Subgroups — C₂×#22609297 #33068220

Order 18688.

790 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988967	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17988868	C₂	1	2	yes	maximal
3	C₂×C₈×D₂₉₂	C₂	1	1	yes	maximal, characteristic
4	#22609297	C₂	1	8	yes	maximal, direct product factor
5	C₂×#17988332	C₂	1	2	yes	maximal, complemented
6	C₂×#335×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₂₉₂	—	2	1	no	—
8	C₂²×Dic₂₉₂	—	2	1	no	—
9	C₂²×C₈×D₇₃	—	2	1	no	—
10	C₂×#14556591	C₂²	1	2	yes	—
11	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
12	#17988967	—	2	2	no	—
13	#17988967	C₂²	1	4	yes	complemented
14	#17988868	—	2	4	no	—
15	#17988868	C₂²	1	8	yes	—
16	C₈×D₂₉₂	—	2	2	no	—
17	C₈×D₂₉₂	C₂²	1	4	yes	—
18	C₂×#14556525	—	2	2	no	—
19	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
20	C₂×#14556513	—	2	2	no	—
21	C₂×#14556503	—	2	1	no	—
22	#17988332	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	—	2	2	no	—
27	C₂²×D₂₉₂	D₄	1	1	yes	characteristic
28	C₂²×C₄×D₇₃	—	2	1	no	—
29	C₂×Dic₂₉₂	—	2	4	no	—
30	C₂×Dic₂₉₂	—	2	4	no	—
31	C₂×Dic₂₉₂	—	4	4	no	—
32	C₂×Dic₂₉₂	—	4	4	no	—
33	C₂²×#10596831	—	2	1	no	—
34	C₂×C₈×D₇₃	—	2	4	no	—
35	C₂×C₈×D₇₃	—	4	4	no	—
36	#14556591	—	2	2	no	—
37	#14556591	—	2	2	no	—
38	#14556591	—	2	2	no	—
39	#14556591	C₂³	1	4	yes	—
40	C₄×D₂₉₂	—	2	1	no	—
41	C₄×D₂₉₂	—	2	1	no	—
42	C₄×D₂₉₂	D₄	1	2	yes	—
43	C₄×D₂₉₂	C₂³	1	2	yes	—
44	C₂×C₄⋊Dic₇₃	—	2	2	no	—
45	C₂×C₄⋊Dic₇₃	—	2	2	no	—
46	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
47	C₂×#11924005	—	2	1	no	—
48	#14556525	—	2	8	no	—
49	C₄⋊#10596831	—	2	1	no	—
50	C₄⋊#10596831	C₂³	1	2	yes	—
51	#14556513	—	2	8	no	—
52	#14556503	—	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	—	2	1	no	—
62	C₂×Dic₁₄₆	—	2	4	no	—
63	C₂×Dic₁₄₆	—	4	4	no	—
64	C₂×Dic₁₄₆	—	4	4	no	—
65	C₂×Dic₁₄₆	—	4	2	no	—
66	C₂×D₂₉₂	—	2	1	no	—
67	C₂×D₂₉₂	—	2	2	no	—
68	C₂×D₂₉₂	Q₁₆	1	4	yes	complemented
69	C₂×D₂₉₂	C₂×D₄	1	2	yes	—
70	C₂²×Dic₇₃	—	2	1	no	—
71	C₂²×Dic₇₃	—	4	2	no	—
72	C₂×C₄×D₇₃	—	2	2	no	—
73	C₂×C₄×D₇₃	—	2	2	no	—
74	C₂×C₄×D₇₃	—	4	2	no	—
75	C₂×C₄×D₇₃	—	4	2	no	—
76	Dic₂₉₂	—	4	8	no	—
77	Dic₂₉₂	—	4	8	no	—
78	C₂×#10596831	—	2	2	no	—
79	C₂×#10596831	—	4	2	no	—
80	C₈×D₇₃	—	4	8	no	—
81	C₄⋊Dic₇₃	—	2	1	no	—
82	C₄⋊Dic₇₃	—	2	4	no	—
83	C₄⋊Dic₇₃	—	2	4	no	—
84	C₄⋊Dic₇₃	—	4	2	no	—
85	C₄⋊Dic₇₃	—	4	2	no	—
86	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
87	#11924005	—	2	2	no	—
88	#11924005	—	2	2	no	—
89	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
92	C₂×C₈×C₇₃	—	2	2	no	—
93	C₂×C₈×C₇₃	—	2	4	no	—
94	C₂×C₈×C₇₃	—	4	4	no	—
95	C₂×C₈×C₇₃	D₈	1	4	yes	—
96	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
98	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₇₃	#41	1	1	yes	characteristic
101	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
102	C₂³×C₇₃	#121	1	1	yes	characteristic
103	C₂²×D₇₃	—	2	2	no	—
104	C₂²×D₇₃	—	2	4	no	—
105	C₂²×D₇₃	—	4	2	no	—
106	C₂²×D₇₃	—	4	2	no	—
107	C₂×C₄×C₇₃	—	2	2	no	—
108	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
109	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
113	C₂×C₄×C₇₃	#135	1	2	yes	—
114	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
115	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
118	C₂×Dic₇₃	—	2	2	no	—
119	C₂×Dic₇₃	—	4	2	no	—
120	C₂×Dic₇₃	—	4	4	no	—
121	C₂×Dic₇₃	—	8	4	no	—
122	D₂₉₂	—	2	2	no	—
123	D₂₉₂	—	2	4	no	—
124	D₂₉₂	C₂×Q₁₆	1	4	yes	complemented
125	C₄×D₇₃	—	4	2	no	—
126	C₄×D₇₃	—	4	2	no	—
127	C₄×D₇₃	—	4	4	no	—
128	Dic₁₄₆	—	4	4	no	—
129	Dic₁₄₆	—	4	4	no	—
130	Dic₁₄₆	—	8	4	no	—
131	C₈×C₇₃	—	2	2	no	—
132	C₈×C₇₃	—	4	4	no	—
133	C₈×C₇₃	C₂×D₈	1	4	yes	—
134	#10596831	—	4	2	no	—
135	C₂²×C₇₃	#459	1	2	yes	—
136	C₂²×C₇₃	#467	1	2	yes	—
137	C₂²×C₇₃	#473	1	2	yes	—
138	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
139	C₂×D₇₃	—	2	4	no	—
140	C₂×D₇₃	—	4	8	no	—
141	C₂×D₇₃	—	4	2	no	—
142	C₂×D₇₃	—	4	2	no	—
143	C₄×C₇₃	—	2	1	no	—
144	C₄×C₇₃	—	2	1	no	—
145	C₄×C₇₃	#459	1	2	yes	—
146	C₄×C₇₃	#467	1	2	yes	—
147	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
148	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
149	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
150	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	Dic₇₃	—	4	2	no	—
152	Dic₇₃	—	8	4	no	—
153	C₂×#1669	—	73	1	no	maximal, Sylow
154	C₂×C₇₃	#1669	1	4	yes	complemented
155	C₂×C₇₃	C₂×#459	1	1	yes	characteristic
156	C₂×C₇₃	C₂×#467	1	1	yes	characteristic
157	C₂×C₇₃	C₂×#473	1	1	yes	characteristic
158	D₇₃	—	4	4	no	—
159	C₂×#494	—	73	1	no	—
160	C₂×#474	—	73	2	no	—
161	C₂×C₈×D₄	—	73	1	no	—
162	#1669	—	73	8	no	—
163	C₂×#335	D₇₃	1	1	yes	characteristic, complemented
164	C₂×#332	—	73	2	no	—
165	C₇₃	C₂×#1669	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₁₆	—	146	1	no	—
167	C₂²×Q₁₆	—	146	1	no	—
168	C₂³×C₈	—	146	1	no	—
169	C₂×C₄⋊Q₈	—	73	2	no	—
170	C₂×C₄×D₄	—	73	1	no	—
171	#494	—	73	4	no	—
172	#494	—	146	2	no	—
173	#474	—	73	8	no	—
174	#474	—	146	4	no	—
175	C₈×D₄	—	73	4	no	—
176	C₈×D₄	—	146	2	no	—
177	C₂×#107	—	146	2	no	—
178	C₂×C₄⋊C₈	—	73	1	no	—
179	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
180	C₂×#102	—	146	2	no	—
181	C₂×C₂²⋊C₈	—	146	1	no	—
182	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
183	#335	C₂×D₇₃	1	4	yes	complemented
184	#332	—	73	8	no	—
185	C₂²×Q₈	—	146	2	no	—
186	C₂²×D₄	—	73	1	no	—
187	C₂³×C₄	—	146	1	no	—
188	C₂×Q₁₆	—	292	4	no	—
189	C₂×Q₁₆	—	292	4	no	—
190	C₂×Q₁₆	—	146	4	no	—
191	C₂×Q₁₆	—	146	4	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	292	4	no	—
194	C₂²×C₈	—	146	1	no	—
195	C₂²×C₈	—	146	4	no	—
196	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
197	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
198	C₄⋊Q₈	—	73	4	no	—
199	C₄⋊Q₈	—	146	2	no	—
200	C₄⋊Q₈	—	146	2	no	—
201	C₄⋊Q₈	—	146	2	no	—
202	C₄×D₄	—	73	2	no	—
203	C₄×D₄	—	73	2	no	—
204	C₄×D₄	—	146	1	no	—
205	C₄×D₄	—	146	1	no	—
206	C₂×C₄⋊C₄	—	73	1	no	—
207	C₂×C₄⋊C₄	—	146	2	no	—
208	C₂×C₄⋊C₄	—	146	2	no	—
209	C₂×C₂²⋊C₄	—	146	1	no	—
210	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
211	#107	—	146	8	no	—
212	C₄⋊C₈	—	2	2	no	—
213	C₄⋊C₈	—	73	2	no	—
214	C₄⋊C₈	—	146	1	no	—
215	C₄⋊C₈	C₂²×D₇₃	1	4	yes	—
216	#102	—	146	8	no	—
217	C₂²⋊C₈	—	146	4	no	—
218	C₄×C₈	D₂₉₂	1	2	yes	—
219	C₄×C₈	C₂²×D₇₃	1	2	yes	—
220	C₂⁴	—	146	1	no	—
221	C₂×Q₈	—	292	2	no	—
222	C₂×Q₈	—	292	4	no	—
223	C₂×Q₈	—	292	4	no	—
224	C₂×Q₈	—	146	4	no	—
225	C₂×D₄	—	73	4	no	—
226	C₂×D₄	—	73	2	no	—
227	C₂×D₄	—	146	1	no	—
228	C₂×D₄	—	146	2	no	—
229	C₂²×C₄	—	292	2	no	—
230	C₂²×C₄	—	292	2	no	—
231	C₂²×C₄	—	292	2	no	—
232	C₂²×C₄	—	146	1	no	—
233	C₂²×C₄	—	146	2	no	—
234	C₂²×C₄	—	146	2	no	—
235	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
237	C₂²×C₄	#11924029	1	1	yes	characteristic
238	Q₁₆	—	292	8	no	—
239	Q₁₆	—	292	8	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	4	4	no	—
243	C₂×C₈	—	292	2	no	—
244	C₂×C₈	—	292	8	no	—
245	C₂×C₈	—	146	2	no	—
246	C₂×C₈	D₅₈₄	1	4	yes	—
247	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
248	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
249	C₄⋊C₄	—	292	2	no	—
250	C₄⋊C₄	—	292	2	no	—
251	C₄⋊C₄	—	73	2	no	—
252	C₄⋊C₄	—	146	4	no	—
253	C₄⋊C₄	—	146	4	no	—
254	C₄⋊C₄	—	146	1	no	—
255	C₂²⋊C₄	—	146	2	no	—
256	C₂²⋊C₄	—	146	2	no	—
257	C₄²	D₄×D₇₃	1	1	yes	characteristic
258	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
259	C₄²	#11924029	1	1	yes	characteristic
260	C₄²	C₂³×D₇₃	1	1	yes	characteristic
261	C₂³	—	292	2	no	—
262	C₂³	—	292	2	no	—
263	C₂³	—	146	2	no	—
264	C₂³	—	146	4	no	—
265	C₂³	#14556566	1	1	yes	characteristic
266	Q₈	—	292	4	no	—
267	Q₈	—	292	4	no	—
268	Q₈	—	584	4	no	—
269	D₄	—	73	4	no	—
270	D₄	—	146	4	no	—
271	D₄	—	146	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	292	2	no	—
274	C₂×C₄	—	292	2	no	—
275	C₂×C₄	—	292	4	no	—
276	C₂×C₄	—	292	2	no	—
277	C₂×C₄	—	292	4	no	—
278	C₂×C₄	—	584	4	no	—
279	C₂×C₄	—	146	2	no	—
280	C₂×C₄	#14556566	1	1	yes	characteristic
281	C₂×C₄	#14556566	1	1	yes	characteristic
282	C₂×C₄	#14556566	1	1	yes	characteristic
283	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
284	C₂×C₄	Q₁₆×D₇₃	1	2	yes	—
285	C₂×C₄	#14556610	1	2	yes	—
286	C₂×C₄	#14556619	1	2	yes	—
287	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
288	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
289	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	4	4	no	—
292	C₈	—	292	2	no	—
293	C₈	C₂×D₅₈₄	1	4	yes	—
294	C₂²	—	292	8	no	—
295	C₂²	—	292	2	no	—
296	C₂²	—	292	2	no	—
297	C₂²	—	146	4	no	—
298	C₂²	#17988783	1	2	yes	—
299	C₂²	#17988829	1	2	yes	—
300	C₂²	#17988863	1	2	yes	—
301	C₂²	C₂×#14556566	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	292	2	no	—
305	C₄	—	584	4	no	—
306	C₄	#17988783	1	2	yes	—
307	C₄	#17988829	1	2	yes	—
308	C₄	C₂²×D₅₈₄	1	1	yes	characteristic
309	C₄	C₂×Q₁₆×D₇₃	1	1	yes	characteristic
310	C₄	C₂×#14556610	1	1	yes	characteristic
311	C₄	C₂×#14556619	1	1	yes	characteristic
312	C₂	—	292	4	no	—
313	C₂	#22609297	1	4	yes	direct product factor
314	C₂	C₂×#17988783	1	1	yes	characteristic
315	C₂	C₂×#17988829	1	1	yes	characteristic
316	C₂	C₂×#17988863	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	C₂²×D₂₉₂	D₄
hypercenter	C₂×#335	D₇₃
hypocenter	C₇₃	C₂×#1669
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
1. C₂×#17988967	C₂	semidirect
4. #22609297	C₂	direct (N × H)
5. C₂×#17988332	C₂	semidirect
6. C₂×#335×C₇₃	C₂	semidirect
13. #17988967	C₂²	semidirect
22. #17988332	C₂²	semidirect
25. #335×C₇₃	C₂²	semidirect
68. C₂×D₂₉₂	Q₁₆	semidirect
124. D₂₉₂	C₂×Q₁₆	semidirect
154. C₂×C₇₃	#1669	semidirect
163. C₂×#335	D₇₃	semidirect
165. C₇₃	C₂×#1669	semidirect
183. #335	C₂×D₇₃	semidirect
313. C₂	#22609297	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×#17988967	2
C₂×#17988868	2
C₂×C₈×D₂₉₂	2
#22609297	2
C₂×#17988332	2
C₂×#335×C₇₃	2
C₂×#1669	73

Subgroups — C2×#22609297 #33068220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22609297 #33068220