Subgroups — C₂×#22611798 #33239276

Order 18688.

734 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556534	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17988669	C₂	1	1	yes	maximal, characteristic
4	#22611798	C₂	1	8	yes	maximal, direct product factor
5	C₂×#17988412	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988389	C₂	1	1	yes	maximal, characteristic
7	C₂×#17988362	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#358×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₈₄	C₂²	1	1	yes	characteristic
10	C₂×Q₈×Dic₇₃	—	2	1	no	—
11	C₂×#14556553	—	2	1	no	—
12	C₂×#14556534	C₄	1	8	yes	complemented
13	C₂×#14556534	C₂²	1	4	yes	—
14	C₂²×#11924003	C₂²	1	1	yes	characteristic
15	Q₁₆×Dic₇₃	—	2	2	no	—
16	Q₁₆×Dic₇₃	C₂²	1	4	yes	complemented
17	#17988669	—	2	2	no	—
18	#17988669	C₂²	1	4	yes	—
19	C₂×#14556525	C₂²	1	1	yes	characteristic
20	C₂×#14556515	—	2	1	no	—
21	C₂×#14556511	—	2	1	no	—
22	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
23	#17988412	C₂²	1	4	yes	complemented
24	#17988389	C₂²	1	4	yes	—
25	#17988362	C₂²	1	4	yes	complemented
26	C₂²×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
28	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
29	#358×C₇₃	C₂²	1	4	yes	complemented
30	C₂²×D₂₉₂	—	2	1	no	—
31	C₂²×C₄×D₇₃	—	4	1	no	—
32	C₂×D₅₈₄	—	2	1	no	—
33	C₂×D₅₈₄	—	2	2	no	—
34	C₂×D₅₈₄	C₂×C₄	1	4	yes	—
35	C₂×D₅₈₄	C₂³	1	2	yes	—
36	C₂²×#10596831	D₄	1	1	yes	characteristic
37	Q₈×Dic₇₃	—	2	4	no	—
38	Q₈×Dic₇₃	—	4	2	no	—
39	#14556553	—	2	4	no	—
40	#14556553	—	4	2	no	—
41	C₂×C₄⋊Dic₇₃	—	2	1	no	—
42	C₂×C₄⋊Dic₇₃	—	4	1	no	—
43	C₂×#11924005	—	4	1	no	—
44	C₂×C₄×Dic₇₃	—	4	1	no	—
45	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
46	#14556534	C₂×C₄	1	16	yes	complemented
47	C₂×#11924003	C₂×C₄	1	4	yes	complemented
48	C₂×#11924003	C₂³	1	2	yes	—
49	#14556525	—	2	1	no	—
50	#14556525	C₂³	1	2	yes	—
51	#14556515	—	2	4	no	—
52	#14556511	—	2	4	no	—
53	C₈×Dic₇₃	D₄	1	2	yes	—
54	C₈×Dic₇₃	C₂³	1	2	yes	—
55	C₂²×Q₈×C₇₃	—	2	1	no	—
56	C₂×Q₁₆×C₇₃	—	2	1	no	—
57	C₂×Q₁₆×C₇₃	—	2	2	no	—
58	C₂×Q₁₆×C₇₃	C₂×C₄	1	4	yes	complemented
59	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
60	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
62	C₂×C₁₆×C₇₃	—	2	2	no	—
63	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
64	#107×C₇₃	—	2	1	no	—
65	#107×C₇₃	C₂³	1	2	yes	—
66	C₂³×D₇₃	—	4	1	no	—
67	C₂×D₂₉₂	—	2	4	no	—
68	C₂×D₂₉₂	—	2	2	no	—
69	C₂×D₂₉₂	—	4	1	no	—
70	C₂×D₂₉₂	—	4	2	no	—
71	C₂²×Dic₇₃	—	4	1	no	—
72	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
73	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
74	C₂×C₄×D₇₃	—	4	4	no	—
75	C₂×C₄×D₇₃	—	8	4	no	—
76	D₅₈₄	—	2	4	no	—
77	D₅₈₄	—	2	2	no	—
78	D₅₈₄	C₂²×C₄	1	4	yes	—
79	C₂×#10596831	—	2	2	no	—
80	C₂×#10596831	C₂×D₄	1	2	yes	—
81	C₄⋊Dic₇₃	—	2	2	no	—
82	C₄⋊Dic₇₃	—	4	1	no	—
83	C₄⋊Dic₇₃	—	4	4	no	—
84	#11924005	—	4	4	no	—
85	C₄×Dic₇₃	—	4	4	no	—
86	C₄×Dic₇₃	D₈	1	2	yes	—
87	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
88	#11924003	C₂²×C₄	1	4	yes	complemented
89	C₂×Q₈×C₇₃	—	2	4	no	—
90	C₂×Q₈×C₇₃	—	2	2	no	—
91	C₂×Q₈×C₇₃	—	4	1	no	—
92	C₂×Q₈×C₇₃	—	4	2	no	—
93	C₂²×C₄×C₇₃	—	4	1	no	—
94	C₂²×C₄×C₇₃	—	4	1	no	—
95	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
96	Q₁₆×C₇₃	—	2	4	no	—
97	Q₁₆×C₇₃	—	2	2	no	—
98	Q₁₆×C₇₃	C₂²×C₄	1	4	yes	complemented
99	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₇₃	#41	1	2	yes	—
102	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₇₃	—	2	2	no	—
104	C₄⋊C₄×C₇₃	—	4	1	no	—
105	C₁₆×C₇₃	—	2	2	no	—
106	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
107	C₂²×D₇₃	—	4	2	no	—
108	C₂²×D₇₃	—	4	4	no	—
109	C₂²×D₇₃	—	8	2	no	—
110	C₂²×D₇₃	—	8	2	no	—
111	Q₈×C₇₃	—	2	4	no	—
112	Q₈×C₇₃	—	4	2	no	—
113	Q₈×C₇₃	—	4	4	no	—
114	C₂×C₄×C₇₃	—	4	2	no	—
115	C₂×C₄×C₇₃	—	4	4	no	—
116	C₂×C₄×C₇₃	—	4	2	no	—
117	C₂×C₄×C₇₃	—	8	2	no	—
118	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₇₃	#135	1	2	yes	—
121	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×Dic₇₃	—	2	2	no	—
123	C₂×Dic₇₃	—	2	2	no	—
124	C₂×Dic₇₃	—	4	2	no	—
125	C₂×Dic₇₃	—	8	2	no	—
126	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
127	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
128	D₂₉₂	—	2	4	no	—
129	D₂₉₂	—	4	2	no	—
130	D₂₉₂	—	4	4	no	—
131	C₄×D₇₃	—	8	8	no	—
132	C₈×C₇₃	C₄×D₄	1	2	yes	—
133	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
134	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
135	#10596831	—	2	2	no	—
136	C₂²×C₇₃	C₄×D₈	1	2	yes	—
137	C₂²×C₇₃	#509	1	2	yes	—
138	C₂²×C₇₃	#510	1	2	yes	—
139	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
140	C₂×D₇₃	—	4	4	no	—
141	C₂×D₇₃	—	8	2	no	—
142	C₂×D₇₃	—	8	2	no	—
143	C₂×D₇₃	—	8	8	no	—
144	C₄×C₇₃	—	4	4	no	—
145	C₄×C₇₃	—	8	2	no	—
146	C₄×C₇₃	C₄×D₈	1	2	yes	—
147	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
149	Dic₇₃	—	2	2	no	—
150	Dic₇₃	—	2	2	no	—
151	Dic₇₃	—	8	2	no	—
152	C₂×#2175	—	73	1	no	maximal, Sylow
153	C₂×C₇₃	#2175	1	4	yes	complemented
154	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
155	C₂×C₇₃	C₂×#509	1	1	yes	characteristic
156	C₂×C₇₃	C₂×#510	1	1	yes	characteristic
157	D₇₃	—	8	4	no	—
158	C₂²×SD₃₂	—	73	1	no	—
159	C₂×C₄×Q₁₆	—	73	1	no	—
160	C₂×C₄×D₈	—	73	1	no	—
161	#2175	—	73	8	no	—
162	C₂×#366	—	73	1	no	—
163	C₂×#358	D₇₃	1	1	yes	characteristic, complemented
164	C₂×#357	—	73	1	no	—
165	C₂×#346	—	73	1	no	—
166	C₇₃	C₂×#2175	1	1	yes	characteristic, complemented, Sylow
167	C₂²×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
168	C₂²×D₈	—	73	1	no	—
169	C₂×C₄×Q₈	—	146	1	no	—
170	C₂×C₄×D₄	—	146	1	no	—
171	C₂×SD₃₂	—	73	8	no	—
172	C₂×SD₃₂	—	73	4	no	—
173	C₂²×C₁₆	—	73	1	no	—
174	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
175	C₄×Q₁₆	—	73	4	no	—
176	C₄×Q₁₆	—	146	2	no	—
177	C₄×D₈	—	73	4	no	—
178	C₄×D₈	—	146	2	no	—
179	C₂×#107	—	73	1	no	—
180	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
181	C₂×#103	—	146	1	no	—
182	C₂×#102	—	146	1	no	—
183	C₂×C₄×C₈	—	73	1	no	—
184	#366	—	73	4	no	—
185	#358	C₂×D₇₃	1	4	yes	complemented
186	#357	—	73	4	no	—
187	#346	—	73	4	no	—
188	C₂²×Q₈	—	2	1	no	—
189	C₂²×D₄	—	146	1	no	—
190	C₂³×C₄	—	292	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	—	2	2	no	—
193	C₂×Q₁₆	C₄×D₇₃	1	4	yes	complemented
194	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
195	C₂×D₈	—	73	4	no	—
196	C₂×D₈	—	73	2	no	—
197	C₂×D₈	—	146	1	no	—
198	C₂×D₈	—	146	2	no	—
199	C₂²×C₈	—	73	1	no	—
200	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
201	C₄×Q₈	—	292	2	no	—
202	C₄×Q₈	—	146	4	no	—
203	C₄×D₄	—	292	2	no	—
204	C₄×D₄	—	146	4	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	292	1	no	—
207	C₂×C₄⋊C₄	—	146	1	no	—
208	C₂×C₂²⋊C₄	—	292	1	no	—
209	C₂×C₄²	—	292	1	no	—
210	C₂×C₄²	—	73	1	no	—
211	SD₃₂	—	73	16	no	—
212	C₂×C₁₆	—	2	2	no	—
213	C₂×C₁₆	—	73	4	no	—
214	C₂×C₁₆	—	73	2	no	—
215	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
216	#107	—	2	1	no	—
217	#107	—	73	2	no	—
218	#107	—	146	1	no	—
219	#107	C₂²×D₇₃	1	2	yes	—
220	#103	—	146	4	no	—
221	#102	—	146	4	no	—
222	C₄×C₈	—	73	2	no	—
223	C₄×C₈	—	73	2	no	—
224	C₂⁴	—	292	1	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×D₄	—	292	1	no	—
230	C₂×D₄	—	292	2	no	—
231	C₂×D₄	—	146	4	no	—
232	C₂×D₄	—	146	2	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	292	4	no	—
236	C₂²×C₄	—	292	1	no	—
237	C₂²×C₄	—	584	4	no	—
238	C₂²×C₄	—	73	1	no	—
239	C₂²×C₄	—	73	1	no	—
240	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
241	Q₁₆	—	2	4	no	—
242	Q₁₆	—	2	2	no	—
243	Q₁₆	C₂×C₄×D₇₃	1	4	yes	complemented
244	D₈	—	73	4	no	—
245	D₈	—	146	4	no	—
246	D₈	—	146	2	no	—
247	C₂×C₈	—	73	2	no	—
248	C₂×C₈	—	146	2	no	—
249	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
250	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
251	C₂×C₈	#11924027	1	2	yes	—
252	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	292	4	no	—
256	C₄⋊C₄	—	292	1	no	—
257	C₄⋊C₄	—	146	2	no	—
258	C₂²⋊C₄	—	292	4	no	—
259	C₄²	—	292	4	no	—
260	C₄²	—	73	2	no	—
261	C₄²	—	73	2	no	—
262	C₁₆	—	2	2	no	—
263	C₁₆	—	73	4	no	—
264	C₂³	—	292	4	no	—
265	C₂³	—	292	2	no	—
266	C₂³	—	584	2	no	—
267	C₂³	—	584	2	no	—
268	C₂³	D₈×D₇₃	1	1	yes	characteristic
269	Q₈	—	2	4	no	—
270	Q₈	—	4	4	no	—
271	Q₈	—	4	2	no	—
272	D₄	—	292	4	no	—
273	D₄	—	292	2	no	—
274	D₄	—	146	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	—	292	2	no	—
280	C₂×C₄	—	584	8	no	—
281	C₂×C₄	—	584	2	no	—
282	C₂×C₄	—	73	2	no	—
283	C₂×C₄	—	73	2	no	—
284	C₂×C₄	—	146	2	no	—
285	C₂×C₄	—	146	2	no	—
286	C₂×C₄	#14556551	1	2	yes	—
287	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
288	C₂×C₄	#14556612	1	2	yes	—
289	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
290	C₈	—	146	2	no	—
291	C₈	#14556551	1	2	yes	—
292	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
293	C₈	C₂×#11924027	1	1	yes	characteristic
294	C₂²	—	292	4	no	—
295	C₂²	—	584	2	no	—
296	C₂²	—	584	2	no	—
297	C₂²	—	584	8	no	—
298	C₂²	#17988667	1	2	yes	—
299	C₂²	#17989024	1	2	yes	—
300	C₂²	#17989029	1	2	yes	—
301	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
302	C₄	—	4	4	no	—
303	C₄	—	8	2	no	—
304	C₄	—	584	2	no	—
305	C₄	—	146	2	no	—
306	C₄	—	146	2	no	—
307	C₄	#17988667	1	2	yes	—
308	C₄	C₂×#14556551	1	1	yes	characteristic
309	C₄	C₂×#14556612	1	1	yes	characteristic
310	C₂	—	584	4	no	—
311	C₂	#22611798	1	4	yes	direct product factor
312	C₂	C₂×#17988667	1	1	yes	characteristic
313	C₂	C₂×#17989024	1	1	yes	characteristic
314	C₂	C₂×#17989029	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₇₃
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#358×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₈
prime radical	C₂²×D₅₈₄	C₂²
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×#358	D₇₃
hypocenter	C₇₃	C₂×#2175
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#358

Complements

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

N	H ≅ G/N	type
2. C₂×Q₁₆×Dic₇₃	C₂	semidirect
4. #22611798	C₂	direct (N × H)
5. C₂×#17988412	C₂	semidirect
7. C₂×#17988362	C₂	semidirect
8. C₂×#358×C₇₃	C₂	semidirect
12. C₂×#14556534	C₄	semidirect
16. Q₁₆×Dic₇₃	C₂²	semidirect
23. #17988412	C₂²	semidirect
25. #17988362	C₂²	semidirect
29. #358×C₇₃	C₂²	semidirect
46. #14556534	C₂×C₄	semidirect
47. C₂×#11924003	C₂×C₄	semidirect
58. C₂×Q₁₆×C₇₃	C₂×C₄	semidirect
88. #11924003	C₂²×C₄	semidirect
98. Q₁₆×C₇₃	C₂²×C₄	semidirect
153. C₂×C₇₃	#2175	semidirect
163. C₂×#358	D₇₃	semidirect
166. C₇₃	C₂×#2175	semidirect
185. #358	C₂×D₇₃	semidirect
193. C₂×Q₁₆	C₄×D₇₃	semidirect
243. Q₁₆	C₂×C₄×D₇₃	semidirect
311. C₂	#22611798	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×#14556534	2
C₂×Q₁₆×Dic₇₃	2
C₂×#17988669	2
#22611798	2
C₂×#17988412	2
C₂×#17988389	2
C₂×#17988362	2
C₂×#358×C₇₃	2
C₂×#2175	73

Subgroups — C2×#22611798 #33239276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22611798 #33239276