Subgroups — C₂×#23774059 #35158113

Order 20224.

1842 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099175	C₂	1	2	yes	maximal
2	#23774059	C₂	1	16	yes	maximal, direct product factor
3	C₂×#19098965	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19098943	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098932	C₂	1	2	yes	maximal, complemented
6	C₂×#19098821	C₂	1	2	yes	maximal
7	C₂×#19098804	C₂	1	2	yes	maximal
8	C₂×#19098671	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Dic₃₁₆	C₂	1	2	yes	maximal, complemented
10	C₂×#405×C₇₉	C₂	1	1	yes	maximal, characteristic
11	C₂²×Dic₃₁₆	C₂²	1	2	yes	—
12	C₂²×Dic₃₁₆	C₂²	1	2	yes	—
13	#19099175	—	2	8	no	—
14	#19099175	C₂²	1	16	yes	—
15	C₂×#14579161	C₂²	1	2	yes	—
16	C₂×#14579154	—	2	4	no	—
17	C₂×#14579153	C₂²	1	2	yes	—
18	C₂×#14579151	—	2	4	no	—
19	C₂×#14579140	C₂²	1	4	yes	—
20	C₂×C₄×Dic₁₅₈	C₂²	1	4	yes	—
21	#19098965	C₂²	1	8	yes	complemented
22	#19098943	C₂²	1	8	yes	complemented
23	#19098932	C₂²	1	16	yes	complemented
24	#19098821	C₂²	1	16	yes	—
25	#19098804	C₂²	1	16	yes	—
26	#19098671	C₂²	1	16	yes	complemented
27	C₄×Dic₃₁₆	C₂²	1	16	yes	complemented
28	C₂×#14579095	C₂²	1	2	yes	—
29	C₂×#14579093	C₂²	1	2	yes	—
30	C₂×#14579086	C₂²	1	4	yes	—
31	C₂×#14579086	C₂²	1	4	yes	—
32	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
33	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
34	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
35	#405×C₇₉	C₂²	1	8	yes	—
36	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
37	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
40	C₂²×Dic₁₅₈	C₂³	1	4	yes	—
41	C₂×Dic₃₁₆	—	2	8	no	—
42	C₂×Dic₃₁₆	—	2	8	no	—
43	C₂×Dic₃₁₆	C₂³	1	8	yes	—
44	C₂×Dic₃₁₆	C₂³	1	8	yes	—
45	#14579161	—	2	4	no	—
46	#14579161	C₂³	1	8	yes	—
47	#14579154	—	2	32	no	—
48	#14579153	—	2	4	no	—
49	#14579153	C₂³	1	8	yes	—
50	#14579151	—	2	32	no	—
51	#14579140	—	2	8	no	—
52	#14579140	C₂³	1	16	yes	—
53	C₄×Dic₁₅₈	—	2	8	no	—
54	C₄×Dic₁₅₈	C₂³	1	16	yes	—
55	C₂×C₄⋊Dic₇₉	—	2	4	no	—
56	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
57	C₂×Dic₇₉⋊C₄	—	2	4	no	—
58	C₂×Dic₇₉⋊C₄	—	2	4	no	—
59	C₂×Dic₇₉⋊C₄	—	2	4	no	—
60	C₂×C₂²⋊Dic₇₉	—	2	4	no	—
61	C₂×C₄×Dic₇₉	—	2	4	no	—
62	#14579095	C₂³	1	8	yes	—
63	#14579093	C₂³	1	8	yes	—
64	#14579086	C₂³	1	16	yes	—
65	#14579086	C₂³	1	16	yes	—
66	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₇₉	—	2	4	no	—
68	C₂×M₁₆×C₇₉	C₂³	1	4	yes	—
69	C₂²×C₈×C₇₉	—	2	4	no	—
70	C₂²×C₈×C₇₉	C₂³	1	4	yes	—
71	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
75	#117×C₇₉	D₄	1	4	yes	—
76	#117×C₇₉	C₂³	1	4	yes	—
77	C₂×C₄⋊C₄×C₇₉	D₄	1	2	yes	—
78	C₂×C₂²⋊C₄×C₇₉	D₄	1	2	yes	—
79	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
81	#97×C₇₉	C₂³	1	4	yes	—
82	#97×C₇₉	C₂³	1	4	yes	—
83	C₄×C₈×C₇₉	C₂³	1	4	yes	—
84	C₄×C₈×C₇₉	C₂³	1	4	yes	—
85	C₂×Dic₁₅₈	—	2	4	no	—
86	C₂×Dic₁₅₈	—	2	8	no	—
87	C₂×Dic₁₅₈	—	2	8	no	—
88	C₂×Dic₁₅₈	C₂⁴	1	8	yes	—
89	C₂²×Dic₇₉	—	2	4	no	—
90	C₂²×Dic₇₉	—	2	4	no	—
91	Dic₃₁₆	—	2	16	no	—
92	Dic₃₁₆	—	2	16	no	—
93	C₄⋊Dic₇₉	—	2	16	no	—
94	C₄⋊Dic₇₉	—	2	4	no	—
95	C₄⋊Dic₇₉	C₂⁴	1	8	yes	—
96	Dic₇₉⋊C₄	—	2	16	no	—
97	Dic₇₉⋊C₄	—	2	16	no	—
98	Dic₇₉⋊C₄	—	2	16	no	—
99	C₂²⋊Dic₇₉	—	2	16	no	—
100	C₄×Dic₇₉	—	2	16	no	—
101	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₇₉	—	2	2	no	—
103	C₂²×C₄×C₇₉	—	2	2	no	—
104	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₇₉	C₂⁴	1	2	yes	—
109	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
110	M₁₆×C₇₉	—	2	8	no	—
111	C₂×C₈×C₇₉	—	2	8	no	—
112	C₂×C₈×C₇₉	—	2	2	no	—
113	C₂×C₈×C₇₉	—	2	2	no	—
114	C₂×C₈×C₇₉	#41	1	4	yes	—
115	C₂×C₈×C₇₉	#41	1	4	yes	—
116	C₂×C₈×C₇₉	C₂⁴	1	2	yes	—
117	C₂×C₈×C₇₉	C₂⁴	1	2	yes	—
118	C₂×C₈×C₇₉	C₂⁴	1	2	yes	—
119	C₂×C₈×C₇₉	C₂⁴	1	2	yes	—
120	C₄⋊C₄×C₇₉	C₂×D₄	1	8	yes	—
121	C₂²⋊C₄×C₇₉	C₂×D₄	1	8	yes	—
122	C₄²×C₇₉	C₂×D₄	1	2	yes	—
123	C₄²×C₇₉	C₂×D₄	1	2	yes	—
124	C₄²×C₇₉	C₂⁴	1	2	yes	—
125	C₄²×C₇₉	C₂⁴	1	2	yes	—
126	C₂³×C₇₉	—	2	2	no	—
127	C₂³×C₇₉	—	2	4	no	—
128	C₂³×C₇₉	C₂²×D₄	1	2	yes	—
129	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇₉	—	2	2	no	—
131	C₂×C₄×C₇₉	—	2	4	no	—
132	C₂×C₄×C₇₉	—	2	4	no	—
133	C₂×C₄×C₇₉	—	2	4	no	—
134	C₂×C₄×C₇₉	—	2	2	no	—
135	C₂×C₄×C₇₉	—	2	2	no	—
136	C₂×C₄×C₇₉	C₂²×D₄	1	4	yes	—
137	C₂×C₄×C₇₉	C₂²×D₄	1	2	yes	—
138	C₂×C₄×C₇₉	C₂²×D₄	1	4	yes	—
139	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
141	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
142	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
143	C₂×Dic₇₉	—	2	8	no	—
144	C₂×Dic₇₉	—	2	8	no	—
145	C₂×Dic₇₉	—	4	8	no	—
146	C₂×Dic₇₉	—	4	8	no	—
147	Dic₁₅₈	—	2	8	no	—
148	Dic₁₅₈	—	2	8	no	—
149	Dic₁₅₈	—	4	8	no	—
150	C₈×C₇₉	—	2	2	no	—
151	C₈×C₇₉	—	2	2	no	—
152	C₈×C₇₉	C₂×#41	1	4	yes	—
153	C₈×C₇₉	C₂×#41	1	4	yes	—
154	C₂²×C₇₉	—	2	2	no	—
155	C₂²×C₇₉	—	2	4	no	—
156	C₂²×C₇₉	—	4	4	no	—
157	C₂²×C₇₉	#532	1	2	yes	—
158	C₂²×C₇₉	#578	1	4	yes	—
159	C₂²×C₇₉	C₂³×D₄	1	1	yes	characteristic
160	C₄×C₇₉	—	2	2	no	—
161	C₄×C₇₉	—	2	4	no	—
162	C₄×C₇₉	—	2	4	no	—
163	C₄×C₇₉	#532	1	2	yes	—
164	C₄×C₇₉	C₂²×#41	1	1	yes	characteristic
165	C₄×C₇₉	C₂²×#41	1	1	yes	characteristic
166	Dic₇₉	—	4	8	no	—
167	Dic₇₉	—	4	8	no	—
168	C₂×#3177	—	79	1	no	maximal, Sylow
169	C₂×C₇₉	—	4	2	no	—
170	C₂×C₇₉	#3177	1	4	yes	complemented
171	C₂×C₇₉	C₂×#532	1	1	yes	characteristic
172	C₂×C₇₉	C₂×#578	1	2	yes	—
173	C₂×#541	—	79	2	no	—
174	#3177	—	79	16	no	—
175	C₂×#499	—	79	1	no	—
176	C₂×#494	—	79	1	no	—
177	C₂×#491	—	79	2	no	—
178	C₂×#470	—	79	2	no	—
179	C₂×#467	—	79	2	no	—
180	C₂×#441	—	79	2	no	—
181	C₂×C₄×Q₁₆	—	79	2	no	—
182	C₂×#405	D₇₉	1	1	yes	characteristic
183	C₇₉	C₂×#3177	1	1	yes	characteristic, complemented, Sylow
184	C₂²×Q₁₆	—	79	2	no	—
185	C₂²×Q₁₆	—	79	2	no	—
186	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
187	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
188	#541	—	79	16	no	—
189	#541	—	158	8	no	—
190	C₂×C₄⋊Q₈	—	79	2	no	—
191	C₂×#126	—	158	4	no	—
192	C₂×#125	—	79	2	no	—
193	C₂×#125	—	158	4	no	—
194	C₂×C₂²⋊Q₈	—	79	4	no	—
195	C₂×C₄×Q₈	—	79	4	no	—
196	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
197	#499	—	79	8	no	—
198	#494	—	79	8	no	—
199	#491	—	79	16	no	—
200	#470	—	79	16	no	—
201	#467	—	79	16	no	—
202	#441	—	79	16	no	—
203	C₄×Q₁₆	—	79	16	no	—
204	C₂×#107	—	79	2	no	—
205	C₂×#106	—	79	2	no	—
206	C₂×#103	—	79	4	no	—
207	C₂×#103	—	79	4	no	—
208	#405	C₂×D₇₉	1	8	yes	—
209	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
210	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
211	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
212	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
213	C₂²×Q₈	—	79	4	no	—
214	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
215	C₂×Q₁₆	—	79	8	no	—
216	C₂×Q₁₆	—	79	8	no	—
217	C₂×Q₁₆	—	158	8	no	—
218	C₂×Q₁₆	—	158	8	no	—
219	C₂×M₁₆	—	2	4	no	—
220	C₂×M₁₆	C₂²×D₇₉	1	4	yes	—
221	C₂²×C₈	—	2	4	no	—
222	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
223	C₂²×C₈	C₂²×D₇₉	1	4	yes	—
224	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
226	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
227	C₄⋊Q₈	—	79	8	no	—
228	C₄⋊Q₈	—	158	4	no	—
229	#126	—	158	32	no	—
230	#125	—	79	8	no	—
231	#125	—	158	32	no	—
232	#125	—	158	4	no	—
233	C₂²⋊Q₈	—	79	16	no	—
234	C₂²⋊Q₈	—	158	8	no	—
235	C₄×Q₈	—	79	16	no	—
236	C₄×Q₈	—	158	8	no	—
237	#117	D₃₁₆	1	4	yes	—
238	#117	C₂²×D₇₉	1	4	yes	—
239	C₂×C₄⋊C₄	—	79	4	no	—
240	C₂×C₄⋊C₄	—	158	4	no	—
241	C₂×C₄⋊C₄	—	158	4	no	—
242	C₂×C₄⋊C₄	—	158	4	no	—
243	C₂×C₄⋊C₄	—	158	4	no	—
244	C₂×C₄⋊C₄	D₃₁₆	1	2	yes	—
245	C₂×C₂²⋊C₄	—	158	4	no	—
246	C₂×C₂²⋊C₄	D₃₁₆	1	2	yes	—
247	C₂×C₄²	—	158	4	no	—
248	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
249	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
250	#107	—	79	8	no	—
251	#106	—	79	8	no	—
252	#103	—	79	16	no	—
253	#103	—	79	16	no	—
254	#97	C₂²×D₇₉	1	4	yes	—
255	#97	C₂²×D₇₉	1	4	yes	—
256	C₄×C₈	C₂²×D₇₉	1	4	yes	—
257	C₄×C₈	C₂²×D₇₉	1	4	yes	—
258	C₂⁴	C₂×D₃₁₆	1	1	yes	characteristic
259	C₂×Q₈	—	79	8	no	—
260	C₂×Q₈	—	158	8	no	—
261	C₂×Q₈	—	158	8	no	—
262	C₂×Q₈	—	158	4	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	158	4	no	—
266	C₂²×C₄	—	158	4	no	—
267	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
268	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
269	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
270	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
271	C₂²×C₄	C₂³×D₇₉	1	2	yes	—
272	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
273	Q₁₆	—	158	16	no	—
274	Q₁₆	—	158	16	no	—
275	M₁₆	—	2	8	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	8	no	—
279	C₂×C₈	#11929624	1	4	yes	—
280	C₂×C₈	#11929624	1	4	yes	—
281	C₂×C₈	C₂³×D₇₉	1	2	yes	—
282	C₂×C₈	C₂³×D₇₉	1	2	yes	—
283	C₂×C₈	C₂³×D₇₉	1	2	yes	—
284	C₂×C₈	C₂³×D₇₉	1	2	yes	—
285	C₄⋊C₄	—	79	8	no	—
286	C₄⋊C₄	—	158	16	no	—
287	C₄⋊C₄	—	158	16	no	—
288	C₄⋊C₄	—	158	16	no	—
289	C₄⋊C₄	—	158	4	no	—
290	C₄⋊C₄	—	158	16	no	—
291	C₄⋊C₄	C₂×D₃₁₆	1	8	yes	—
292	C₂²⋊C₄	—	158	16	no	—
293	C₂²⋊C₄	C₂×D₃₁₆	1	8	yes	—
294	C₄²	—	158	16	no	—
295	C₄²	C₂×D₃₁₆	1	2	yes	—
296	C₄²	C₂×D₃₁₆	1	2	yes	—
297	C₄²	C₂³×D₇₉	1	2	yes	—
298	C₄²	C₂³×D₇₉	1	2	yes	—
299	C₂³	—	2	2	no	—
300	C₂³	—	2	4	no	—
301	C₂³	C₂²×D₃₁₆	1	2	yes	—
302	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
303	Q₈	—	316	8	no	—
304	Q₈	—	158	8	no	—
305	Q₈	—	158	8	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	316	8	no	—
313	C₂×C₄	—	316	8	no	—
314	C₂×C₄	—	158	8	no	—
315	C₂×C₄	—	158	8	no	—
316	C₂×C₄	C₂²×D₃₁₆	1	4	yes	—
317	C₂×C₄	C₂²×D₃₁₆	1	4	yes	—
318	C₂×C₄	C₂²×D₃₁₆	1	2	yes	—
319	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
320	C₂×C₄	C₂×#11929624	1	2	yes	—
321	C₂×C₄	C₂×#11929624	1	2	yes	—
322	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
323	C₈	—	2	2	no	—
324	C₈	—	2	2	no	—
325	C₈	C₂×#11929624	1	4	yes	—
326	C₈	C₂×#11929624	1	4	yes	—
327	C₂²	—	2	4	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	4	4	no	—
330	C₂²	#19099113	1	2	yes	—
331	C₂²	#19099380	1	4	yes	—
332	C₂²	C₂³×D₃₁₆	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	2	4	no	—
335	C₄	—	2	2	no	—
336	C₄	—	316	8	no	—
337	C₄	—	316	8	no	—
338	C₄	#19099113	1	2	yes	—
339	C₄	C₂²×#11929624	1	1	yes	characteristic
340	C₄	C₂²×#11929624	1	1	yes	characteristic
341	C₂	—	4	2	no	—
342	C₂	#23774059	1	4	yes	direct product factor
343	C₂	C₂×#19099113	1	1	yes	characteristic
344	C₂	C₂×#19099380	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₉
Fitting	C₂×#405×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×D₄
prime radical	C₂⁴×C₇₉	C₂×D₄
hypercenter	C₂×#405	D₇₉
hypocenter	C₇₉	C₂×#3177
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #23774059	C₂	direct (N × H)
3. C₂×#19098965	C₂	semidirect
4. C₂×#19098943	C₂	semidirect
5. C₂×#19098932	C₂	semidirect
8. C₂×#19098671	C₂	semidirect
9. C₂×C₄×Dic₃₁₆	C₂	semidirect
21. #19098965	C₂²	semidirect
22. #19098943	C₂²	semidirect
23. #19098932	C₂²	semidirect
26. #19098671	C₂²	semidirect
27. C₄×Dic₃₁₆	C₂²	semidirect
170. C₂×C₇₉	#3177	semidirect
183. C₇₉	C₂×#3177	semidirect
342. C₂	#23774059	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×#19099175	2
#23774059	2
C₂×#19098965	2
C₂×#19098943	2
C₂×#19098932	2
C₂×#19098821	2
C₂×#19098804	2
C₂×#19098671	2
C₂×C₄×Dic₃₁₆	2
C₂×#405×C₇₉	2
C₂×#3177	79

Subgroups — C2×#23774059 #35158113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23774059 #35158113