Subgroups — C₂×#23764974 #34318233

Order 20224.

710 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098586	C₂	1	1	yes	maximal, characteristic
2	#23764974	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19098334	C₂	1	2	yes	maximal
4	C₂×#19098324	C₂	1	2	yes	maximal
5	C₂×#19098320	C₂	1	1	yes	maximal, characteristic
6	C₂×#430×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11929608	C₂²	1	1	yes	characteristic
8	C₂×#14579120	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
10	#19098586	—	2	2	no	—
11	#19098586	C₂²	1	4	yes	—
12	C₂×Dic₇₉⋊C₈	—	2	2	no	—
13	#19098334	C₂²	1	8	yes	—
14	#19098324	C₂²	1	8	yes	—
15	#19098320	C₂²	1	4	yes	—
16	C₂²×M₁₆×C₇₉	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
18	#430×C₇₉	—	2	2	no	—
19	#430×C₇₉	C₂²	1	4	yes	—
20	C₂×#108×C₇₉	—	2	2	no	—
21	C₂³×Dic₇₉	—	2	1	no	—
22	C₂×#11929608	—	2	1	no	—
23	C₂×#11929608	—	2	2	no	—
24	C₂×#11929608	—	2	2	no	—
25	C₂×#11929608	C₂³	1	2	yes	—
26	C₂²×C₇₉⋊C₈	—	2	1	no	—
27	#14579120	—	2	2	no	—
28	#14579120	—	2	2	no	—
29	#14579120	—	2	2	no	—
30	#14579120	C₂³	1	4	yes	—
31	C₂×C₄⋊Dic₇₉	—	2	2	no	—
32	C₂×C₂²⋊Dic₇₉	—	2	2	no	—
33	C₂×C₄×Dic₇₉	—	2	1	no	—
34	C₂×C₄×Dic₇₉	—	2	1	no	—
35	C₂×C₄×Dic₇₉	—	2	1	no	—
36	C₂×C₄×Dic₇₉	—	2	2	no	—
37	C₂×C₄×Dic₇₉	Q₈	1	1	yes	characteristic
38	C₂×C₄×Dic₇₉	Q₈	1	2	yes	—
39	C₂×C₄×Dic₇₉	Q₈	1	1	yes	characteristic
40	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
41	Dic₇₉⋊C₈	—	2	8	no	—
42	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₇₉	—	2	2	no	—
44	C₂×M₁₆×C₇₉	—	2	2	no	—
45	C₂×M₁₆×C₇₉	—	2	2	no	—
46	C₂×M₁₆×C₇₉	—	2	2	no	—
47	C₂×M₁₆×C₇₉	—	2	2	no	—
48	C₂×M₁₆×C₇₉	—	2	2	no	—
49	C₂×M₁₆×C₇₉	—	2	2	no	—
50	C₂×M₁₆×C₇₉	D₄	1	2	yes	—
51	C₂×M₁₆×C₇₉	C₂³	1	4	yes	—
52	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
53	C₂²×C₈×C₇₉	—	2	2	no	—
54	C₂²×C₈×C₇₉	D₄	1	2	yes	—
55	#108×C₇₉	—	2	8	no	—
56	C₂²×Dic₇₉	—	2	1	no	—
57	C₂²×Dic₇₉	—	2	1	no	—
58	C₂²×Dic₇₉	—	2	2	no	—
59	C₂²×Dic₇₉	—	2	2	no	—
60	C₂²×Dic₇₉	—	4	4	no	—
61	C₂²×Dic₇₉	—	4	2	no	—
62	#11929608	—	2	4	no	—
63	#11929608	—	4	2	no	—
64	C₂×C₇₉⋊C₈	—	2	2	no	—
65	C₂×C₇₉⋊C₈	—	4	1	no	—
66	C₂×C₇₉⋊C₈	—	4	1	no	—
67	C₄⋊Dic₇₉	—	2	4	no	—
68	C₄⋊Dic₇₉	—	4	2	no	—
69	C₂²⋊Dic₇₉	—	2	4	no	—
70	C₂²⋊Dic₇₉	—	4	2	no	—
71	C₄×Dic₇₉	—	2	1	no	—
72	C₄×Dic₇₉	—	2	2	no	—
73	C₄×Dic₇₉	—	2	1	no	—
74	C₄×Dic₇₉	—	2	2	no	—
75	C₄×Dic₇₉	—	2	4	no	—
76	C₄×Dic₇₉	—	4	2	no	—
77	C₄×Dic₇₉	C₂×Q₈	1	2	yes	—
78	C₄×Dic₇₉	C₂×Q₈	1	2	yes	—
79	C₂⁴×C₇₉	#41	1	1	yes	characteristic
80	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
85	C₂²×C₄×C₇₉	#41	1	2	yes	—
86	C₂²×C₄×C₇₉	#41	1	2	yes	—
87	C₂²×C₄×C₇₉	#41	1	2	yes	—
88	C₂²×C₄×C₇₉	#41	1	2	yes	—
89	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
90	M₁₆×C₇₉	—	2	4	no	—
91	M₁₆×C₇₉	—	2	4	no	—
92	M₁₆×C₇₉	—	2	4	no	—
93	M₁₆×C₇₉	—	2	4	no	—
94	M₁₆×C₇₉	—	4	2	no	—
95	M₁₆×C₇₉	—	4	2	no	—
96	C₂×C₈×C₇₉	—	2	2	no	—
97	C₂×C₈×C₇₉	—	2	2	no	—
98	C₂×C₈×C₇₉	—	2	4	no	—
99	C₂×C₈×C₇₉	—	4	2	no	—
100	C₂×C₈×C₇₉	—	4	2	no	—
101	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
102	C₂³×C₇₉	—	4	2	no	—
103	C₂³×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂³×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂³×C₇₉	#126	1	2	yes	—
106	C₂³×C₇₉	#126	1	2	yes	—
107	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
111	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
112	C₂×C₄×C₇₉	#123	1	2	yes	—
113	C₂×C₄×C₇₉	#124	1	2	yes	—
114	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
115	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
116	C₂×C₄×C₇₉	#125	1	2	yes	—
117	C₂×C₄×C₇₉	#125	1	2	yes	—
118	C₂×C₄×C₇₉	#125	1	2	yes	—
119	C₂×C₄×C₇₉	#125	1	2	yes	—
120	C₂×C₄×C₇₉	#126	1	2	yes	—
121	C₂×C₄×C₇₉	#126	1	2	yes	—
122	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
125	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
126	C₂×Dic₇₉	—	2	4	no	—
127	C₂×Dic₇₉	—	2	2	no	—
128	C₂×Dic₇₉	—	2	2	no	—
129	C₂×Dic₇₉	—	4	4	no	—
130	C₂×Dic₇₉	—	4	4	no	—
131	C₂×Dic₇₉	—	4	2	no	—
132	C₂×Dic₇₉	—	4	4	no	—
133	C₂×Dic₇₉	—	8	4	no	—
134	C₈×C₇₉	—	2	4	no	—
135	C₈×C₇₉	—	4	4	no	—
136	C₇₉⋊C₈	—	4	2	no	—
137	C₂²×C₇₉	—	2	2	no	—
138	C₂²×C₇₉	—	2	4	no	—
139	C₂²×C₇₉	—	4	1	no	—
140	C₂²×C₇₉	—	4	2	no	—
141	C₂²×C₇₉	—	4	1	no	—
142	C₂²×C₇₉	#400	1	1	yes	characteristic
143	C₂²×C₇₉	#400	1	2	yes	—
144	C₂²×C₇₉	#400	1	1	yes	characteristic
145	C₂²×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
146	C₂²×C₇₉	C₂×#126	1	2	yes	—
147	C₄×C₇₉	#400	1	1	yes	characteristic
148	C₄×C₇₉	#400	1	2	yes	—
149	C₄×C₇₉	#400	1	1	yes	characteristic
150	C₄×C₇₉	C₂×#123	1	1	yes	characteristic
151	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
152	C₄×C₇₉	C₂×#125	1	2	yes	—
153	Dic₇₉	—	2	4	no	—
154	Dic₇₉	—	4	2	no	—
155	Dic₇₉	—	8	4	no	—
156	C₂×#2102	—	79	1	no	maximal, Sylow
157	C₂×C₇₉	—	2	1	no	—
158	C₂×C₇₉	—	2	2	no	—
159	C₂×C₇₉	—	2	1	no	—
160	C₂×C₇₉	—	2	2	no	—
161	C₂×C₇₉	#2102	1	2	yes	complemented
162	C₂×C₇₉	C₂×#400	1	1	yes	characteristic
163	C₂×#430	D₇₉	1	1	yes	characteristic
164	C₂×#423	—	79	1	no	—
165	#2102	—	79	8	no	—
166	C₂×#344	—	79	2	no	—
167	C₂×#339	—	79	2	no	—
168	C₂×H(3,ℤ/4)	—	79	1	no	—
169	C₇₉	C₂×#2102	1	1	yes	characteristic, complemented, Sylow
170	C₂²×M₁₆	—	79	1	no	—
171	C₂²×M₁₆	C₂×D₇₉	1	2	yes	—
172	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
173	C₂×#117	—	79	2	no	—
174	C₂²×C₄²	—	79	1	no	—
175	#430	—	2	2	no	—
176	#430	C₂×D₇₉	1	4	yes	—
177	C₂×#108	—	2	2	no	—
178	#423	—	79	4	no	—
179	#423	—	158	2	no	—
180	C₂×C₄⋊C₈	—	158	2	no	—
181	#344	—	79	8	no	—
182	#339	—	79	8	no	—
183	H(3,ℤ/4)	—	79	4	no	—
184	C₂³×C₄	—	158	1	no	—
185	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	79	2	no	—
194	C₂×M₁₆	—	158	2	no	—
195	C₂×M₁₆	—	158	2	no	—
196	C₂×M₁₆	—	158	1	no	—
197	C₂×M₁₆	C₂²⋊D₇₉	1	2	yes	—
198	C₂×M₁₆	C₂²×D₇₉	1	4	yes	—
199	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	158	1	no	—
202	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
203	#117	—	79	4	no	—
204	#117	—	158	2	no	—
205	#117	—	158	2	no	—
206	#117	—	158	2	no	—
207	C₂×C₄⋊C₄	—	158	2	no	—
208	C₂×C₂²⋊C₄	—	158	2	no	—
209	C₂×C₄²	—	79	1	no	—
210	C₂×C₄²	—	79	2	no	—
211	C₂×C₄²	—	79	1	no	—
212	C₂×C₄²	—	79	2	no	—
213	C₂×C₄²	—	158	2	no	—
214	C₂×C₄²	—	158	1	no	—
215	C₂×C₄²	—	158	1	no	—
216	C₂×C₄²	—	158	1	no	—
217	#108	—	2	8	no	—
218	C₄⋊C₈	—	158	8	no	—
219	C₂⁴	#11929625	1	1	yes	characteristic
220	C₂²×C₄	—	316	4	no	—
221	C₂²×C₄	—	316	2	no	—
222	C₂²×C₄	—	158	2	no	—
223	C₂²×C₄	—	158	1	no	—
224	C₂²×C₄	—	158	2	no	—
225	C₂²×C₄	—	158	1	no	—
226	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
228	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
229	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
230	C₂²×C₄	#11929623	1	2	yes	—
231	C₂²×C₄	#11929623	1	2	yes	—
232	C₂²×C₄	#11929624	1	2	yes	—
233	C₂²×C₄	#11929624	1	2	yes	—
234	C₂²×C₄	#11929625	1	1	yes	characteristic
235	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	4	2	no	—
241	M₁₆	—	4	2	no	—
242	M₁₆	—	316	2	no	—
243	M₁₆	—	158	4	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	316	1	no	—
250	C₂×C₈	—	316	1	no	—
251	C₂×C₈	—	158	2	no	—
252	C₂×C₈	C₂×C₂²⋊D₇₉	1	4	yes	—
253	C₄⋊C₄	—	316	2	no	—
254	C₄⋊C₄	—	158	4	no	—
255	C₂²⋊C₄	—	316	2	no	—
256	C₂²⋊C₄	—	158	4	no	—
257	C₄²	—	316	2	no	—
258	C₄²	—	79	2	no	—
259	C₄²	—	79	2	no	—
260	C₄²	—	158	4	no	—
261	C₄²	—	158	1	no	—
262	C₄²	—	158	2	no	—
263	C₄²	—	158	1	no	—
264	C₄²	—	158	2	no	—
265	C₂³	—	4	2	no	—
266	C₂³	#14579137	1	1	yes	characteristic
267	C₂³	#14579137	1	1	yes	characteristic
268	C₂³	#14579156	1	2	yes	—
269	C₂³	#14579156	1	2	yes	—
270	C₂³	C₂×#11929625	1	1	yes	characteristic
271	C₂×C₄	—	316	2	no	—
272	C₂×C₄	—	316	4	no	—
273	C₂×C₄	—	316	4	no	—
274	C₂×C₄	—	316	4	no	—
275	C₂×C₄	—	632	4	no	—
276	C₂×C₄	—	158	2	no	—
277	C₂×C₄	—	158	4	no	—
278	C₂×C₄	—	158	2	no	—
279	C₂×C₄	#14579137	1	1	yes	characteristic
280	C₂×C₄	#14579137	1	1	yes	characteristic
281	C₂×C₄	#14579144	1	1	yes	characteristic
282	C₂×C₄	#14579144	1	1	yes	characteristic
283	C₂×C₄	#14579144	1	2	yes	—
284	C₂×C₄	#14579148	1	2	yes	—
285	C₂×C₄	#14579148	1	1	yes	characteristic
286	C₂×C₄	#14579148	1	1	yes	characteristic
287	C₂×C₄	#14579151	1	2	yes	—
288	C₂×C₄	#14579151	1	2	yes	—
289	C₂×C₄	#14579151	1	2	yes	—
290	C₂×C₄	#14579151	1	2	yes	—
291	C₂×C₄	#14579156	1	2	yes	—
292	C₂×C₄	#14579156	1	2	yes	—
293	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
295	C₂×C₄	C₂×#11929623	1	2	yes	—
296	C₂×C₄	C₂×#11929624	1	2	yes	—
297	C₈	—	2	4	no	—
298	C₈	—	4	4	no	—
299	C₈	—	316	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	#19098501	1	1	yes	characteristic
306	C₂²	#19098501	1	2	yes	—
307	C₂²	#19098501	1	1	yes	characteristic
308	C₂²	C₂×#14579137	1	1	yes	characteristic
309	C₂²	C₂×#14579156	1	2	yes	—
310	C₄	—	316	2	no	—
311	C₄	—	632	4	no	—
312	C₄	—	158	4	no	—
313	C₄	#19098501	1	1	yes	characteristic
314	C₄	#19098501	1	2	yes	—
315	C₄	#19098501	1	1	yes	characteristic
316	C₄	C₂×#14579144	1	1	yes	characteristic
317	C₄	C₂×#14579148	1	1	yes	characteristic
318	C₄	C₂×#14579151	1	2	yes	—
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	2	no	—
323	C₂	#23764974	1	2	yes	direct product factor
324	C₂	C₂×#19098501	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579148
commutator, cosocle	C₂²×C₄×C₇₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×#430×C₇₉	C₂
socle	C₂²×C₇₉	#400
prime radical	C₂⁴×C₇₉	#41
hypercenter	C₂×#430	D₇₉
hypocenter	C₇₉	C₂×#2102
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
2. #23764974	C₂	direct (N × H)
161. C₂×C₇₉	#2102	semidirect
169. C₇₉	C₂×#2102	semidirect
323. C₂	#23764974	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×#19098586	2
#23764974	2
C₂×#19098334	2
C₂×#19098324	2
C₂×#19098320	2
C₂×#430×C₇₉	2
C₂×#2102	79

Subgroups — C2×#23764974 #34318233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23764974 #34318233