Subgroups — C₂×#22612122 #33240754

Order 18688.

638 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×#17988823	C₂	1	1	yes	maximal, characteristic
2	#22612122	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17988414	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#17988412	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17988387	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988387	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊#11924003	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#427×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₈₄	C₂²	1	1	yes	characteristic
10	C₂×#14556565	—	2	1	no	—
11	C₂²×#11924003	C₂²	1	1	yes	characteristic
12	C₂²×#11924003	C₂²	1	1	yes	characteristic
13	#17988823	—	2	2	no	—
14	#17988823	C₂²	1	4	yes	—
15	C₂×#14556525	C₂²	1	1	yes	characteristic
16	C₂×#14556513	—	2	1	no	—
17	#17988414	C₂²	1	4	yes	complemented
18	#17988412	C₂²	1	4	yes	complemented
19	#17988387	C₂²	1	4	yes	complemented
20	#17988387	C₂²	1	4	yes	complemented
21	C₂²⋊#11924003	C₂²	1	4	yes	complemented
22	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×#117×C₇₃	—	2	1	no	—
24	#427×C₇₃	—	2	2	no	—
25	#427×C₇₃	C₂²	1	4	yes	complemented
26	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
27	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
28	C₂×#106×C₇₃	—	2	1	no	—
29	C₂²×D₂₉₂	—	2	1	no	—
30	C₂²×C₂²⋊D₇₃	—	4	1	no	—
31	C₂×D₅₈₄	—	2	1	no	—
32	C₂×D₅₈₄	—	2	2	no	—
33	C₂×D₅₈₄	—	2	2	no	—
34	C₂×D₅₈₄	C₂³	1	2	yes	—
35	#14556565	—	2	4	no	—
36	#14556565	—	4	2	no	—
37	C₂×C₄⋊Dic₇₃	—	2	1	no	—
38	C₂×#11924005	—	4	1	no	—
39	C₂×#11924003	—	2	2	no	—
40	C₂×#11924003	—	2	2	no	—
41	C₂×#11924003	C₂³	1	2	yes	—
42	C₂×#11924003	C₂³	1	2	yes	—
43	#14556525	—	2	1	no	—
44	#14556525	C₂³	1	2	yes	—
45	#14556513	—	2	4	no	—
46	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₃	—	2	2	no	—
48	C₂²×C₈×C₇₃	—	2	2	no	—
49	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₇₃	D₄	1	2	yes	—
51	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
53	#117×C₇₃	—	2	4	no	—
54	#117×C₇₃	—	4	2	no	—
55	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
57	C₂×C₂²⋊C₄×C₇₃	—	4	1	no	—
58	C₂×C₄²×C₇₃	—	4	1	no	—
59	#107×C₇₃	—	2	1	no	—
60	#107×C₇₃	—	2	1	no	—
61	#107×C₇₃	C₂³	1	2	yes	—
62	#107×C₇₃	C₂³	1	2	yes	—
63	#106×C₇₃	—	2	4	no	—
64	C₂³×D₇₃	—	4	1	no	—
65	C₂×D₂₉₂	—	2	2	no	—
66	C₂×D₂₉₂	—	4	2	no	—
67	C₂×D₂₉₂	—	4	2	no	—
68	C₂×D₂₉₂	—	4	1	no	—
69	C₂×C₂²⋊D₇₃	—	4	4	no	—
70	C₂×C₂²⋊D₇₃	—	8	4	no	—
71	C₂²×Dic₇₃	—	4	1	no	—
72	D₅₈₄	—	2	2	no	—
73	D₅₈₄	—	2	2	no	—
74	D₅₈₄	—	4	2	no	—
75	C₄⋊Dic₇₃	—	2	2	no	—
76	C₄⋊Dic₇₃	—	4	1	no	—
77	#11924005	—	4	4	no	—
78	#11924003	—	2	2	no	—
79	#11924003	—	2	2	no	—
80	C₂⁴×C₇₃	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₇₃	—	2	2	no	—
82	C₂²×C₄×C₇₃	—	2	2	no	—
83	C₂²×C₄×C₇₃	—	4	1	no	—
84	C₂²×C₄×C₇₃	—	4	1	no	—
85	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₇₃	D₈	1	2	yes	—
87	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₇₃	—	2	2	no	—
90	C₂×C₈×C₇₃	—	2	4	no	—
91	C₂×C₈×C₇₃	—	2	2	no	—
92	C₂×C₈×C₇₃	—	2	2	no	—
93	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₇₃	#41	1	2	yes	—
96	C₂×C₈×C₇₃	#41	1	2	yes	—
97	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₇₃	—	2	2	no	—
99	C₄⋊C₄×C₇₃	—	2	2	no	—
100	C₄⋊C₄×C₇₃	—	4	1	no	—
101	C₄⋊C₄×C₇₃	—	4	1	no	—
102	C₂²⋊C₄×C₇₃	—	4	4	no	—
103	C₄²×C₇₃	—	4	4	no	—
104	C₂³×C₇₃	—	2	2	no	—
105	C₂³×C₇₃	—	2	2	no	—
106	C₂³×C₇₃	—	2	2	no	—
107	C₂³×C₇₃	C₂×D₈	1	2	yes	—
108	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
109	C₂²×D₇₃	—	4	2	no	—
110	C₂²×D₇₃	—	8	2	no	—
111	C₂²×D₇₃	—	8	2	no	—
112	C₂²×D₇₃	—	8	2	no	—
113	C₂×C₄×C₇₃	—	2	2	no	—
114	C₂×C₄×C₇₃	—	2	2	no	—
115	C₂×C₄×C₇₃	—	2	2	no	—
116	C₂×C₄×C₇₃	—	2	4	no	—
117	C₂×C₄×C₇₃	—	4	2	no	—
118	C₂×C₄×C₇₃	—	4	2	no	—
119	C₂×C₄×C₇₃	—	8	2	no	—
120	C₂×C₄×C₇₃	—	8	2	no	—
121	C₂×C₄×C₇₃	#123	1	2	yes	—
122	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₇₃	#137	1	2	yes	—
125	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
126	C₂²⋊D₇₃	—	8	8	no	—
127	C₂×Dic₇₃	—	4	2	no	—
128	C₂×Dic₇₃	—	8	2	no	—
129	D₂₉₂	—	4	2	no	—
130	D₂₉₂	—	4	2	no	—
131	D₂₉₂	—	8	2	no	—
132	C₈×C₇₃	—	2	2	no	—
133	C₈×C₇₃	#123	1	2	yes	—
134	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
135	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
136	C₂²×C₇₃	—	2	2	no	—
137	C₂²×C₇₃	—	2	2	no	—
138	C₂²×C₇₃	—	2	2	no	—
139	C₂²×C₇₃	—	4	4	no	—
140	C₂²×C₇₃	#480	1	2	yes	—
141	C₂²×C₇₃	#508	1	2	yes	—
142	C₂²×C₇₃	#509	1	2	yes	—
143	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
144	C₂×D₇₃	—	8	2	no	—
145	C₂×D₇₃	—	8	2	no	—
146	C₂×D₇₃	—	8	2	no	—
147	C₂×D₇₃	—	16	4	no	—
148	C₄×C₇₃	—	2	2	no	—
149	C₄×C₇₃	—	8	2	no	—
150	C₄×C₇₃	—	8	2	no	—
151	C₄×C₇₃	#480	1	2	yes	—
152	C₄×C₇₃	C₂×#123	1	1	yes	characteristic
153	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
154	Dic₇₃	—	8	2	no	—
155	C₂×#2234	—	73	1	no	maximal, Sylow
156	C₂×C₇₃	—	4	2	no	—
157	C₂×C₇₃	#2234	1	4	yes	complemented
158	C₂×C₇₃	C₂×#480	1	1	yes	characteristic
159	C₂×C₇₃	C₂×#508	1	1	yes	characteristic
160	C₂×C₇₃	C₂×#509	1	1	yes	characteristic
161	D₇₃	—	16	2	no	—
162	C₂×#466	—	73	1	no	—
163	C₂×#427	D₇₃	1	1	yes	characteristic, complemented
164	#2234	—	73	8	no	—
165	C₂×#367	—	73	1	no	—
166	C₂×#366	—	73	1	no	—
167	C₂×#357	—	73	1	no	—
168	C₂×#357	—	73	1	no	—
169	C₂×C₂²⋊C₁₆	—	73	1	no	—
170	C₇₃	C₂×#2234	1	1	yes	characteristic, complemented, Sylow
171	C₂²×D₈	—	73	1	no	—
172	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
173	C₂×#121	—	146	1	no	—
174	C₂×#117	—	2	1	no	—
175	C₂²×C₁₆	—	73	1	no	—
176	C₂²×C₁₆	—	73	1	no	—
177	#466	—	73	4	no	—
178	#466	—	146	2	no	—
179	#427	—	2	2	no	—
180	#427	C₂×D₇₃	1	4	yes	complemented
181	C₂×#107	—	73	1	no	—
182	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
183	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
184	C₂×#106	—	2	1	no	—
185	C₂×#102	—	146	1	no	—
186	#367	—	73	4	no	—
187	#366	—	73	4	no	—
188	#357	—	73	4	no	—
189	#357	—	73	4	no	—
190	C₂²⋊C₁₆	—	73	4	no	—
191	C₂²×D₄	—	292	1	no	—
192	C₂²×D₄	—	146	1	no	—
193	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
194	C₂×D₈	—	73	2	no	—
195	C₂×D₈	—	146	1	no	—
196	C₂×D₈	—	146	2	no	—
197	C₂×D₈	—	146	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
201	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
202	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
203	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
204	#121	—	292	2	no	—
205	#121	—	146	4	no	—
206	#117	—	2	4	no	—
207	#117	—	4	2	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	146	1	no	—
211	C₂×C₂²⋊C₄	—	4	1	no	—
212	C₂×C₂²⋊C₄	—	292	1	no	—
213	C₂×C₄²	—	4	1	no	—
214	C₂×C₁₆	—	73	2	no	—
215	C₂×C₁₆	—	73	2	no	—
216	C₂×C₁₆	—	146	2	no	—
217	C₂×C₁₆	—	146	2	no	—
218	#107	—	2	1	no	—
219	#107	—	2	1	no	—
220	#107	—	73	2	no	—
221	#107	—	146	1	no	—
222	#107	C₂²×D₇₃	1	2	yes	—
223	#107	C₂²×D₇₃	1	2	yes	—
224	#106	—	2	4	no	—
225	#102	—	146	4	no	—
226	C₂⁴	—	292	1	no	—
227	C₂⁴	#11924013	1	1	yes	characteristic
228	C₂×D₄	—	292	4	no	—
229	C₂×D₄	—	292	2	no	—
230	C₂×D₄	—	292	1	no	—
231	C₂×D₄	—	292	2	no	—
232	C₂×D₄	—	584	4	no	—
233	C₂×D₄	—	146	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	292	1	no	—
239	C₂²×C₄	#11924013	1	2	yes	—
240	C₂²×C₄	#11924013	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
242	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
243	D₈	—	292	2	no	—
244	D₈	—	146	2	no	—
245	D₈	—	146	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
251	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
252	C₂×C₈	#11924028	1	2	yes	—
253	C₂×C₈	#11924028	1	2	yes	—
254	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	292	1	no	—
260	C₄⋊C₄	—	146	2	no	—
261	C₂²⋊C₄	—	4	4	no	—
262	C₂²⋊C₄	—	292	4	no	—
263	C₄²	—	4	4	no	—
264	C₁₆	—	146	2	no	—
265	C₁₆	—	146	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	292	2	no	—
270	C₂³	—	584	2	no	—
271	C₂³	—	584	2	no	—
272	C₂³	—	584	2	no	—
273	C₂³	C₂×#11924013	1	2	yes	—
274	C₂³	C₂×#11924013	1	1	yes	characteristic
275	D₄	—	292	2	no	—
276	D₄	—	292	2	no	—
277	D₄	—	584	8	no	—
278	D₄	—	584	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	292	2	no	—
288	C₂×C₄	—	584	2	no	—
289	C₂×C₄	#14556574	1	2	yes	—
290	C₂×C₄	C₂×#11924013	1	2	yes	—
291	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
292	C₂×C₄	#14556622	1	2	yes	—
293	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	#14556574	1	2	yes	—
296	C₈	C₂×#11924028	1	1	yes	characteristic
297	C₈	C₂×#11924028	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	584	2	no	—
303	C₂²	—	584	2	no	—
304	C₂²	—	584	2	no	—
305	C₂²	—	1168	4	no	—
306	C₂²	#17988900	1	2	yes	—
307	C₂²	#17989017	1	2	yes	—
308	C₂²	#17989022	1	2	yes	—
309	C₂²	C₂²×#11924013	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	584	2	no	—
314	C₄	#17988900	1	2	yes	—
315	C₄	C₂×#14556574	1	1	yes	characteristic
316	C₄	C₂×#14556622	1	1	yes	characteristic
317	C₂	—	4	2	no	—
318	C₂	—	1168	2	no	—
319	C₂	#22612122	1	4	yes	direct product factor
320	C₂	C₂×#17988900	1	1	yes	characteristic
321	C₂	C₂×#17989017	1	1	yes	characteristic
322	C₂	C₂×#17989022	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11924013
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#427×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₈
prime radical	C₂×#17988823	C₂
hypercenter	C₂×#427	D₇₃
hypocenter	C₇₃	C₂×#2234
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. #22612122	C₂	direct (N × H)
3. C₂×#17988414	C₂	semidirect
4. C₂×#17988412	C₂	semidirect
5. C₂×#17988387	C₂	semidirect
6. C₂×#17988387	C₂	semidirect
7. C₂×C₂²⋊#11924003	C₂	semidirect
8. C₂×#427×C₇₃	C₂	semidirect
17. #17988414	C₂²	semidirect
18. #17988412	C₂²	semidirect
19. #17988387	C₂²	semidirect
20. #17988387	C₂²	semidirect
21. C₂²⋊#11924003	C₂²	semidirect
25. #427×C₇₃	C₂²	semidirect
157. C₂×C₇₃	#2234	semidirect
163. C₂×#427	D₇₃	semidirect
170. C₇₃	C₂×#2234	semidirect
180. #427	C₂×D₇₃	semidirect
319. C₂	#22612122	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×#17988823	2
#22612122	2
C₂×#17988414	2
C₂×#17988412	2
C₂×#17988387	2
C₂×C₂²⋊#11924003	2
C₂×#427×C₇₃	2
C₂×#2234	73

Subgroups — C2×#22612122 #33240754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22612122 #33240754