Subgroups — C₂×#22609292 #33068313

Order 18688.

870 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988964	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988788	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#22609292	C₂	1	8	yes	maximal, direct product factor
5	C₂×#17988330	C₂	1	2	yes	maximal, complemented
6	C₂×#335×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11924017	—	2	2	no	—
8	C₂²×D₅₈₄	—	2	1	no	—
9	C₂²×D₅₈₄	—	2	1	no	—
10	C₂×#14556588	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
12	#17988964	—	2	2	no	—
13	#17988964	C₂²	1	4	yes	—
14	#17988788	—	2	4	no	—
15	#17988788	C₂²	1	8	yes	complemented
16	C₈×Dic₁₄₆	—	2	2	no	—
17	C₈×Dic₁₄₆	C₂²	1	4	yes	complemented
18	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
19	C₂×#14556520	—	2	1	no	—
20	C₂×#14556513	—	2	2	no	—
21	C₂×C₈×Dic₇₃	—	2	1	no	—
22	#17988330	C₂²	1	8	yes	complemented
23	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
25	#335×C₇₃	C₂²	1	4	yes	complemented
26	C₂²×Dic₁₄₆	D₄	1	1	yes	characteristic
27	C₂²×D₂₉₂	—	2	2	no	—
28	C₂²×D₂₉₂	—	2	2	no	—
29	C₂²×D₂₉₂	—	2	2	no	—
30	C₂×#11924017	—	2	8	no	—
31	C₂×#11924017	—	4	8	no	—
32	C₂×D₅₈₄	—	2	4	no	—
33	C₂×D₅₈₄	—	2	4	no	—
34	C₂×D₅₈₄	—	4	4	no	—
35	C₂×D₅₈₄	—	4	4	no	—
36	C₂²×#10596831	—	2	1	no	—
37	#14556588	—	2	2	no	—
38	#14556588	—	2	2	no	—
39	#14556588	—	2	2	no	—
40	#14556588	C₂³	1	4	yes	—
41	C₄×Dic₁₄₆	—	2	1	no	—
42	C₄×Dic₁₄₆	—	2	1	no	—
43	C₄×Dic₁₄₆	D₄	1	2	yes	—
44	C₄×Dic₁₄₆	C₂³	1	2	yes	—
45	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
46	C₂×#11924007	—	2	1	no	—
47	C₂×C₄×Dic₇₃	—	2	1	no	—
48	C₄⋊#10596831	—	2	1	no	—
49	C₄⋊#10596831	C₂³	1	2	yes	—
50	#14556520	—	2	4	no	—
51	#14556513	—	2	8	no	—
52	C₈×Dic₇₃	—	2	4	no	—
53	C₂²×C₈×C₇₃	—	2	2	no	—
54	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₇₃	—	2	2	no	—
58	C₄⋊C₈×C₇₃	C₂³	1	4	yes	—
59	C₄×C₈×C₇₃	D₄	1	2	yes	—
60	C₄×C₈×C₇₃	C₂³	1	2	yes	—
61	C₂³×D₇₃	—	4	2	no	—
62	C₂×Dic₁₄₆	—	2	1	no	—
63	C₂×Dic₁₄₆	—	2	2	no	—
64	C₂×Dic₁₄₆	D₈	1	4	yes	complemented
65	C₂×Dic₁₄₆	C₂×D₄	1	2	yes	—
66	C₂×D₂₉₂	—	2	4	no	—
67	C₂×D₂₉₂	—	2	4	no	—
68	C₂×D₂₉₂	—	2	4	no	—
69	C₂×D₂₉₂	—	4	4	no	—
70	C₂×D₂₉₂	—	4	4	no	—
71	C₂×D₂₉₂	—	4	4	no	—
72	C₂×D₂₉₂	—	4	4	no	—
73	C₂×D₂₉₂	—	4	2	no	—
74	C₂×D₂₉₂	—	4	2	no	—
75	C₂×D₂₉₂	—	4	2	no	—
76	C₂×D₂₉₂	—	8	4	no	—
77	C₂²×Dic₇₃	—	2	1	no	—
78	C₂²×Dic₇₃	—	2	1	no	—
79	#11924017	—	4	16	no	—
80	D₅₈₄	—	4	8	no	—
81	D₅₈₄	—	4	8	no	—
82	C₂×#10596831	—	2	2	no	—
83	C₂×#10596831	—	4	2	no	—
84	C₄⋊Dic₇₃	—	2	1	no	—
85	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
86	#11924007	—	2	2	no	—
87	#11924007	—	2	2	no	—
88	C₄×Dic₇₃	—	2	2	no	—
89	C₄×Dic₇₃	—	2	2	no	—
90	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
93	C₂×C₈×C₇₃	—	2	2	no	—
94	C₂×C₈×C₇₃	—	2	4	no	—
95	C₂×C₈×C₇₃	—	4	4	no	—
96	C₂×C₈×C₇₃	SD₁₆	1	4	yes	complemented
97	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
99	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₇₃	#41	1	1	yes	characteristic
102	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
103	C₂³×C₇₃	#121	1	1	yes	characteristic
104	C₂²×D₇₃	—	4	4	no	—
105	C₂²×D₇₃	—	8	4	no	—
106	C₂²×D₇₃	—	8	4	no	—
107	C₂²×D₇₃	—	8	4	no	—
108	C₂×C₄×C₇₃	—	2	2	no	—
109	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
114	C₂×C₄×C₇₃	#135	1	2	yes	—
115	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
119	C₂×Dic₇₃	—	2	2	no	—
120	C₂×Dic₇₃	—	2	4	no	—
121	C₂×Dic₇₃	—	2	2	no	—
122	C₂×Dic₇₃	—	4	2	no	—
123	D₂₉₂	—	4	4	no	—
124	D₂₉₂	—	4	4	no	—
125	D₂₉₂	—	4	4	no	—
126	D₂₉₂	—	4	4	no	—
127	D₂₉₂	—	8	4	no	—
128	D₂₉₂	—	8	4	no	—
129	D₂₉₂	—	8	4	no	—
130	D₂₉₂	—	8	4	no	—
131	Dic₁₄₆	—	2	2	no	—
132	Dic₁₄₆	—	2	4	no	—
133	Dic₁₄₆	C₂×D₈	1	4	yes	complemented
134	C₈×C₇₃	—	2	2	no	—
135	C₈×C₇₃	—	4	4	no	—
136	C₈×C₇₃	C₂×SD₁₆	1	4	yes	complemented
137	#10596831	—	4	2	no	—
138	C₂²×C₇₃	#460	1	2	yes	—
139	C₂²×C₇₃	#466	1	2	yes	—
140	C₂²×C₇₃	#472	1	2	yes	—
141	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
142	C₂×D₇₃	—	8	4	no	—
143	C₂×D₇₃	—	8	4	no	—
144	C₂×D₇₃	—	8	4	no	—
145	C₂×D₇₃	—	16	8	no	—
146	C₄×C₇₃	—	2	1	no	—
147	C₄×C₇₃	—	2	1	no	—
148	C₄×C₇₃	#460	1	2	yes	—
149	C₄×C₇₃	#466	1	2	yes	—
150	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
151	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
152	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
153	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
154	Dic₇₃	—	2	4	no	—
155	Dic₇₃	—	4	2	no	—
156	C₂×#1668	—	73	1	no	maximal, Sylow
157	C₂×C₇₃	#1668	1	4	yes	complemented
158	C₂×C₇₃	C₂×#460	1	1	yes	characteristic
159	C₂×C₇₃	C₂×#466	1	1	yes	characteristic
160	C₂×C₇₃	C₂×#472	1	1	yes	characteristic
161	D₇₃	—	16	4	no	—
162	C₂×#493	—	73	1	no	—
163	C₂×#460	—	73	2	no	—
164	C₂×C₈×Q₈	—	73	1	no	—
165	#1668	—	73	8	no	—
166	C₂×#335	D₇₃	1	1	yes	characteristic, complemented
167	C₂×#331	—	73	2	no	—
168	C₇₃	C₂×#1668	1	1	yes	characteristic, complemented, Sylow
169	C₂²×SD₁₆	—	146	2	no	—
170	C₂²×D₈	—	146	1	no	—
171	C₂²×D₈	—	146	1	no	—
172	C₂×#127	—	73	2	no	—
173	C₂×C₄×Q₈	—	73	1	no	—
174	#493	—	73	4	no	—
175	#493	—	146	2	no	—
176	#460	—	73	8	no	—
177	#460	—	146	4	no	—
178	C₈×Q₈	—	73	4	no	—
179	C₈×Q₈	—	146	2	no	—
180	C₂×C₄⋊C₈	—	73	1	no	—
181	C₂×C₄⋊C₈	—	146	1	no	—
182	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
183	C₂×#102	—	146	2	no	—
184	C₂×C₄×C₈	—	146	1	no	—
185	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
186	#335	C₂×D₇₃	1	4	yes	complemented
187	#331	—	73	8	no	—
188	C₂²×Q₈	—	73	1	no	—
189	C₂²×D₄	—	146	2	no	—
190	C₂²×D₄	—	146	2	no	—
191	C₂²×D₄	—	146	2	no	—
192	C₂×SD₁₆	—	292	8	no	—
193	C₂×SD₁₆	—	146	8	no	—
194	C₂×D₈	—	292	4	no	—
195	C₂×D₈	—	292	4	no	—
196	C₂×D₈	—	146	4	no	—
197	C₂×D₈	—	146	4	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	146	1	no	—
200	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
201	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
202	#127	—	73	4	no	—
203	#127	—	146	2	no	—
204	#127	—	146	2	no	—
205	#127	—	146	2	no	—
206	C₄×Q₈	—	73	2	no	—
207	C₄×Q₈	—	73	2	no	—
208	C₄×Q₈	—	146	1	no	—
209	C₄×Q₈	—	146	1	no	—
210	C₂×C₄⋊C₄	—	73	1	no	—
211	C₂×C₄⋊C₄	—	146	1	no	—
212	C₂×C₄²	—	146	1	no	—
213	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
214	C₄⋊C₈	—	2	2	no	—
215	C₄⋊C₈	—	73	2	no	—
216	C₄⋊C₈	—	146	4	no	—
217	C₄⋊C₈	—	146	1	no	—
218	C₄⋊C₈	C₂²×D₇₃	1	4	yes	—
219	#102	—	146	8	no	—
220	C₄×C₈	—	146	4	no	—
221	C₄×C₈	D₂₉₂	1	2	yes	—
222	C₄×C₈	C₂²×D₇₃	1	2	yes	—
223	C₂⁴	—	292	2	no	—
224	C₂×Q₈	—	73	4	no	—
225	C₂×Q₈	—	73	2	no	—
226	C₂×Q₈	—	146	1	no	—
227	C₂×Q₈	—	146	2	no	—
228	C₂×D₄	—	292	2	no	—
229	C₂×D₄	—	292	2	no	—
230	C₂×D₄	—	292	2	no	—
231	C₂×D₄	—	292	4	no	—
232	C₂×D₄	—	292	4	no	—
233	C₂×D₄	—	292	4	no	—
234	C₂×D₄	—	292	4	no	—
235	C₂×D₄	—	584	4	no	—
236	C₂×D₄	—	146	4	no	—
237	C₂×D₄	—	146	4	no	—
238	C₂×D₄	—	146	4	no	—
239	C₂²×C₄	—	146	1	no	—
240	C₂²×C₄	—	146	1	no	—
241	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
243	C₂²×C₄	#11924029	1	1	yes	characteristic
244	SD₁₆	—	292	16	no	—
245	D₈	—	292	8	no	—
246	D₈	—	292	8	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	4	4	no	—
250	C₂×C₈	—	292	2	no	—
251	C₂×C₈	—	146	2	no	—
252	C₂×C₈	#11924017	1	4	yes	complemented
253	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
254	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
255	C₄⋊C₄	—	73	2	no	—
256	C₄⋊C₄	—	146	2	no	—
257	C₄⋊C₄	—	146	1	no	—
258	C₄⋊C₄	—	146	2	no	—
259	C₄²	—	146	2	no	—
260	C₄²	—	146	2	no	—
261	C₄²	D₄×D₇₃	1	1	yes	characteristic
262	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
263	C₄²	#11924029	1	1	yes	characteristic
264	C₄²	C₂³×D₇₃	1	1	yes	characteristic
265	C₂³	—	292	4	no	—
266	C₂³	—	584	4	no	—
267	C₂³	—	584	4	no	—
268	C₂³	—	584	4	no	—
269	C₂³	#14556566	1	1	yes	characteristic
270	Q₈	—	73	4	no	—
271	Q₈	—	146	4	no	—
272	Q₈	—	146	2	no	—
273	D₄	—	292	4	no	—
274	D₄	—	292	4	no	—
275	D₄	—	292	4	no	—
276	D₄	—	292	4	no	—
277	D₄	—	584	4	no	—
278	D₄	—	584	4	no	—
279	D₄	—	584	4	no	—
280	D₄	—	584	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	292	2	no	—
283	C₂×C₄	—	146	2	no	—
284	C₂×C₄	—	146	4	no	—
285	C₂×C₄	—	146	2	no	—
286	C₂×C₄	#14556566	1	1	yes	characteristic
287	C₂×C₄	#14556566	1	1	yes	characteristic
288	C₂×C₄	#14556566	1	1	yes	characteristic
289	C₂×C₄	D₈×D₇₃	1	2	yes	—
290	C₂×C₄	C₂×#11924017	1	2	yes	—
291	C₂×C₄	#14556614	1	2	yes	—
292	C₂×C₄	#14556619	1	2	yes	—
293	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
295	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	4	4	no	—
298	C₈	—	292	2	no	—
299	C₈	C₂×#11924017	1	4	yes	complemented
300	C₂²	—	584	4	no	—
301	C₂²	—	584	4	no	—
302	C₂²	—	584	4	no	—
303	C₂²	—	1168	8	no	—
304	C₂²	#17988788	1	2	yes	—
305	C₂²	#17988824	1	2	yes	—
306	C₂²	#17988858	1	2	yes	—
307	C₂²	C₂×#14556566	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	292	2	no	—
311	C₄	—	146	4	no	—
312	C₄	#17988788	1	2	yes	—
313	C₄	#17988824	1	2	yes	—
314	C₄	C₂×D₈×D₇₃	1	1	yes	characteristic
315	C₄	C₂²×#11924017	1	1	yes	characteristic
316	C₄	C₂×#14556614	1	1	yes	characteristic
317	C₄	C₂×#14556619	1	1	yes	characteristic
318	C₂	—	1168	4	no	—
319	C₂	#22609292	1	4	yes	direct product factor
320	C₂	C₂×#17988788	1	1	yes	characteristic
321	C₂	C₂×#17988824	1	1	yes	characteristic
322	C₂	C₂×#17988858	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556566
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×#335×C₇₃	C₂
socle	C₂³×C₇₃	#121
prime radical	C₂×#17988964	C₂
hypercenter	C₂×#335	D₇₃
hypocenter	C₇₃	C₂×#1668
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
2. C₂×#17988788	C₂	semidirect
3. C₂×C₈×Dic₁₄₆	C₂	semidirect
4. #22609292	C₂	direct (N × H)
5. C₂×#17988330	C₂	semidirect
6. C₂×#335×C₇₃	C₂	semidirect
15. #17988788	C₂²	semidirect
17. C₈×Dic₁₄₆	C₂²	semidirect
22. #17988330	C₂²	semidirect
25. #335×C₇₃	C₂²	semidirect
64. C₂×Dic₁₄₆	D₈	semidirect
96. C₂×C₈×C₇₃	SD₁₆	semidirect
133. Dic₁₄₆	C₂×D₈	semidirect
136. C₈×C₇₃	C₂×SD₁₆	semidirect
157. C₂×C₇₃	#1668	semidirect
166. C₂×#335	D₇₃	semidirect
168. C₇₃	C₂×#1668	semidirect
186. #335	C₂×D₇₃	semidirect
252. C₂×C₈	#11924017	semidirect
299. C₈	C₂×#11924017	semidirect
319. C₂	#22609292	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×#17988964	2
C₂×#17988788	2
C₂×C₈×Dic₁₄₆	2
#22609292	2
C₂×#17988330	2
C₂×#335×C₇₃	2
C₂×#1668	73

Subgroups — C2×#22609292 #33068313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22609292 #33068313