Subgroups — C₂×#23765524 #34442831

Order 20224.

638 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579107	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19098981	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098839	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098835	C₂	1	1	yes	maximal, characteristic
5	#23765524	C₂	1	8	yes	maximal, direct product factor
6	C₂×#19098392	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19098380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#362×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14579194	—	2	1	no	—
10	C₂×#14579192	—	2	1	no	—
11	C₂²×Dic₃₁₆	C₂²	1	1	yes	characteristic
12	C₂×#14579167	C₂²	1	1	yes	characteristic
13	C₂×#14579107	—	2	1	no	—
14	C₂×#14579107	—	2	1	no	—
15	C₂×#14579107	—	2	2	no	—
16	C₂×#14579107	C₂²	1	4	yes	complemented
17	#19098981	—	2	2	no	—
18	#19098981	C₂²	1	4	yes	—
19	C₂×#14579100	—	2	1	no	—
20	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₇₉	—	2	1	no	—
22	#19098839	—	2	2	no	—
23	#19098839	C₂²	1	4	yes	—
24	#19098835	—	2	2	no	—
25	#19098835	C₂²	1	4	yes	—
26	C₂×#14579097	C₂²	1	1	yes	characteristic
27	C₂×#14579080	—	2	1	no	—
28	C₂×#14579079	—	2	1	no	—
29	#19098392	C₂²	1	4	yes	complemented
30	#19098380	C₂²	1	4	yes	complemented
31	C₂²×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
32	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
33	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
34	#362×C₇₉	C₂²	1	4	yes	complemented
35	C₂×#11929624	D₄	1	1	yes	characteristic
36	C₂²×Dic₁₅₈	—	2	1	no	—
37	#14579194	—	2	4	no	—
38	#14579194	—	4	2	no	—
39	#14579192	—	2	4	no	—
40	#14579192	—	4	2	no	—
41	C₂×Dic₃₁₆	—	2	1	no	—
42	C₂×Dic₃₁₆	—	2	1	no	—
43	C₂×Dic₃₁₆	—	2	1	no	—
44	C₂×Dic₃₁₆	—	2	1	no	—
45	C₂×Dic₃₁₆	—	2	1	no	—
46	C₂×Dic₃₁₆	—	4	1	no	—
47	C₂×Dic₃₁₆	C₂³	1	2	yes	—
48	C₂×#11929614	—	4	1	no	—
49	C₂×#11929613	—	4	1	no	—
50	C₂×#11929611	—	4	1	no	—
51	#14579167	—	2	1	no	—
52	#14579167	—	2	1	no	—
53	#14579167	D₄	1	2	yes	—
54	#14579167	C₂³	1	2	yes	—
55	C₂×#11929608	—	2	1	no	—
56	C₂×#11929608	D₄	1	1	yes	characteristic
57	C₂×#11929607	—	2	1	no	—
58	C₂×C₈×D₇₉	—	2	1	no	—
59	#14579107	—	2	4	no	—
60	#14579107	—	2	4	no	—
61	#14579100	—	2	4	no	—
62	C₂×C₇₉⋊C₁₆	D₄	1	1	yes	characteristic
63	C₂×C₇₉⋊C₁₆	D₄	1	1	yes	characteristic
64	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
65	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
66	C₁₆×D₇₉	—	2	4	no	—
67	#14579097	—	2	1	no	—
68	#14579097	C₂³	1	2	yes	—
69	#14579080	—	2	4	no	—
70	#14579079	—	2	4	no	—
71	C₂²×Q₈×C₇₉	—	2	1	no	—
72	C₂×Q₁₆×C₇₉	—	2	1	no	—
73	C₂×Q₁₆×C₇₉	—	2	1	no	—
74	C₂×Q₁₆×C₇₉	—	2	1	no	—
75	C₂×Q₁₆×C₇₉	—	2	1	no	—
76	C₂×Q₁₆×C₇₉	—	2	1	no	—
77	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
78	C₂×M₁₆×C₇₉	—	2	1	no	—
79	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
80	M₃₂×C₇₉	—	2	1	no	—
81	M₃₂×C₇₉	C₂³	1	2	yes	—
82	C₂×C₁₆×C₇₉	—	2	1	no	—
83	#108×C₇₉	—	2	1	no	—
84	#108×C₇₉	C₂³	1	2	yes	—
85	#11929624	—	2	1	no	—
86	#11929624	—	2	1	no	—
87	#11929624	D₈	1	2	yes	—
88	#11929624	C₂×D₄	1	2	yes	—
89	C₂×Dic₁₅₈	—	2	2	no	—
90	C₂×Dic₁₅₈	—	4	1	no	—
91	C₂×Dic₁₅₈	—	4	1	no	—
92	C₂×Dic₁₅₈	—	4	1	no	—
93	C₂×Dic₁₅₈	—	4	1	no	—
94	C₂×Dic₁₅₈	—	4	1	no	—
95	C₂×Dic₁₅₈	D₈	1	1	yes	characteristic
96	C₂×D₃₁₆	D₈	1	1	yes	characteristic
97	C₂×C₂²⋊D₇₉	—	2	1	no	—
98	C₂²×Dic₇₉	—	4	1	no	—
99	C₂×C₄×D₇₉	—	2	1	no	—
100	Dic₃₁₆	—	2	2	no	—
101	Dic₃₁₆	—	2	2	no	—
102	Dic₃₁₆	—	4	4	no	—
103	Dic₃₁₆	—	4	1	no	—
104	Dic₃₁₆	—	4	1	no	—
105	#11929614	—	4	4	no	—
106	#11929613	—	4	4	no	—
107	#11929611	—	4	4	no	—
108	#11929608	—	2	1	no	—
109	#11929608	—	2	2	no	—
110	#11929608	—	4	1	no	—
111	#11929608	C₂×D₄	1	2	yes	—
112	#11929607	—	2	2	no	—
113	#11929607	—	2	2	no	—
114	C₂×C₇₉⋊C₈	—	2	1	no	—
115	C₂×C₇₉⋊C₈	—	4	1	no	—
116	C₈×D₇₉	—	2	2	no	—
117	C₈×D₇₉	—	2	2	no	—
118	C₇₉⋊C₁₆	C₂×D₄	1	2	yes	—
119	C₇₉⋊C₁₆	C₂×D₄	1	2	yes	—
120	C₂×Q₈×C₇₉	—	2	2	no	—
121	C₂×Q₈×C₇₉	—	4	1	no	—
122	C₂×Q₈×C₇₉	—	4	1	no	—
123	C₂×Q₈×C₇₉	—	4	1	no	—
124	C₂×Q₈×C₇₉	—	4	1	no	—
125	C₂×Q₈×C₇₉	—	4	1	no	—
126	C₂²×C₄×C₇₉	—	4	1	no	—
127	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
128	Q₁₆×C₇₉	—	2	2	no	—
129	Q₁₆×C₇₉	—	2	2	no	—
130	Q₁₆×C₇₉	—	4	1	no	—
131	Q₁₆×C₇₉	—	4	1	no	—
132	M₁₆×C₇₉	—	2	2	no	—
133	M₁₆×C₇₉	—	4	1	no	—
134	C₂×C₈×C₇₉	—	4	1	no	—
135	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
139	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
140	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
141	C₁₆×C₇₉	—	2	2	no	—
142	C₂³×C₇₉	#135	1	1	yes	characteristic
143	C₂²×D₇₉	—	2	1	no	—
144	Q₈×C₇₉	—	4	2	no	—
145	Q₈×C₇₉	—	4	2	no	—
146	Q₈×C₇₉	—	8	1	no	—
147	Q₈×C₇₉	—	8	1	no	—
148	C₂×C₄×C₇₉	—	4	2	no	—
149	C₂×C₄×C₇₉	—	8	1	no	—
150	C₂×C₄×C₇₉	—	8	1	no	—
151	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
152	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
153	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
156	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
157	C₂²⋊D₇₉	—	2	2	no	—
158	C₂²⋊D₇₉	—	2	2	no	—
159	C₂×Dic₇₉	—	2	1	no	—
160	C₂×Dic₇₉	—	4	2	no	—
161	C₂×Dic₇₉	—	8	1	no	—
162	C₂×Dic₇₉	—	8	1	no	—
163	D₃₁₆	—	2	1	no	—
164	D₃₁₆	C₂×D₈	1	2	yes	—
165	C₄×D₇₉	—	2	2	no	—
166	C₄×D₇₉	—	2	2	no	—
167	Dic₁₅₈	—	2	1	no	—
168	Dic₁₅₈	—	4	2	no	—
169	Dic₁₅₈	—	4	2	no	—
170	Dic₁₅₈	—	8	1	no	—
171	Dic₁₅₈	—	8	1	no	—
172	Dic₁₅₈	C₂×D₈	1	2	yes	—
173	C₈×C₇₉	—	4	2	no	—
174	C₈×C₇₉	#121	1	1	yes	characteristic
175	C₈×C₇₉	#121	1	1	yes	characteristic
176	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
177	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
178	C₇₉⋊C₈	—	2	2	no	—
179	C₇₉⋊C₈	—	4	2	no	—
180	C₂²×C₇₉	—	2	2	no	—
181	C₂²×C₇₉	#466	1	1	yes	characteristic
182	C₂²×C₇₉	#466	1	1	yes	characteristic
183	C₂²×C₇₉	C₂×#135	1	1	yes	characteristic
184	C₂×D₇₉	—	2	2	no	—
185	C₂×D₇₉	—	4	2	no	—
186	C₄×C₇₉	—	8	2	no	—
187	C₄×C₇₉	#466	1	1	yes	characteristic
188	C₄×C₇₉	#466	1	1	yes	characteristic
189	C₄×C₇₉	C₂×#121	1	1	yes	characteristic
190	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
191	Dic₇₉	—	2	2	no	—
192	Dic₇₉	—	8	2	no	—
193	C₂×#2222	—	79	1	no	maximal, Sylow
194	C₂×C₇₉	—	2	1	no	—
195	C₂×C₇₉	—	2	1	no	—
196	C₂×C₇₉	#2222	1	2	yes	complemented
197	C₂×C₇₉	C₂×#466	1	1	yes	characteristic
198	D₇₉	—	4	2	no	—
199	C₂²×Q₃₂	—	79	1	no	—
200	C₂×#504	—	79	1	no	—
201	C₂×#473	—	79	1	no	—
202	C₂×#473	—	79	1	no	—
203	#2222	—	79	8	no	—
204	C₂×#368	—	79	1	no	—
205	C₂×#362	—	79	1	no	—
206	C₂×#362	D₇₉	1	1	yes	characteristic, complemented
207	C₇₉	C₂×#2222	1	1	yes	characteristic, complemented, Sylow
208	C₂×#137	—	158	1	no	—
209	C₂×#137	—	158	1	no	—
210	C₂²×Q₁₆	—	79	1	no	—
211	C₂²×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
212	C₂×#131	—	79	1	no	—
213	C₂×Q₃₂	—	79	4	no	—
214	C₂×Q₃₂	—	158	1	no	—
215	C₂×Q₃₂	—	158	1	no	—
216	C₂×Q₃₂	—	158	2	no	—
217	#504	—	79	4	no	—
218	#504	—	158	2	no	—
219	C₂×M₃₂	—	158	1	no	—
220	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
221	C₂²×C₁₆	—	79	1	no	—
222	C₂²×C₁₆	—	158	1	no	—
223	#473	—	79	4	no	—
224	#473	—	79	4	no	—
225	#473	—	158	2	no	—
226	#473	—	158	2	no	—
227	C₂×#108	—	79	1	no	—
228	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
229	C₂×#101	—	158	1	no	—
230	C₂×#101	—	158	1	no	—
231	#368	—	79	4	no	—
232	#362	—	79	4	no	—
233	#362	C₂×D₇₉	1	4	yes	complemented
234	C₂×#41	—	79	1	no	—
235	C₂²×Q₈	—	2	1	no	—
236	C₂²×Q₈	—	158	1	no	—
237	#137	—	316	2	no	—
238	#137	—	316	2	no	—
239	#137	—	158	4	no	—
240	#137	—	158	4	no	—
241	C₂×Q₁₆	—	2	1	no	—
242	C₂×Q₁₆	—	2	1	no	—
243	C₂×Q₁₆	—	2	1	no	—
244	C₂×Q₁₆	—	2	1	no	—
245	C₂×Q₁₆	—	2	1	no	—
246	C₂×Q₁₆	—	316	1	no	—
247	C₂×Q₁₆	—	316	1	no	—
248	C₂×Q₁₆	—	79	2	no	—
249	C₂×Q₁₆	—	158	1	no	—
250	C₂×Q₁₆	—	158	1	no	—
251	C₂×Q₁₆	—	158	1	no	—
252	C₂×Q₁₆	—	158	1	no	—
253	C₂×Q₁₆	—	158	1	no	—
254	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
255	C₂×SD₁₆	—	316	1	no	—
256	C₂×SD₁₆	—	316	1	no	—
257	#131	—	79	2	no	—
258	#131	—	79	2	no	—
259	#131	—	158	1	no	—
260	#131	—	158	1	no	—
261	C₂×M₁₆	—	2	1	no	—
262	C₂×M₁₆	—	79	1	no	—
263	C₂×M₁₆	—	158	1	no	—
264	C₂×M₁₆	—	158	1	no	—
265	C₂²×C₈	—	158	1	no	—
266	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
267	Q₃₂	—	158	4	no	—
268	Q₃₂	—	158	4	no	—
269	M₃₂	—	2	1	no	—
270	M₃₂	—	158	4	no	—
271	M₃₂	C₂²×D₇₉	1	2	yes	—
272	C₂×C₁₆	—	2	1	no	—
273	C₂×C₁₆	—	79	1	no	—
274	C₂×C₁₆	—	79	1	no	—
275	C₂×C₁₆	—	79	2	no	—
276	C₂×C₁₆	—	79	2	no	—
277	C₂×C₁₆	—	158	4	no	—
278	#108	—	2	1	no	—
279	#108	—	79	2	no	—
280	#108	—	158	1	no	—
281	#108	C₂²×D₇₉	1	2	yes	—
282	#101	—	158	4	no	—
283	#101	—	158	4	no	—
284	#41	—	79	2	no	—
285	#41	—	79	2	no	—
286	#41	—	158	1	no	—
287	#41	—	158	1	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	—	4	1	no	—
290	C₂×Q₈	—	4	1	no	—
291	C₂×Q₈	—	4	1	no	—
292	C₂×Q₈	—	4	1	no	—
293	C₂×Q₈	—	4	1	no	—
294	C₂×Q₈	—	316	1	no	—
295	C₂×Q₈	—	316	1	no	—
296	C₂×Q₈	—	316	1	no	—
297	C₂×Q₈	—	316	1	no	—
298	C₂×Q₈	—	316	1	no	—
299	C₂×Q₈	—	79	1	no	—
300	C₂×Q₈	—	158	2	no	—
301	C₂×D₄	—	79	1	no	—
302	C₂×D₄	—	158	1	no	—
303	C₂²×C₄	—	4	1	no	—
304	C₂²×C₄	—	316	1	no	—
305	C₂²×C₄	—	158	1	no	—
306	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
307	Q₁₆	—	2	2	no	—
308	Q₁₆	—	2	2	no	—
309	Q₁₆	—	4	1	no	—
310	Q₁₆	—	4	1	no	—
311	Q₁₆	—	316	4	no	—
312	Q₁₆	—	316	4	no	—
313	Q₁₆	—	316	1	no	—
314	Q₁₆	—	316	1	no	—
315	Q₁₆	—	158	2	no	—
316	Q₁₆	—	158	2	no	—
317	SD₁₆	—	316	4	no	—
318	SD₁₆	—	316	4	no	—
319	M₁₆	—	2	2	no	—
320	M₁₆	—	4	1	no	—
321	M₁₆	—	316	1	no	—
322	M₁₆	—	79	2	no	—
323	M₁₆	—	158	2	no	—
324	M₁₆	—	158	1	no	—
325	M₁₆	—	158	2	no	—
326	M₁₆	—	158	2	no	—
327	C₂×C₈	—	4	1	no	—
328	C₂×C₈	—	316	1	no	—
329	C₂×C₈	—	158	1	no	—
330	C₂×C₈	—	158	2	no	—
331	C₂×C₈	—	158	2	no	—
332	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
333	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
334	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
335	C₂×C₈	#11929624	1	1	yes	characteristic
336	C₂×C₈	#11929624	1	1	yes	characteristic
337	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
338	C₁₆	—	2	2	no	—
339	C₁₆	—	79	2	no	—
340	C₁₆	—	79	2	no	—
341	C₂³	—	158	1	no	—
342	C₂³	#14579182	1	1	yes	characteristic
343	Q₈	—	4	2	no	—
344	Q₈	—	4	2	no	—
345	Q₈	—	8	1	no	—
346	Q₈	—	8	1	no	—
347	Q₈	—	316	2	no	—
348	Q₈	—	316	2	no	—
349	Q₈	—	79	2	no	—
350	Q₈	—	632	1	no	—
351	Q₈	—	632	1	no	—
352	Q₈	—	158	1	no	—
353	D₄	—	79	2	no	—
354	D₄	—	158	2	no	—
355	D₄	—	158	2	no	—
356	D₄	—	158	1	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	8	1	no	—
359	C₂×C₄	—	8	1	no	—
360	C₂×C₄	—	316	2	no	—
361	C₂×C₄	—	632	1	no	—
362	C₂×C₄	—	632	1	no	—
363	C₂×C₄	—	158	1	no	—
364	C₂×C₄	—	158	2	no	—
365	C₂×C₄	—	158	2	no	—
366	C₂×C₄	#14579134	1	1	yes	characteristic
367	C₂×C₄	#14579134	1	1	yes	characteristic
368	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
369	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
370	C₂×C₄	#14579182	1	1	yes	characteristic
371	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
372	C₈	—	4	2	no	—
373	C₈	—	316	2	no	—
374	C₈	—	158	2	no	—
375	C₈	#14579134	1	1	yes	characteristic
376	C₈	#14579134	1	1	yes	characteristic
377	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
378	C₈	C₂×#11929624	1	1	yes	characteristic
379	C₂²	—	2	2	no	—
380	C₂²	—	316	2	no	—
381	C₂²	—	158	2	no	—
382	C₂²	#19098798	1	1	yes	characteristic
383	C₂²	#19098798	1	1	yes	characteristic
384	C₂²	C₂×#14579182	1	1	yes	characteristic
385	C₄	—	8	2	no	—
386	C₄	—	632	2	no	—
387	C₄	—	158	2	no	—
388	C₄	#19098798	1	1	yes	characteristic
389	C₄	#19098798	1	1	yes	characteristic
390	C₄	C₂×#14579134	1	1	yes	characteristic
391	C₄	C₂×D₈×D₇₉	1	1	yes	characteristic
392	C₂	—	2	1	no	—
393	C₂	—	2	1	no	—
394	C₂	—	316	2	no	—
395	C₂	#23765524	1	2	yes	direct product factor
396	C₂	C₂×#19098798	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098798
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#362×C₇₉	C₂
socle	C₂²×C₇₉	#466
prime radical	C₂×#11929624	D₄
hypercenter	C₂×#362	D₇₉
hypocenter	C₇₉	C₂×#2222
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#362

Complements

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

N	H ≅ G/N	type
1. C₂²×#14579107	C₂	semidirect
5. #23765524	C₂	direct (N × H)
6. C₂×#19098392	C₂	semidirect
7. C₂×#19098380	C₂	semidirect
8. C₂×#362×C₇₉	C₂	semidirect
16. C₂×#14579107	C₂²	semidirect
29. #19098392	C₂²	semidirect
30. #19098380	C₂²	semidirect
34. #362×C₇₉	C₂²	semidirect
196. C₂×C₇₉	#2222	semidirect
206. C₂×#362	D₇₉	semidirect
207. C₇₉	C₂×#2222	semidirect
233. #362	C₂×D₇₉	semidirect
395. C₂	#23765524	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×#14579107	2
C₂×#19098981	2
C₂×#19098839	2
C₂×#19098835	2
#23765524	2
C₂×#19098392	2
C₂×#19098380	2
C₂×#362×C₇₉	2
C₂×#2222	79

Subgroups — C2×#23765524 #34442831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765524 #34442831