇₉⋊C₁₆	—	2	2	no	—
88	C₇₉⋊C₁₆	#41	1	4	yes	complemented
89	#41×C₇₉	—	8	4	no	—
90	C₂×Q₈×C₇₉	—	4	1	no	—
91	C₂×D₄×C₇₉	—	2	2	no	—
92	C₂×D₄×C₇₉	—	4	2	no	—
93	C₂×D₄×C₇₉	—	4	1	no	—
94	C₂×D₄×C₇₉	—	4	1	no	—
95	C₂²×C₄×C₇₉	—	4	2	no	—
96	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
97	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
98	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
99	D₈×C₇₉	—	2	4	no	—
100	D₈×C₇₉	—	4	2	no	—
101	C₂×C₈×C₇₉	—	2	2	no	—
102	C₂×C₈×C₇₉	—	2	2	no	—
103	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₇₉	—	2	2	no	—
108	C₄⋊C₄×C₇₉	—	4	1	no	—
109	C₂²⋊C₄×C₇₉	—	4	2	no	—
110	C₂²⋊C₄×C₇₉	—	4	2	no	—
111	C₄²×C₇₉	D₈	1	1	yes	characteristic
112	C₄²×C₇₉	D₈	1	1	yes	characteristic
113	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
115	C₂³×C₇₉	—	4	2	no	—
116	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
117	C₂²×D₇₉	—	4	2	no	—
118	Q₈×C₇₉	—	8	2	no	—
119	D₄×C₇₉	—	4	2	no	—
120	D₄×C₇₉	—	4	2	no	—
121	D₄×C₇₉	—	8	4	no	—
122	D₄×C₇₉	—	8	2	no	—
123	D₄×C₇₉	—	8	2	no	—
124	C₂×C₄×C₇₉	—	2	1	no	—
125	C₂×C₄×C₇₉	—	2	1	no	—
126	C₂×C₄×C₇₉	—	2	2	no	—
127	C₂×C₄×C₇₉	—	4	2	no	—
128	C₂×C₄×C₇₉	—	8	4	no	—
129	C₂×C₄×C₇₉	D₁₆	1	4	yes	complemented
130	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
136	C₂²⋊D₇₉	—	8	4	no	—
137	C₂×Dic₇₉	—	4	2	no	—
138	D₃₁₆	—	4	2	no	—
139	D₃₁₆	—	4	2	no	—
140	D₃₁₆	—	8	2	no	—
141	D₃₁₆	—	8	2	no	—
142	C₄×D₇₉	—	8	4	no	—
143	Dic₁₅₈	—	8	2	no	—
144	C₈×C₇₉	—	2	1	no	—
145	C₈×C₇₉	—	2	1	no	—
146	C₈×C₇₉	#135	1	2	yes	—
147	C₈×C₇₉	C₂×#41	1	2	yes	—
148	C₂²×C₇₉	—	2	1	no	—
149	C₂²×C₇₉	—	2	1	no	—
150	C₂²×C₇₉	—	8	2	no	—
151	C₂²×C₇₉	—	8	2	no	—
152	C₂²×C₇₉	—	8	2	no	—
153	C₂²×C₇₉	C₂×D₁₆	1	2	yes	—
154	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₇₉	—	8	2	no	—
156	C₂×D₇₉	—	8	2	no	—
157	C₂×D₇₉	—	8	2	no	—
158	C₄×C₇₉	—	2	1	no	—
159	C₄×C₇₉	—	2	1	no	—
160	C₄×C₇₉	—	8	2	no	—
161	C₄×C₇₉	C₂×D₁₆	1	2	yes	—
162	C₄×C₇₉	C₂×D₁₆	1	2	yes	—
163	C₄×C₇₉	#508	1	2	yes	—
164	C₄×C₇₉	#532	1	1	yes	characteristic
165	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
166	Dic₇₉	—	8	2	no	—
167	#33733	—	79	1	no	maximal, Sylow
168	C₂×C₇₉	—	2	2	no	—
169	C₂×C₇₉	—	16	2	no	—
170	C₂×C₇₉	#3169	1	1	yes	characteristic
171	C₂×C₇₉	C₂²×D₁₆	1	1	yes	characteristic
172	C₂×C₇₉	C₂×#508	1	1	yes	characteristic
173	D₇₉	—	16	2	no	—
174	#3169	—	79	1	no	—
175	#3169	D₇₉	1	1	yes	characteristic, complemented
176	#2252	—	79	1	no	—
177	#2246	—	79	1	no	—
178	#2240	—	79	1	no	—
179	#2240	—	79	1	no	—
180	#2215	—	79	4	no	—
181	C₄×D₁₆	—	79	4	no	—
182	C₂×C₄×C₁₆	—	79	1	no	—
183	C₇₉	#33733	1	1	yes	characteristic, complemented, Sylow
184	#532	—	2	1	no	—
185	#532	—	158	1	no	—
186	C₂×D₁₆	—	79	4	no	—
187	C₂²×C₁₆	—	79	2	no	—
188	#500	—	79	1	no	—
189	#500	C₂×D₇₉	1	1	yes	characteristic
190	#493	—	79	1	no	—
191	#493	C₂×D₇₉	1	1	yes	characteristic
192	#486	—	2	1	no	—
193	#486	—	158	1	no	—
194	#466	—	2	1	no	—
195	#466	—	79	2	no	—
196	#466	—	158	1	no	—
197	#466	C₂×D₇₉	1	2	yes	—
198	C₄×D₈	—	2	1	no	—
199	C₄×D₈	—	79	2	no	—
200	C₄×D₈	—	158	1	no	—
201	C₄×D₈	C₂×D₇₉	1	2	yes	—
202	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
203	#366	—	79	4	no	—
204	#357	—	79	4	no	—
205	#357	—	79	4	no	—
206	C₄×C₁₆	—	79	2	no	—
207	C₄×C₁₆	—	79	1	no	—
208	C₄×C₁₆	—	79	1	no	—
209	C₂×#41	—	4	1	no	—
210	C₂×#41	—	316	1	no	—
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	—	79	2	no	—
213	C₂×D₈	—	158	1	no	—
214	C₂×D₈	C₂²×D₇₉	1	2	yes	—
215	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
216	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	158	1	no	—
219	#127	—	2	1	no	—
220	#127	—	158	1	no	—
221	#124	—	4	1	no	—
222	#124	—	316	1	no	—
223	#121	—	2	2	no	—
224	#121	—	4	1	no	—
225	#121	—	316	1	no	—
226	#121	—	158	2	no	—
227	C₄×D₄	—	2	2	no	—
228	C₄×D₄	—	4	1	no	—
229	C₄×D₄	—	316	1	no	—
230	C₄×D₄	—	158	2	no	—
231	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
232	D₁₆	—	158	8	no	—
233	C₂×C₁₆	—	79	2	no	—
234	C₂×C₁₆	—	79	2	no	—
235	C₂×C₁₆	—	158	4	no	—
236	#107	—	2	1	no	—
237	#107	—	79	2	no	—
238	#107	—	158	1	no	—
239	#107	C₂²×D₇₉	1	2	yes	—
240	#102	—	2	2	no	—
241	#102	—	2	2	no	—
242	#102	—	158	2	no	—
243	#102	—	158	2	no	—
244	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
245	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
248	#41	—	8	4	no	—
249	#41	—	632	4	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	316	1	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	316	2	no	—
257	C₂×D₄	—	316	1	no	—
258	C₂×D₄	—	316	1	no	—
259	C₂×D₄	—	158	2	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	316	2	no	—
262	C₂²×C₄	#11929609	1	1	yes	characteristic
263	C₂²×C₄	#11929609	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
265	D₈	—	2	4	no	—
266	D₈	—	4	2	no	—
267	D₈	—	316	2	no	—
268	D₈	—	158	4	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
273	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	316	1	no	—
278	C₄⋊C₄	—	158	2	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	316	2	no	—
282	C₂²⋊C₄	—	316	2	no	—
283	C₄²	#11929609	1	1	yes	characteristic
284	C₄²	#11929609	1	1	yes	characteristic
285	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
286	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
287	C₁₆	—	79	4	no	—
288	C₁₆	—	158	2	no	—
289	C₂³	—	4	2	no	—
290	C₂³	—	316	2	no	—
291	C₂³	C₂×#11929609	1	1	yes	characteristic
292	Q₈	—	8	2	no	—
293	Q₈	—	632	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	8	4	no	—
297	D₄	—	8	2	no	—
298	D₄	—	8	2	no	—
299	D₄	—	316	2	no	—
300	D₄	—	316	2	no	—
301	D₄	—	632	4	no	—
302	D₄	—	632	2	no	—
303	D₄	—	632	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	—	316	2	no	—
310	C₂×C₄	—	632	4	no	—
311	C₂×C₄	#14579102	1	4	yes	complemented
312	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
313	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
314	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
315	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
316	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	#14579181	1	2	yes	—
321	C₈	#41×D₇₉	1	2	yes	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	632	2	no	—
328	C₂²	—	632	2	no	—
329	C₂²	—	632	2	no	—
330	C₂²	C₂×#14579102	1	2	yes	—
331	C₂²	C₂²×#11929609	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	632	2	no	—
336	C₄	C₂×#14579102	1	2	yes	—
337	C₄	C₂×#14579102	1	2	yes	—
338	C₄	#19098993	1	2	yes	—
339	C₄	#19099109	1	1	yes	characteristic
340	C₄	C₂×#14579181	1	1	yes	characteristic
341	C₂	—	2	2	no	—
342	C₂	—	16	2	no	—
343	C₂	—	1264	2	no	—
344	C₂	#23773975	1	1	yes	characteristic
345	C₂	C₂²×#14579102	1	1	yes	characteristic
346	C₂	C₂×#19098993	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#3169	C₂
socle	C₂²×C₇₉	C₂²×D₈
hypercenter	#3169	D₇₉
hypocenter	C₇₉	#33733
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #23773979	C₂	semidirect
2. #23765673	C₂	semidirect
3. #23765649	C₂	semidirect
4. #23765625	C₂	semidirect
5. #23765623	C₂	semidirect
6. #23765484	C₂	semidirect
7. C₄×#14579102	C₂	semidirect
8. C₂×C₄×C₇₉⋊C₁₆	C₂	semidirect
9. C₇₉×#3169	C₂	semidirect
88. C₇₉⋊C₁₆	#41	semidirect
129. C₂×C₄×C₇₉	D₁₆	semidirect
175. #3169	D₇₉	semidirect
183. C₇₉	#33733	semidirect
311. C₂×C₄	#14579102	semidirect

Maximal subgroups

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

subgroup	index
#23773979	2
#23765673	2
#23765649	2
#23765625	2
#23765623	2
#23765484	2
C₄×#14579102	2
C₂×C₄×C₇₉⋊C₁₆	2
C₇₉×#3169	2
#33733	79