Subgroups — C₂×#23765293 #34441704

Order 20224.

646 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099006	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098686	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098671	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765293	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19098386	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19098380	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19098374	C₂	1	1	yes	maximal, characteristic
8	C₂×#347×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14579183	C₂²	1	1	yes	characteristic
10	C₂²×Dic₃₁₆	C₂²	1	1	yes	characteristic
11	C₂×#14579167	—	2	1	no	—
12	C₂×C₄×Dic₁₅₈	—	2	1	no	—
13	#19099006	C₄	1	4	yes	complemented
14	#19099006	C₂²	1	4	yes	—
15	C₂×Dic₆₃₂	C₄	1	2	yes	complemented
16	C₂×#14579106	C₄	1	2	yes	complemented
17	#19098686	—	2	2	no	—
18	#19098686	C₂²	1	4	yes	—
19	#19098671	—	2	2	no	—
20	#19098671	C₂²	1	4	yes	complemented
21	C₂×#14579097	C₂²	1	1	yes	characteristic
22	C₂×#14579093	C₂²	1	1	yes	characteristic
23	C₂×#14579089	—	2	1	no	—
24	C₂×#14579086	—	2	1	no	—
25	#19098386	C₂²	1	4	yes	complemented
26	#19098380	C₂²	1	4	yes	complemented
27	#19098374	C₂²	1	4	yes	—
28	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
29	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
30	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
31	#347×C₇₉	C₂²	1	4	yes	complemented
32	C₂×#11929624	—	2	1	no	—
33	C₂²×Dic₁₅₈	—	2	1	no	—
34	#14579183	—	2	1	no	—
35	#14579183	—	2	1	no	—
36	#14579183	C₂×C₄	1	2	yes	—
37	#14579183	C₂³	1	2	yes	—
38	C₂×Dic₃₁₆	—	2	1	no	—
39	C₂×Dic₃₁₆	—	2	1	no	—
40	C₂×Dic₃₁₆	—	2	1	no	—
41	C₂×Dic₃₁₆	C₂×C₄	1	2	yes	—
42	C₂×Dic₃₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂×Dic₃₁₆	C₂×C₄	1	2	yes	—
44	C₂×Dic₃₁₆	C₂³	1	2	yes	—
45	C₂×#11929613	—	2	1	no	—
46	C₂×D₆₃₂	C₂×C₄	1	1	yes	characteristic
47	#14579167	—	2	4	no	—
48	#14579167	—	4	2	no	—
49	C₂×#11929608	—	2	1	no	—
50	C₂×#11929607	—	4	1	no	—
51	C₂×C₈×D₇₉	—	4	1	no	—
52	C₄×Dic₁₅₈	—	2	4	no	—
53	C₄×Dic₁₅₈	—	4	2	no	—
54	C₂×C₄⋊Dic₇₉	—	2	1	no	—
55	C₂×Dic₇₉⋊C₄	—	4	1	no	—
56	C₂×C₄×Dic₇₉	—	4	1	no	—
57	Dic₆₃₂	C₂×C₄	1	8	yes	complemented
58	#14579106	C₂×C₄	1	8	yes	complemented
59	#14579097	—	2	1	no	—
60	#14579097	C₂³	1	2	yes	—
61	#14579093	—	2	1	no	—
62	#14579093	C₂³	1	2	yes	—
63	#14579089	—	2	4	no	—
64	#14579086	—	2	4	no	—
65	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
68	M₃₂×C₇₉	—	2	1	no	—
69	M₃₂×C₇₉	C₂×C₄	1	2	yes	—
70	M₃₂×C₇₉	C₂³	1	2	yes	—
71	M₃₂×C₇₉	C₂³	1	2	yes	—
72	C₂×C₁₆×C₇₉	—	2	1	no	—
73	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
74	#97×C₇₉	D₄	1	2	yes	—
75	#97×C₇₉	C₂³	1	2	yes	—
76	#11929624	—	2	2	no	—
77	#11929624	—	2	2	no	—
78	#11929624	—	4	1	no	—
79	#11929624	—	4	1	no	—
80	C₂×Dic₁₅₈	—	2	2	no	—
81	C₂×Dic₁₅₈	—	2	2	no	—
82	C₂×Dic₁₅₈	—	2	1	no	—
83	C₂×Dic₁₅₈	—	2	2	no	—
84	C₂×Dic₁₅₈	—	4	1	no	—
85	C₂×Dic₁₅₈	—	4	1	no	—
86	C₂×Dic₁₅₈	—	4	1	no	—
87	C₂×D₃₁₆	—	2	1	no	—
88	C₂×C₂²⋊D₇₉	—	4	1	no	—
89	C₂²×Dic₇₉	—	4	1	no	—
90	C₂²×Dic₇₉	—	4	1	no	—
91	C₂×C₄×D₇₉	—	4	1	no	—
92	Dic₃₁₆	—	2	2	no	—
93	Dic₃₁₆	—	2	2	no	—
94	Dic₃₁₆	—	2	1	no	—
95	Dic₃₁₆	—	2	2	no	—
96	Dic₃₁₆	C₂²×C₄	1	4	yes	—
97	Dic₃₁₆	C₂²×C₄	1	2	yes	—
98	#11929613	—	2	2	no	—
99	#11929613	—	2	2	no	—
100	D₆₃₂	—	2	1	no	—
101	D₆₃₂	C₂²×C₄	1	2	yes	—
102	#11929608	—	2	2	no	—
103	#11929608	—	4	1	no	—
104	#11929607	—	4	4	no	—
105	C₂×C₇₉⋊C₈	—	4	1	no	—
106	C₈×D₇₉	—	4	4	no	—
107	C₄⋊Dic₇₉	—	2	2	no	—
108	C₄⋊Dic₇₉	—	4	1	no	—
109	Dic₇₉⋊C₄	—	4	4	no	—
110	C₄×Dic₇₉	—	4	4	no	—
111	C₂²×C₄×C₇₉	—	2	1	no	—
112	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₇₉	—	2	1	no	—
114	C₂×C₈×C₇₉	—	2	1	no	—
115	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
120	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
121	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
122	C₄²×C₇₉	—	2	1	no	—
123	C₄²×C₇₉	C₂×D₄	1	2	yes	—
124	C₁₆×C₇₉	—	2	2	no	—
125	C₁₆×C₇₉	C₂²×C₄	1	4	yes	—
126	C₂³×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂²×D₇₉	—	4	1	no	—
128	C₂×C₄×C₇₉	—	2	2	no	—
129	C₂×C₄×C₇₉	—	4	2	no	—
130	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
136	C₂²⋊D₇₉	—	4	2	no	—
137	C₂²⋊D₇₉	—	4	2	no	—
138	C₂×Dic₇₉	—	4	2	no	—
139	C₂×Dic₇₉	—	4	1	no	—
140	C₂×Dic₇₉	—	4	2	no	—
141	C₂×Dic₇₉	—	4	2	no	—
142	C₂×Dic₇₉	—	4	2	no	—
143	C₂×Dic₇₉	—	8	2	no	—
144	D₃₁₆	—	2	2	no	—
145	D₃₁₆	—	4	1	no	—
146	C₄×D₇₉	—	4	2	no	—
147	C₄×D₇₉	—	4	2	no	—
148	Dic₁₅₈	—	2	2	no	—
149	Dic₁₅₈	—	2	4	no	—
150	Dic₁₅₈	—	4	2	no	—
151	Dic₁₅₈	—	4	2	no	—
152	Dic₁₅₈	—	4	2	no	—
153	Dic₁₅₈	—	4	1	no	—
154	C₈×C₇₉	—	2	2	no	—
155	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
156	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
157	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
158	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
159	C₇₉⋊C₈	—	4	2	no	—
160	C₂²×C₇₉	—	2	2	no	—
161	C₂²×C₇₉	#442	1	1	yes	characteristic
162	C₂²×C₇₉	#442	1	1	yes	characteristic
163	C₂²×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₂×D₇₉	—	4	2	no	—
165	C₂×D₇₉	—	8	2	no	—
166	C₄×C₇₉	—	4	2	no	—
167	C₄×C₇₉	#442	1	1	yes	characteristic
168	C₄×C₇₉	#442	1	1	yes	characteristic
169	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
170	C₄×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
171	Dic₇₉	—	4	2	no	—
172	Dic₇₉	—	4	4	no	—
173	Dic₇₉	—	8	2	no	—
174	C₂×#2180	—	79	1	no	maximal, Sylow
175	C₂×C₇₉	—	2	1	no	—
176	C₂×C₇₉	—	2	1	no	—
177	C₂×C₇₉	#2180	1	2	yes	complemented
178	C₂×C₇₉	C₂×#442	1	1	yes	characteristic
179	D₇₉	—	8	2	no	—
180	C₂×#510	—	79	1	no	—
181	C₂×#444	—	79	1	no	—
182	C₂×#441	—	79	1	no	—
183	#2180	—	79	8	no	—
184	C₂×#365	—	79	1	no	—
185	C₂×#362	—	79	1	no	—
186	C₂×#360	—	79	1	no	—
187	C₂×#347	D₇₉	1	1	yes	characteristic, complemented
188	C₇₉	C₂×#2180	1	1	yes	characteristic, complemented, Sylow
189	C₂×#135	—	79	1	no	—
190	C₂²×Q₁₆	—	79	1	no	—
191	C₂×#131	—	158	1	no	—
192	C₂×C₄×Q₈	—	158	1	no	—
193	#510	—	79	4	no	—
194	#510	—	79	4	no	—
195	C₂×Q₃₂	—	79	2	no	—
196	C₂×SD₃₂	—	79	2	no	—
197	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
198	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
199	#444	—	79	4	no	—
200	#444	—	158	2	no	—
201	#441	—	79	4	no	—
202	#441	—	158	2	no	—
203	C₂×#108	—	79	1	no	—
204	C₂×#106	—	79	1	no	—
205	C₂×#104	—	158	1	no	—
206	C₂×#103	—	158	1	no	—
207	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
208	#365	—	79	4	no	—
209	#362	—	79	4	no	—
210	#360	—	79	4	no	—
211	#347	C₂×D₇₉	1	4	yes	complemented
212	C₂×#41	—	158	1	no	—
213	C₂²×Q₈	—	158	1	no	—
214	#135	—	79	2	no	—
215	#135	—	79	2	no	—
216	#135	—	158	1	no	—
217	#135	—	158	1	no	—
218	C₂×Q₁₆	—	79	2	no	—
219	C₂×Q₁₆	—	79	2	no	—
220	C₂×Q₁₆	—	79	1	no	—
221	C₂×Q₁₆	—	79	2	no	—
222	C₂×Q₁₆	—	158	1	no	—
223	C₂×Q₁₆	—	158	1	no	—
224	C₂×Q₁₆	—	158	1	no	—
225	C₂×SD₁₆	—	158	1	no	—
226	C₂×D₈	—	79	1	no	—
227	#131	—	316	2	no	—
228	#131	—	158	4	no	—
229	C₂×M₁₆	—	316	1	no	—
230	C₂×M₁₆	—	158	1	no	—
231	C₂²×C₈	—	316	1	no	—
232	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
234	C₄×Q₈	—	316	2	no	—
235	C₄×Q₈	—	158	4	no	—
236	C₂×C₄⋊C₄	—	316	1	no	—
237	C₂×C₄⋊C₄	—	158	1	no	—
238	C₂×C₄²	—	316	1	no	—
239	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
240	Q₃₂	—	79	8	no	—
241	SD₃₂	—	79	8	no	—
242	M₃₂	—	2	1	no	—
243	M₃₂	C₄×D₇₉	1	2	yes	—
244	M₃₂	C₂²×D₇₉	1	2	yes	—
245	M₃₂	C₂²×D₇₉	1	2	yes	—
246	C₂×C₁₆	—	2	1	no	—
247	C₂×C₁₆	C₄×D₇₉	1	2	yes	—
248	#108	—	79	2	no	—
249	#108	—	158	1	no	—
250	#106	—	79	2	no	—
251	#106	—	158	1	no	—
252	#104	—	158	4	no	—
253	#103	—	158	4	no	—
254	#97	D₃₁₆	1	2	yes	—
255	#97	C₂²×D₇₉	1	2	yes	—
256	#41	—	316	1	no	—
257	#41	—	316	1	no	—
258	#41	—	158	2	no	—
259	#41	—	158	2	no	—
260	C₂×Q₈	—	316	1	no	—
261	C₂×Q₈	—	316	1	no	—
262	C₂×Q₈	—	316	1	no	—
263	C₂×Q₈	—	158	1	no	—
264	C₂×Q₈	—	158	2	no	—
265	C₂×Q₈	—	158	2	no	—
266	C₂×Q₈	—	158	2	no	—
267	C₂×D₄	—	316	1	no	—
268	C₂×D₄	—	158	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	316	1	no	—
271	C₂²×C₄	—	316	1	no	—
272	C₂²×C₄	—	316	1	no	—
273	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
274	Q₁₆	—	79	4	no	—
275	Q₁₆	—	79	2	no	—
276	Q₁₆	—	158	2	no	—
277	Q₁₆	—	158	2	no	—
278	Q₁₆	—	158	2	no	—
279	Q₁₆	—	158	1	no	—
280	SD₁₆	—	158	2	no	—
281	SD₁₆	—	158	2	no	—
282	D₈	—	79	2	no	—
283	D₈	—	158	1	no	—
284	M₁₆	—	316	4	no	—
285	M₁₆	—	316	1	no	—
286	M₁₆	—	158	2	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	316	4	no	—
290	C₂×C₈	—	316	1	no	—
291	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
293	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
294	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
295	C₂×C₈	#11929624	1	1	yes	characteristic
296	C₂×C₈	#11929624	1	1	yes	characteristic
297	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
298	C₄⋊C₄	—	316	4	no	—
299	C₄⋊C₄	—	316	1	no	—
300	C₄⋊C₄	—	158	2	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	316	4	no	—
303	C₄²	C₂×D₃₁₆	1	2	yes	—
304	C₁₆	—	2	2	no	—
305	C₁₆	C₂×C₄×D₇₉	1	4	yes	—
306	C₂³	—	316	1	no	—
307	C₂³	#14579189	1	1	yes	characteristic
308	Q₈	—	316	2	no	—
309	Q₈	—	316	2	no	—
310	Q₈	—	316	1	no	—
311	Q₈	—	316	2	no	—
312	Q₈	—	158	2	no	—
313	Q₈	—	158	4	no	—
314	D₄	—	316	2	no	—
315	D₄	—	316	2	no	—
316	D₄	—	316	1	no	—
317	D₄	—	158	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	316	2	no	—
321	C₂×C₄	—	316	1	no	—
322	C₂×C₄	—	316	2	no	—
323	C₂×C₄	—	316	2	no	—
324	C₂×C₄	—	316	2	no	—
325	C₂×C₄	—	316	2	no	—
326	C₂×C₄	—	316	2	no	—
327	C₂×C₄	—	632	2	no	—
328	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
329	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
330	C₂×C₄	#14579189	1	1	yes	characteristic
331	C₂×C₄	#14579189	1	1	yes	characteristic
332	C₂×C₄	#14579189	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	316	2	no	—
336	C₈	C₄×D₃₁₆	1	1	yes	characteristic
337	C₈	C₄×D₃₁₆	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₇₉	1	1	yes	characteristic
339	C₈	C₂×#11929624	1	1	yes	characteristic
340	C₂²	—	2	2	no	—
341	C₂²	—	316	2	no	—
342	C₂²	—	632	2	no	—
343	C₂²	#19098676	1	1	yes	characteristic
344	C₂²	#19098676	1	1	yes	characteristic
345	C₂²	C₂×#14579189	1	1	yes	characteristic
346	C₄	—	4	2	no	—
347	C₄	—	316	2	no	—
348	C₄	—	316	4	no	—
349	C₄	—	632	2	no	—
350	C₄	#19098676	1	1	yes	characteristic
351	C₄	#19098676	1	1	yes	characteristic
352	C₄	C₂×C₄×D₃₁₆	1	1	yes	characteristic
353	C₄	C₂×#14579189	1	1	yes	characteristic
354	C₂	—	2	1	no	—
355	C₂	—	2	1	no	—
356	C₂	—	632	2	no	—
357	C₂	#23765293	1	2	yes	direct product factor
358	C₂	C₂×#19098676	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098676
commutator	C₈×C₇₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#347×C₇₉	C₂
socle	C₂²×C₇₉	#442
prime radical	C₂×#14579183	C₂²
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	C₂×#347	D₇₉
hypocenter	C₇₉	C₂×#2180
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
3. C₂×#19098671	C₂	semidirect
4. #23765293	C₂	direct (N × H)
5. C₂×#19098386	C₂	semidirect
6. C₂×#19098380	C₂	semidirect
8. C₂×#347×C₇₉	C₂	semidirect
13. #19099006	C₄	semidirect
15. C₂×Dic₆₃₂	C₄	semidirect
16. C₂×#14579106	C₄	semidirect
20. #19098671	C₂²	semidirect
25. #19098386	C₂²	semidirect
26. #19098380	C₂²	semidirect
31. #347×C₇₉	C₂²	semidirect
57. Dic₆₃₂	C₂×C₄	semidirect
58. #14579106	C₂×C₄	semidirect
177. C₂×C₇₉	#2180	semidirect
187. C₂×#347	D₇₉	semidirect
188. C₇₉	C₂×#2180	semidirect
211. #347	C₂×D₇₉	semidirect
357. C₂	#23765293	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×#19099006	2
C₂×#19098686	2
C₂×#19098671	2
#23765293	2
C₂×#19098386	2
C₂×#19098380	2
C₂×#19098374	2
C₂×#347×C₇₉	2
C₂×#2180	79

Subgroups — C2×#23765293 #34441704

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765293 #34441704