Subgroups — C₂×#23765522 #34442829

Order 20224.

638 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098979	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098835	C₂	1	2	yes	maximal
3	#23765522	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098393	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098379	C₂	1	2	yes	maximal, complemented
6	C₂²×Q₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14579192	—	2	2	no	—
8	C₂×#14579167	C₂²	1	1	yes	characteristic
9	#19098979	—	2	2	no	—
10	#19098979	C₂²	1	4	yes	—
11	C₂×#14579101	C₂²	1	1	yes	characteristic
12	C₂×#14579100	—	2	1	no	—
13	C₂×C₁₆×D₇₉	—	2	1	no	—
14	#19098835	—	2	4	no	—
15	#19098835	C₂²	1	8	yes	—
16	C₂×#14579097	C₂²	1	2	yes	—
17	C₂×#14579080	—	2	2	no	—
18	#19098393	C₂²	1	4	yes	complemented
19	#19098379	C₂²	1	8	yes	complemented
20	C₂²×Q₁₆×C₇₉	C₂²	1	2	yes	—
21	C₂×Q₃₂×C₇₉	—	2	1	no	—
22	C₂×Q₃₂×C₇₉	—	2	1	no	—
23	C₂×Q₃₂×C₇₉	—	2	2	no	—
24	C₂×Q₃₂×C₇₉	C₂²	1	4	yes	complemented
25	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
26	C₂×#11929624	D₄	1	1	yes	characteristic
27	#14579192	—	2	8	no	—
28	#14579192	—	4	4	no	—
29	C₂×#11929614	—	4	2	no	—
30	C₂×#11929611	—	4	2	no	—
31	#14579167	—	2	1	no	—
32	#14579167	—	2	1	no	—
33	#14579167	D₄	1	2	yes	—
34	#14579167	C₂³	1	2	yes	—
35	C₂×#11929608	—	2	2	no	—
36	C₂×#11929608	D₄	1	1	yes	characteristic
37	C₂×#11929607	—	2	1	no	—
38	C₂×C₈×D₇₉	—	2	1	no	—
39	#14579101	—	2	1	no	—
40	#14579101	C₂³	1	2	yes	—
41	#14579100	—	2	4	no	—
42	C₂×C₇₉⋊C₁₆	—	2	1	no	—
43	C₁₆×D₇₉	—	2	4	no	—
44	#14579097	—	2	2	no	—
45	#14579097	C₂³	1	4	yes	—
46	#14579080	—	2	8	no	—
47	C₂²×Q₈×C₇₉	—	2	2	no	—
48	C₂×Q₁₆×C₇₉	—	2	2	no	—
49	C₂×Q₁₆×C₇₉	—	2	2	no	—
50	C₂×Q₁₆×C₇₉	—	2	2	no	—
51	C₂×Q₁₆×C₇₉	—	2	2	no	—
52	C₂×Q₁₆×C₇₉	—	2	2	no	—
53	C₂×Q₁₆×C₇₉	C₂³	1	4	yes	—
54	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
55	Q₃₂×C₇₉	—	2	4	no	—
56	Q₃₂×C₇₉	—	2	4	no	—
57	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
58	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
59	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
60	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
61	#11929624	—	2	1	no	—
62	#11929624	—	2	1	no	—
63	#11929624	D₈	1	2	yes	—
64	#11929624	C₂×D₄	1	2	yes	—
65	C₂×Dic₁₅₈	D₈	1	1	yes	characteristic
66	C₂×D₃₁₆	D₈	1	1	yes	characteristic
67	C₂×C₂²⋊D₇₉	—	2	1	no	—
68	C₂×C₄×D₇₉	—	2	1	no	—
69	#11929614	—	4	8	no	—
70	#11929611	—	4	8	no	—
71	#11929608	—	2	1	no	—
72	#11929608	—	2	4	no	—
73	#11929608	—	4	2	no	—
74	#11929608	C₂×D₄	1	2	yes	—
75	#11929607	—	2	2	no	—
76	#11929607	—	2	2	no	—
77	C₂×C₇₉⋊C₈	—	2	1	no	—
78	C₂×C₇₉⋊C₈	—	4	2	no	—
79	C₈×D₇₉	—	2	2	no	—
80	C₈×D₇₉	—	2	2	no	—
81	C₇₉⋊C₁₆	—	2	2	no	—
82	C₂×Q₈×C₇₉	—	2	4	no	—
83	C₂×Q₈×C₇₉	—	4	2	no	—
84	C₂×Q₈×C₇₉	—	4	2	no	—
85	C₂×Q₈×C₇₉	—	4	2	no	—
86	C₂×Q₈×C₇₉	—	4	2	no	—
87	C₂×Q₈×C₇₉	—	4	2	no	—
88	C₂²×C₄×C₇₉	—	4	2	no	—
89	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
90	Q₁₆×C₇₉	—	2	4	no	—
91	Q₁₆×C₇₉	—	2	4	no	—
92	Q₁₆×C₇₉	—	4	2	no	—
93	Q₁₆×C₇₉	—	4	2	no	—
94	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
98	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
99	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
100	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
101	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
102	C₂³×C₇₉	#135	1	1	yes	characteristic
103	C₂²×D₇₉	—	2	1	no	—
104	Q₈×C₇₉	—	4	4	no	—
105	Q₈×C₇₉	—	4	4	no	—
106	Q₈×C₇₉	—	8	2	no	—
107	Q₈×C₇₉	—	8	2	no	—
108	C₂×C₄×C₇₉	—	4	4	no	—
109	C₂×C₄×C₇₉	—	8	2	no	—
110	C₂×C₄×C₇₉	—	8	2	no	—
111	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
112	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
117	C₂²⋊D₇₉	—	2	2	no	—
118	C₂²⋊D₇₉	—	2	2	no	—
119	C₂×Dic₇₉	—	2	1	no	—
120	D₃₁₆	—	2	1	no	—
121	D₃₁₆	C₂×D₈	1	2	yes	—
122	C₄×D₇₉	—	2	2	no	—
123	C₄×D₇₉	—	2	2	no	—
124	Dic₁₅₈	—	2	1	no	—
125	Dic₁₅₈	C₂×D₈	1	2	yes	—
126	C₈×C₇₉	#121	1	1	yes	characteristic
127	C₈×C₇₉	#121	1	1	yes	characteristic
128	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
129	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
130	C₇₉⋊C₈	—	2	2	no	—
131	C₇₉⋊C₈	—	4	4	no	—
132	C₂²×C₇₉	—	2	2	no	—
133	C₂²×C₇₉	#466	1	1	yes	characteristic
134	C₂²×C₇₉	#466	1	1	yes	characteristic
135	C₂²×C₇₉	C₂×#135	1	1	yes	characteristic
136	C₂×D₇₉	—	2	2	no	—
137	C₂×D₇₉	—	4	2	no	—
138	C₄×C₇₉	—	8	4	no	—
139	C₄×C₇₉	#466	1	1	yes	characteristic
140	C₄×C₇₉	#466	1	1	yes	characteristic
141	C₄×C₇₉	C₂×#121	1	1	yes	characteristic
142	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
143	Dic₇₉	—	2	2	no	—
144	C₂×#2222	—	79	1	no	maximal, Sylow
145	C₂×C₇₉	—	2	1	no	—
146	C₂×C₇₉	—	2	1	no	—
147	C₂×C₇₉	#2222	1	2	yes	complemented
148	C₂×C₇₉	C₂×#466	1	1	yes	characteristic
149	D₇₉	—	4	2	no	—
150	C₂²×Q₃₂	D₇₉	1	1	yes	characteristic, complemented
151	C₂×#504	—	79	1	no	—
152	C₂×#473	—	79	2	no	—
153	#2222	—	79	8	no	—
154	C₂×#368	—	79	1	no	—
155	C₂×#362	—	79	2	no	—
156	C₇₉	C₂×#2222	1	1	yes	characteristic, complemented, Sylow
157	C₂×#137	—	158	2	no	—
158	C₂²×Q₁₆	C₂×D₇₉	1	2	yes	—
159	C₂×#131	—	79	1	no	—
160	C₂×Q₃₂	—	2	1	no	—
161	C₂×Q₃₂	—	2	1	no	—
162	C₂×Q₃₂	—	2	2	no	—
163	C₂×Q₃₂	C₂×D₇₉	1	4	yes	complemented
164	#504	—	79	4	no	—
165	#504	—	158	2	no	—
166	C₂×M₃₂	—	79	1	no	—
167	C₂×M₃₂	—	158	1	no	—
168	C₂²×C₁₆	—	158	1	no	—
169	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
170	#473	—	79	8	no	—
171	#473	—	158	4	no	—
172	C₂×#108	—	79	2	no	—
173	C₂×#101	—	158	2	no	—
174	#368	—	79	4	no	—
175	#362	—	79	8	no	—
176	C₂×#41	—	79	1	no	—
177	C₂²×Q₈	—	2	2	no	—
178	#137	—	316	4	no	—
179	#137	—	158	8	no	—
180	C₂×Q₁₆	—	2	2	no	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	2	2	no	—
185	C₂×Q₁₆	—	316	2	no	—
186	C₂×Q₁₆	C₂²×D₇₉	1	4	yes	—
187	C₂×SD₁₆	—	316	2	no	—
188	#131	—	79	2	no	—
189	#131	—	79	2	no	—
190	#131	—	158	1	no	—
191	#131	—	158	1	no	—
192	C₂×M₁₆	—	79	1	no	—
193	C₂×M₁₆	—	158	2	no	—
194	C₂×M₁₆	—	158	1	no	—
195	C₂²×C₈	—	158	1	no	—
196	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
197	Q₃₂	—	2	4	no	—
198	Q₃₂	—	2	4	no	—
199	M₃₂	—	79	2	no	—
200	M₃₂	—	158	1	no	—
201	M₃₂	—	158	4	no	—
202	C₂×C₁₆	—	158	1	no	—
203	C₂×C₁₆	—	158	4	no	—
204	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
205	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
206	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
207	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
208	#108	—	79	4	no	—
209	#108	—	158	2	no	—
210	#101	—	158	8	no	—
211	#41	—	79	2	no	—
212	#41	—	79	2	no	—
213	#41	—	158	1	no	—
214	#41	—	158	1	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	4	2	no	—
221	C₂×Q₈	—	79	1	no	—
222	C₂×D₄	—	79	1	no	—
223	C₂×D₄	—	158	1	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	158	1	no	—
226	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
227	Q₁₆	—	2	4	no	—
228	Q₁₆	—	2	4	no	—
229	Q₁₆	—	4	2	no	—
230	Q₁₆	—	4	2	no	—
231	Q₁₆	—	316	8	no	—
232	SD₁₆	—	316	8	no	—
233	M₁₆	—	316	2	no	—
234	M₁₆	—	79	2	no	—
235	M₁₆	—	158	4	no	—
236	M₁₆	—	158	1	no	—
237	M₁₆	—	158	2	no	—
238	M₁₆	—	158	2	no	—
239	C₂×C₈	—	316	2	no	—
240	C₂×C₈	—	158	1	no	—
241	C₂×C₈	—	158	2	no	—
242	C₂×C₈	—	158	2	no	—
243	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
244	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
245	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
246	C₂×C₈	#11929623	1	1	yes	characteristic
247	C₂×C₈	#11929623	1	1	yes	characteristic
248	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
249	C₁₆	—	158	2	no	—
250	C₁₆	C₂×C₂²⋊D₇₉	1	2	yes	—
251	C₁₆	C₂×C₂²⋊D₇₉	1	2	yes	—
252	C₂³	—	158	1	no	—
253	C₂³	#14579180	1	1	yes	characteristic
254	Q₈	—	4	4	no	—
255	Q₈	—	4	4	no	—
256	Q₈	—	8	2	no	—
257	Q₈	—	8	2	no	—
258	Q₈	—	79	2	no	—
259	Q₈	—	158	1	no	—
260	D₄	—	79	2	no	—
261	D₄	—	158	2	no	—
262	D₄	—	158	2	no	—
263	D₄	—	158	1	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	—	158	1	no	—
268	C₂×C₄	—	158	2	no	—
269	C₂×C₄	—	158	2	no	—
270	C₂×C₄	#14579132	1	1	yes	characteristic
271	C₂×C₄	#14579132	1	1	yes	characteristic
272	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
273	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
274	C₂×C₄	#14579180	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
276	C₈	—	316	4	no	—
277	C₈	—	158	2	no	—
278	C₈	#14579132	1	1	yes	characteristic
279	C₈	#14579132	1	1	yes	characteristic
280	C₈	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
281	C₈	C₂×#11929623	1	1	yes	characteristic
282	C₂²	—	2	2	no	—
283	C₂²	—	316	2	no	—
284	C₂²	—	158	2	no	—
285	C₂²	#19098796	1	1	yes	characteristic
286	C₂²	#19098796	1	1	yes	characteristic
287	C₂²	C₂×#14579180	1	1	yes	characteristic
288	C₄	—	8	4	no	—
289	C₄	—	158	2	no	—
290	C₄	#19098796	1	1	yes	characteristic
291	C₄	#19098796	1	1	yes	characteristic
292	C₄	C₂×#14579132	1	1	yes	characteristic
293	C₄	C₂×D₈×D₇₉	1	1	yes	characteristic
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	—	316	2	no	—
297	C₂	#23765522	1	2	yes	direct product factor
298	C₂	C₂×#19098796	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098796
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂²×Q₃₂×C₇₉	C₂
socle	C₂²×C₇₉	#466
prime radical	C₂×#11929624	D₄
hypercenter	C₂²×Q₃₂	D₇₉
hypocenter	C₇₉	C₂×#2222
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

Complements

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

N	H ≅ G/N	type
3. #23765522	C₂	direct (N × H)
4. C₂×#19098393	C₂	semidirect
5. C₂×#19098379	C₂	semidirect
6. C₂²×Q₃₂×C₇₉	C₂	semidirect
18. #19098393	C₂²	semidirect
19. #19098379	C₂²	semidirect
24. C₂×Q₃₂×C₇₉	C₂²	semidirect
147. C₂×C₇₉	#2222	semidirect
150. C₂²×Q₃₂	D₇₉	semidirect
156. C₇₉	C₂×#2222	semidirect
163. C₂×Q₃₂	C₂×D₇₉	semidirect
297. C₂	#23765522	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×#19098979	2
C₂×#19098835	2
#23765522	2
C₂×#19098393	2
C₂×#19098379	2
C₂²×Q₃₂×C₇₉	2
C₂×#2222	79

Subgroups — C2×#23765522 #34442829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765522 #34442829