Subgroups — C₂×#23761338 #34197394

Order 20224.

518 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098345	C₂	1	2	yes	maximal, complemented
2	#23761338	C₂	1	4	yes	maximal, direct product factor
3	C₂×#408×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579101	—	2	2	no	—
5	#19098345	—	2	2	no	—
6	#19098345	C₂²	1	4	yes	complemented
7	C₂²×#41×C₇₉	C₄	1	1	yes	characteristic
8	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₇₉	C₄	1	1	yes	characteristic
10	#408×C₇₉	—	2	2	no	—
11	#408×C₇₉	C₄	1	2	yes	—
12	#408×C₇₉	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₈×C₇₉	—	2	2	no	—
14	#14579101	—	2	4	no	—
15	#14579101	—	4	2	no	—
16	C₂×C₇₉⋊C₁₆	—	4	2	no	—
17	C₂×#41×C₇₉	—	2	1	no	—
18	C₂×#41×C₇₉	—	2	1	no	—
19	C₂×#41×C₇₉	—	4	1	no	—
20	C₂×#41×C₇₉	—	4	1	no	—
21	C₂×#41×C₇₉	—	4	1	no	—
22	C₂×#41×C₇₉	—	4	1	no	—
23	C₂×#41×C₇₉	C₈	1	2	yes	—
24	C₂×#41×C₇₉	C₂×C₄	1	2	yes	—
25	C₂²×Q₈×C₇₉	C₈	1	1	yes	characteristic
26	C₂²×D₄×C₇₉	—	2	1	no	—
27	C₂²×D₄×C₇₉	C₈	1	1	yes	characteristic
28	C₂³×C₄×C₇₉	—	2	1	no	—
29	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₇₉	—	4	1	no	—
31	C₂×M₁₆×C₇₉	—	4	1	no	—
32	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₇₉	D₄	1	2	yes	—
34	C₂×M₁₆×C₇₉	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₇₉	—	2	1	no	—
36	C₂²×C₈×C₇₉	—	2	1	no	—
37	C₂²×C₈×C₇₉	—	4	1	no	—
38	C₂²×C₈×C₇₉	—	4	1	no	—
39	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₇₉	D₄	1	2	yes	—
41	C₂²⋊C₈×C₇₉	—	2	4	no	—
42	C₂²⋊C₈×C₇₉	—	2	4	no	—
43	C₇₉⋊C₁₆	—	4	4	no	—
44	C₂⁴×C₇₉	—	2	1	no	—
45	C₂⁴×C₇₉	M₁₆	1	1	yes	characteristic
46	#41×C₇₉	—	4	2	no	—
47	#41×C₇₉	—	4	2	no	—
48	#41×C₇₉	—	4	2	no	—
49	#41×C₇₉	—	4	2	no	—
50	#41×C₇₉	—	4	2	no	—
51	#41×C₇₉	—	4	2	no	—
52	#41×C₇₉	—	4	2	no	—
53	#41×C₇₉	—	4	2	no	—
54	C₂×Q₈×C₇₉	—	2	1	no	—
55	C₂×Q₈×C₇₉	—	4	1	no	—
56	C₂×Q₈×C₇₉	—	4	1	no	—
57	C₂×Q₈×C₇₉	C₂×C₈	1	2	yes	—
58	C₂×D₄×C₇₉	—	2	2	no	—
59	C₂×D₄×C₇₉	—	2	2	no	—
60	C₂×D₄×C₇₉	—	2	1	no	—
61	C₂×D₄×C₇₉	—	4	1	no	—
62	C₂×D₄×C₇₉	—	4	1	no	—
63	C₂×D₄×C₇₉	—	4	2	no	—
64	C₂×D₄×C₇₉	—	4	2	no	—
65	C₂×D₄×C₇₉	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₇₉	—	2	1	no	—
67	C₂²×C₄×C₇₉	—	2	1	no	—
68	C₂²×C₄×C₇₉	—	2	1	no	—
69	C₂²×C₄×C₇₉	—	2	1	no	—
70	C₂²×C₄×C₇₉	—	2	1	no	—
71	C₂²×C₄×C₇₉	—	2	2	no	—
72	C₂²×C₄×C₇₉	—	2	2	no	—
73	C₂²×C₄×C₇₉	—	4	1	no	—
74	C₂²×C₄×C₇₉	—	4	1	no	—
75	C₂²×C₄×C₇₉	—	4	1	no	—
76	C₂²×C₄×C₇₉	—	4	1	no	—
77	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
82	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
83	M₁₆×C₇₉	—	4	1	no	—
84	M₁₆×C₇₉	—	4	2	no	—
85	M₁₆×C₇₉	—	4	1	no	—
86	C₂×C₈×C₇₉	—	2	2	no	—
87	C₂×C₈×C₇₉	—	2	2	no	—
88	C₂×C₈×C₇₉	—	2	2	no	—
89	C₂×C₈×C₇₉	—	4	2	no	—
90	C₂×C₈×C₇₉	—	4	2	no	—
91	C₂×C₈×C₇₉	—	4	1	no	—
92	C₂×C₈×C₇₉	—	4	1	no	—
93	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
94	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
95	C₂³×C₇₉	—	2	1	no	—
96	C₂³×C₇₉	—	2	1	no	—
97	C₂³×C₇₉	—	2	2	no	—
98	C₂³×C₇₉	—	4	2	no	—
99	C₂³×C₇₉	—	4	4	no	—
100	C₂³×C₇₉	—	4	1	no	—
101	C₂³×C₇₉	—	4	1	no	—
102	C₂³×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
103	C₂³×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
104	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
105	Q₈×C₇₉	—	4	2	no	—
106	Q₈×C₇₉	—	4	2	no	—
107	D₄×C₇₉	—	4	2	no	—
108	D₄×C₇₉	—	4	2	no	—
109	D₄×C₇₉	—	4	4	no	—
110	D₄×C₇₉	—	4	4	no	—
111	C₂×C₄×C₇₉	—	2	1	no	—
112	C₂×C₄×C₇₉	—	2	1	no	—
113	C₂×C₄×C₇₉	—	2	1	no	—
114	C₂×C₄×C₇₉	—	2	1	no	—
115	C₂×C₄×C₇₉	—	2	1	no	—
116	C₂×C₄×C₇₉	—	2	1	no	—
117	C₂×C₄×C₇₉	—	2	1	no	—
118	C₂×C₄×C₇₉	—	2	1	no	—
119	C₂×C₄×C₇₉	—	2	1	no	—
120	C₂×C₄×C₇₉	—	2	1	no	—
121	C₂×C₄×C₇₉	—	2	2	no	—
122	C₂×C₄×C₇₉	—	4	1	no	—
123	C₂×C₄×C₇₉	—	4	1	no	—
124	C₂×C₄×C₇₉	—	4	2	no	—
125	C₂×C₄×C₇₉	—	4	2	no	—
126	C₂×C₄×C₇₉	—	4	2	no	—
127	C₂×C₄×C₇₉	—	4	2	no	—
128	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
129	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
130	C₂×C₄×C₇₉	#99	1	1	yes	characteristic
131	C₂×C₄×C₇₉	#99	1	1	yes	characteristic
132	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
133	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
134	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
136	C₈×C₇₉	—	2	2	no	—
137	C₈×C₇₉	—	2	2	no	—
138	C₈×C₇₉	—	4	2	no	—
139	C₂²×C₇₉	—	2	2	no	—
140	C₂²×C₇₉	—	2	1	no	—
141	C₂²×C₇₉	—	2	1	no	—
142	C₂²×C₇₉	—	4	1	no	—
143	C₂²×C₇₉	—	4	1	no	—
144	C₂²×C₇₉	—	4	2	no	—
145	C₂²×C₇₉	—	4	4	no	—
146	C₂²×C₇₉	—	4	2	no	—
147	C₂²×C₇₉	—	4	2	no	—
148	C₂²×C₇₉	—	8	2	no	—
149	C₂²×C₇₉	—	8	2	no	—
150	C₂²×C₇₉	#323	1	1	yes	characteristic
151	C₂²×C₇₉	#323	1	1	yes	characteristic
152	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
153	C₄×C₇₉	—	2	1	no	—
154	C₄×C₇₉	—	2	1	no	—
155	C₄×C₇₉	—	4	2	no	—
156	C₄×C₇₉	#323	1	1	yes	characteristic
157	C₄×C₇₉	#323	1	1	yes	characteristic
158	C₄×C₇₉	C₂×#99	1	1	yes	characteristic
159	C₄×C₇₉	C₂×#100	1	1	yes	characteristic
160	C₂×#1321	—	79	1	no	maximal, Sylow
161	C₂×C₇₉	—	2	1	no	—
162	C₂×C₇₉	—	2	1	no	—
163	C₂×C₇₉	—	4	1	no	—
164	C₂×C₇₉	—	4	1	no	—
165	C₂×C₇₉	—	8	2	no	—
166	C₂×C₇₉	#1321	1	2	yes	complemented
167	C₂×C₇₉	C₂×#323	1	1	yes	characteristic
168	C₂×#408	D₇₉	1	1	yes	characteristic
169	C₂×#349	—	79	2	no	—
170	#1321	—	79	4	no	—
171	C₇₉	C₂×#1321	1	1	yes	characteristic, complemented, Sylow
172	C₂²×#41	Dic₇₉	1	1	yes	characteristic
173	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
174	C₂³×C₈	Dic₇₉	1	1	yes	characteristic
175	C₂×M₃₂	—	158	2	no	—
176	#408	—	2	2	no	—
177	#408	Dic₇₉	1	2	yes	—
178	#408	C₂×D₇₉	1	2	yes	—
179	C₂×C₂²⋊C₈	—	2	2	no	—
180	#349	—	79	4	no	—
181	#349	—	158	2	no	—
182	C₂×#41	—	2	1	no	—
183	C₂×#41	—	2	1	no	—
184	C₂×#41	—	4	1	no	—
185	C₂×#41	—	4	1	no	—
186	C₂×#41	—	4	1	no	—
187	C₂×#41	—	4	1	no	—
188	C₂×#41	C₇₉⋊C₈	1	2	yes	—
189	C₂×#41	C₂×Dic₇₉	1	2	yes	—
190	C₂²×Q₈	C₇₉⋊C₈	1	1	yes	characteristic
191	C₂²×D₄	—	2	1	no	—
192	C₂²×D₄	C₇₉⋊C₈	1	1	yes	characteristic
193	C₂³×C₄	—	2	1	no	—
194	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
195	C₂×M₁₆	—	4	1	no	—
196	C₂×M₁₆	—	4	1	no	—
197	C₂×M₁₆	C₂×Dic₇₉	1	1	yes	characteristic
198	C₂×M₁₆	C₂²⋊D₇₉	1	2	yes	—
199	C₂×M₁₆	C₂²×D₇₉	1	1	yes	characteristic
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	4	1	no	—
203	C₂²×C₈	—	4	1	no	—
204	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
205	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
206	M₃₂	—	316	2	no	—
207	M₃₂	—	158	4	no	—
208	C₂×C₁₆	—	316	2	no	—
209	C₂²⋊C₈	—	2	4	no	—
210	C₂²⋊C₈	—	2	4	no	—
211	C₂⁴	—	2	1	no	—
212	C₂⁴	#11929608	1	1	yes	characteristic
213	#41	—	4	2	no	—
214	#41	—	4	2	no	—
215	#41	—	4	2	no	—
216	#41	—	4	2	no	—
217	#41	—	4	2	no	—
218	#41	—	4	2	no	—
219	#41	—	4	2	no	—
220	#41	—	4	2	no	—
221	C₂×Q₈	—	2	1	no	—
222	C₂×Q₈	—	4	1	no	—
223	C₂×Q₈	—	4	1	no	—
224	C₂×Q₈	C₂×C₇₉⋊C₈	1	2	yes	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂×D₄	—	4	1	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	C₂×C₇₉⋊C₈	1	2	yes	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
245	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
248	C₂²×C₄	#11929608	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
250	M₁₆	—	4	2	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
261	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
262	C₁₆	—	316	4	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	—	4	4	no	—
268	C₂³	—	4	1	no	—
269	C₂³	—	4	1	no	—
270	C₂³	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
271	C₂³	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
272	C₂³	C₂×#11929608	1	1	yes	characteristic
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	4	no	—
278	D₄	—	4	4	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
297	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
298	C₂×C₄	#14579076	1	1	yes	characteristic
299	C₂×C₄	#14579076	1	1	yes	characteristic
300	C₂×C₄	#14579078	1	1	yes	characteristic
301	C₂×C₄	#14579078	1	1	yes	characteristic
302	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	2	no	—
306	C₈	—	4	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	4	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	#19098287	1	1	yes	characteristic
319	C₂²	#19098287	1	1	yes	characteristic
320	C₂²	C₂×C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	4	2	no	—
324	C₄	#19098287	1	1	yes	characteristic
325	C₄	#19098287	1	1	yes	characteristic
326	C₄	C₂×#14579076	1	1	yes	characteristic
327	C₄	C₂×#14579078	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	4	1	no	—
331	C₂	—	4	1	no	—
332	C₂	—	8	2	no	—
333	C₂	#23761338	1	2	yes	direct product factor
334	C₂	C₂×#19098287	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579076
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂×M₁₆	C₂²×D₇₉
Fitting	C₂×#408×C₇₉	C₂
socle	C₂²×C₇₉	#323
prime radical	C₂²×#41×C₇₉	C₄
cosocle	C₂×M₁₆×C₇₉	C₂³
hypercenter	C₂×#408	D₇₉
hypocenter	C₇₉	C₂×#1321
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ C₂×#408

Complements

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

N	H ≅ G/N	type
1. C₂×#19098345	C₂	semidirect
2. #23761338	C₂	direct (N × H)
6. #19098345	C₂²	semidirect
166. C₂×C₇₉	#1321	semidirect
171. C₇₉	C₂×#1321	semidirect
333. C₂	#23761338	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19098345	2
#23761338	2
C₂×#408×C₇₉	2
C₂×#1321	79

Subgroups — C2×#23761338 #34197394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23761338 #34197394