Subgroups — C₂×#23764134 #34291597

Order 20224.

1182 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	C₂²×#14579095	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098462	C₂	1	2	yes	maximal
4	#23764134	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19098325	C₂	1	2	yes	maximal
6	C₂²×#107×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₇₉	—	2	2	no	—
9	C₂²×C₄⋊Dic₇₉	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
11	C₂×#14579095	—	2	4	no	—
12	C₂×#14579095	C₂²	1	4	yes	—
13	C₂×C₈×Dic₇₉	—	2	4	no	—
14	C₂×C₈×Dic₇₉	C₂²	1	4	yes	—
15	#19098462	—	2	4	no	—
16	#19098462	C₂²	1	8	yes	—
17	C₂×C₇₉⋊#95	—	2	2	no	—
18	#19098325	C₂²	1	8	yes	—
19	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
21	C₂×#107×C₇₉	C₄	1	8	yes	complemented
22	C₂×#107×C₇₉	C₂²	1	4	yes	—
23	C₂³×Dic₇₉	—	2	2	no	—
24	C₂³×Dic₇₉	D₄	1	1	yes	characteristic
25	C₂³×Dic₇₉	D₄	1	1	yes	characteristic
26	C₂²×C₇₉⋊C₈	—	2	2	no	—
27	C₂²×C₇₉⋊C₈	—	2	2	no	—
28	C₂²×C₇₉⋊C₈	Q₈	1	4	yes	—
29	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
30	C₂×C₄⋊Dic₇₉	—	2	4	no	—
31	C₂×C₄⋊Dic₇₉	—	2	4	no	—
32	C₂×C₄⋊Dic₇₉	—	2	2	no	—
33	C₂×C₄⋊Dic₇₉	—	2	8	no	—
34	C₂×C₄⋊Dic₇₉	—	4	8	no	—
35	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
36	C₂×C₄×Dic₇₉	—	2	2	no	—
37	C₂×C₄×Dic₇₉	—	2	2	no	—
38	C₂×C₄×Dic₇₉	D₄	1	2	yes	—
39	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
40	#14579095	—	2	8	no	—
41	C₈×Dic₇₉	—	2	8	no	—
42	C₇₉⋊#95	—	2	8	no	—
43	C₂³×C₄×C₇₉	—	2	2	no	—
44	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₇₉	D₄	1	4	yes	—
47	C₂²×C₈×C₇₉	Q₈	1	4	yes	—
48	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
50	C₂×C₄⋊C₄×C₇₉	—	2	4	no	—
51	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	8	yes	—
52	C₂×C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
53	#107×C₇₉	C₂×C₄	1	16	yes	complemented
54	C₂²×Dic₇₉	—	2	2	no	—
55	C₂²×Dic₇₉	—	2	2	no	—
56	C₂²×Dic₇₉	—	2	2	no	—
57	C₂²×Dic₇₉	—	2	2	no	—
58	C₂²×Dic₇₉	—	2	4	no	—
59	C₂²×Dic₇₉	—	4	4	no	—
60	C₂²×Dic₇₉	—	4	4	no	—
61	C₂²×Dic₇₉	—	4	4	no	—
62	C₂²×Dic₇₉	D₈	1	4	yes	—
63	C₂²×Dic₇₉	Q₁₆	1	4	yes	—
64	C₂²×Dic₇₉	C₂×D₄	1	2	yes	—
65	C₂²×Dic₇₉	C₂×D₄	1	2	yes	—
66	C₂×C₇₉⋊C₈	—	2	8	no	—
67	C₂×C₇₉⋊C₈	—	2	2	no	—
68	C₂×C₇₉⋊C₈	—	2	2	no	—
69	C₂×C₇₉⋊C₈	C₂×Q₈	1	4	yes	—
70	C₄⋊Dic₇₉	—	2	4	no	—
71	C₄⋊Dic₇₉	—	2	4	no	—
72	C₄⋊Dic₇₉	—	4	16	no	—
73	C₄⋊Dic₇₉	—	4	4	no	—
74	C₄×Dic₇₉	—	2	4	no	—
75	C₄×Dic₇₉	—	2	2	no	—
76	C₄×Dic₇₉	—	2	2	no	—
77	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	—	2	8	no	—
79	C₂²×C₄×C₇₉	—	2	4	no	—
80	C₂²×C₄×C₇₉	—	4	4	no	—
81	C₂²×C₄×C₇₉	—	4	4	no	—
82	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₇₉	C₂×Q₈	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	C₂×C₈×C₇₉	C₄⋊C₄	1	16	yes	complemented
91	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
93	C₂×C₈×C₇₉	C₂×Q₈	1	4	yes	—
94	C₄⋊C₄×C₇₉	—	2	8	no	—
95	C₄⋊C₄×C₇₉	—	2	4	no	—
96	C₄⋊C₄×C₇₉	C₂²×C₄	1	8	yes	—
97	C₂³×C₇₉	C₄×D₄	1	2	yes	—
98	C₂³×C₇₉	#121	1	2	yes	—
99	C₂³×C₇₉	C₄⋊Q₈	1	2	yes	—
100	C₂³×C₇₉	C₂×D₈	1	2	yes	—
101	C₂³×C₇₉	C₂×Q₁₆	1	2	yes	—
102	C₂³×C₇₉	#135	1	2	yes	—
103	C₂³×C₇₉	#135	1	2	yes	—
104	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₉	—	2	8	no	—
106	C₂×C₄×C₇₉	—	4	4	no	—
107	C₂×C₄×C₇₉	—	4	4	no	—
108	C₂×C₄×C₇₉	—	4	16	no	—
109	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₇₉	C₄×Q₈	1	4	yes	—
112	C₂×C₄×C₇₉	#121	1	2	yes	—
113	C₂×C₄×C₇₉	C₂²⋊Q₈	1	4	yes	—
114	C₂×C₄×C₇₉	#125	1	4	yes	—
115	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
122	C₂×Dic₇₉	—	2	2	no	—
123	C₂×Dic₇₉	—	2	2	no	—
124	C₂×Dic₇₉	—	2	2	no	—
125	C₂×Dic₇₉	—	2	2	no	—
126	C₂×Dic₇₉	—	2	8	no	—
127	C₂×Dic₇₉	—	2	8	no	—
128	C₂×Dic₇₉	—	4	4	no	—
129	C₂×Dic₇₉	—	4	4	no	—
130	C₂×Dic₇₉	—	4	4	no	—
131	C₂×Dic₇₉	—	8	8	no	—
132	C₂×Dic₇₉	C₂×D₈	1	4	yes	—
133	C₂×Dic₇₉	C₂×Q₁₆	1	4	yes	—
134	C₈×C₇₉	C₂×C₄⋊C₄	1	8	yes	complemented
135	C₇₉⋊C₈	—	2	4	no	—
136	C₂²×C₇₉	#389	1	4	yes	—
137	C₂²×C₇₉	C₄×D₈	1	4	yes	—
138	C₂²×C₇₉	C₄×Q₁₆	1	4	yes	—
139	C₂²×C₇₉	#466	1	4	yes	—
140	C₂²×C₇₉	#467	1	4	yes	—
141	C₂²×C₇₉	#499	1	4	yes	—
142	C₂²×C₇₉	#500	1	4	yes	—
143	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂²×C₇₉	C₂×#121	1	1	yes	characteristic
145	C₂²×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
146	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
147	C₂²×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
148	C₂²×C₇₉	C₂×#135	1	1	yes	characteristic
149	C₂²×C₇₉	C₂×#135	1	1	yes	characteristic
150	C₄×C₇₉	—	4	8	no	—
151	C₄×C₇₉	#389	1	4	yes	—
152	C₄×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
154	C₄×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
155	C₄×C₇₉	C₂×#125	1	1	yes	characteristic
156	Dic₇₉	—	2	4	no	—
157	Dic₇₉	—	2	4	no	—
158	Dic₇₉	—	8	4	no	—
159	C₂×#1942	—	79	1	no	maximal, Sylow
160	C₂×C₇₉	#1942	1	8	yes	complemented
161	C₂×C₇₉	C₂×#389	1	1	yes	characteristic
162	C₂×C₇₉	C₂×C₄×D₈	1	1	yes	characteristic
163	C₂×C₇₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
164	C₂×C₇₉	C₂×#466	1	1	yes	characteristic
165	C₂×C₇₉	C₂×#467	1	1	yes	characteristic
166	C₂×C₇₉	C₂×#499	1	1	yes	characteristic
167	C₂×C₇₉	C₂×#500	1	1	yes	characteristic
168	C₂²×#107	—	79	1	no	—
169	C₂²×#107	D₇₉	1	1	yes	characteristic
170	C₂²×C₄×C₈	—	79	1	no	—
171	C₂×#389	—	79	2	no	—
172	#1942	—	79	8	no	—
173	C₂×#340	—	79	2	no	—
174	C₇₉	C₂×#1942	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₈	—	79	1	no	—
176	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
177	C₂²×C₄⋊C₄	—	79	2	no	—
178	C₂²×C₄⋊C₄	—	158	2	no	—
179	C₂²×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
180	C₂²×C₄²	—	79	1	no	—
181	C₂×#107	—	79	4	no	—
182	C₂×#107	—	158	4	no	—
183	C₂×#107	Dic₇₉	1	8	yes	complemented
184	C₂×#107	C₂×D₇₉	1	4	yes	—
185	C₂×C₄×C₈	—	79	4	no	—
186	C₂×C₄×C₈	—	158	4	no	—
187	#389	—	79	8	no	—
188	#389	—	158	4	no	—
189	C₂×#95	—	158	2	no	—
190	#340	—	79	8	no	—
191	C₂³×C₄	—	2	2	no	—
192	C₂³×C₄	—	79	1	no	—
193	C₂³×C₄	—	79	1	no	—
194	C₂³×C₄	—	158	2	no	—
195	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
196	C₂²×C₈	—	79	4	no	—
197	C₂²×C₈	—	79	2	no	—
198	C₂²×C₈	—	158	2	no	—
199	C₂²×C₈	—	158	2	no	—
200	C₂²×C₈	Dic₁₅₈	1	4	yes	—
201	C₂²×C₈	D₃₁₆	1	4	yes	—
202	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
203	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	316	8	no	—
207	C₂×C₄⋊C₄	—	79	4	no	—
208	C₂×C₄⋊C₄	—	158	8	no	—
209	C₂×C₄⋊C₄	—	158	2	no	—
210	C₂×C₄⋊C₄	—	158	4	no	—
211	C₂×C₄⋊C₄	—	158	4	no	—
212	C₂×C₄⋊C₄	C₂×Dic₇₉	1	8	yes	—
213	C₂×C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
214	C₂×C₄²	—	79	2	no	—
215	C₂×C₄²	—	79	2	no	—
216	C₂×C₄²	—	158	2	no	—
217	C₂×C₄²	—	158	2	no	—
218	#107	—	158	8	no	—
219	#107	C₂×Dic₇₉	1	16	yes	complemented
220	C₄×C₈	—	158	8	no	—
221	#95	—	158	8	no	—
222	C₂⁴	D₄×D₇₉	1	1	yes	characteristic
223	C₂²×C₄	—	2	8	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	4	4	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	316	4	no	—
228	C₂²×C₄	—	316	4	no	—
229	C₂²×C₄	—	316	4	no	—
230	C₂²×C₄	—	79	4	no	—
231	C₂²×C₄	—	79	4	no	—
232	C₂²×C₄	—	79	2	no	—
233	C₂²×C₄	—	79	2	no	—
234	C₂²×C₄	—	158	4	no	—
235	C₂²×C₄	—	158	2	no	—
236	C₂²×C₄	—	158	2	no	—
237	C₂²×C₄	—	158	2	no	—
238	C₂²×C₄	—	158	2	no	—
239	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
240	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
242	C₂²×C₄	Q₈×D₇₉	1	2	yes	—
243	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
244	C₂²×C₄	#11929623	1	2	yes	—
245	C₂²×C₄	#11929625	1	2	yes	—
246	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
247	C₂×C₈	—	79	4	no	—
248	C₂×C₈	—	158	2	no	—
249	C₂×C₈	—	158	8	no	—
250	C₂×C₈	—	158	2	no	—
251	C₂×C₈	C₄⋊Dic₇₉	1	16	yes	complemented
252	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
253	C₂×C₈	C₂×D₃₁₆	1	4	yes	—
254	C₂×C₈	C₂×Dic₁₅₈	1	4	yes	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	316	16	no	—
258	C₄⋊C₄	—	316	4	no	—
259	C₄⋊C₄	—	158	4	no	—
260	C₄⋊C₄	—	158	4	no	—
261	C₄⋊C₄	C₂²×Dic₇₉	1	8	yes	—
262	C₄²	—	158	2	no	—
263	C₄²	—	158	4	no	—
264	C₄²	—	158	2	no	—
265	C₂³	D₄×Dic₇₉	1	2	yes	—
266	C₂³	#14579136	1	2	yes	—
267	C₂³	#14579160	1	2	yes	—
268	C₂³	D₈×D₇₉	1	2	yes	—
269	C₂³	Q₁₆×D₇₉	1	2	yes	—
270	C₂³	#14579180	1	2	yes	—
271	C₂³	#14579184	1	2	yes	—
272	C₂³	C₂×D₄×D₇₉	1	1	yes	characteristic
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	4	16	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	316	4	no	—
278	C₂×C₄	—	316	4	no	—
279	C₂×C₄	—	316	4	no	—
280	C₂×C₄	—	79	4	no	—
281	C₂×C₄	—	79	4	no	—
282	C₂×C₄	—	632	8	no	—
283	C₂×C₄	—	158	8	no	—
284	C₂×C₄	—	158	2	no	—
285	C₂×C₄	—	158	2	no	—
286	C₂×C₄	—	158	8	no	—
287	C₂×C₄	—	158	2	no	—
288	C₂×C₄	—	158	2	no	—
289	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
290	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
291	C₂×C₄	Q₈×Dic₇₉	1	4	yes	—
292	C₂×C₄	#14579136	1	2	yes	—
293	C₂×C₄	#14579141	1	4	yes	—
294	C₂×C₄	#14579152	1	4	yes	—
295	C₂×C₄	#14579160	1	2	yes	—
296	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
298	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
299	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
300	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
301	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
302	C₈	—	158	4	no	—
303	C₈	C₂×C₄⋊Dic₇₉	1	8	yes	complemented
304	C₂²	#19098462	1	4	yes	—
305	C₂²	D₈×Dic₇₉	1	4	yes	—
306	C₂²	Q₁₆×Dic₇₉	1	4	yes	—
307	C₂²	#19098800	1	4	yes	—
308	C₂²	#19098805	1	4	yes	—
309	C₂²	#19098964	1	4	yes	—
310	C₂²	#19098967	1	4	yes	—
311	C₂²	C₂×D₄×Dic₇₉	1	1	yes	characteristic
312	C₂²	C₂×#14579136	1	1	yes	characteristic
313	C₂²	C₂×#14579160	1	1	yes	characteristic
314	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
315	C₂²	C₂×Q₁₆×D₇₉	1	1	yes	characteristic
316	C₂²	C₂×#14579180	1	1	yes	characteristic
317	C₂²	C₂×#14579184	1	1	yes	characteristic
318	C₄	—	4	8	no	—
319	C₄	—	632	4	no	—
320	C₄	—	158	4	no	—
321	C₄	—	158	4	no	—
322	C₄	#19098462	1	4	yes	—
323	C₄	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
324	C₄	C₂×Q₈×Dic₇₉	1	1	yes	characteristic
325	C₄	C₂×#14579141	1	1	yes	characteristic
326	C₄	C₂×#14579152	1	1	yes	characteristic
327	C₂	#23764134	1	8	yes	direct product factor
328	C₂	C₂×#19098462	1	1	yes	characteristic
329	C₂	C₂×D₈×Dic₇₉	1	1	yes	characteristic
330	C₂	C₂×Q₁₆×Dic₇₉	1	1	yes	characteristic
331	C₂	C₂×#19098800	1	1	yes	characteristic
332	C₂	C₂×#19098805	1	1	yes	characteristic
333	C₂	C₂×#19098964	1	1	yes	characteristic
334	C₂	C₂×#19098967	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂²×#107×C₇₉	C₂
socle, prime radical	C₂⁴×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂²×#107	D₇₉
hypocenter	C₇₉	C₂×#1942
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
4. #23764134	C₂	direct (N × H)
21. C₂×#107×C₇₉	C₄	semidirect
53. #107×C₇₉	C₂×C₄	semidirect
90. C₂×C₈×C₇₉	C₄⋊C₄	semidirect
134. C₈×C₇₉	C₂×C₄⋊C₄	semidirect
160. C₂×C₇₉	#1942	semidirect
174. C₇₉	C₂×#1942	semidirect
183. C₂×#107	Dic₇₉	semidirect
219. #107	C₂×Dic₇₉	semidirect
251. C₂×C₈	C₄⋊Dic₇₉	semidirect
303. C₈	C₂×C₄⋊Dic₇₉	semidirect
327. C₂	#23764134	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14579095	2
C₂²×C₈×Dic₇₉	2
C₂×#19098462	2
#23764134	2
C₂×#19098325	2
C₂²×#107×C₇₉	2
C₂×#1942	79

Subgroups — C2×#23764134 #34291597

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23764134 #34291597