Subgroups — C₂×#22609331 #33068248

Order 18688.

726 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988992	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17988878	C₂	1	2	yes	maximal
3	C₂×C₈×D₂₉₂	C₂	1	1	yes	maximal, characteristic
4	#22609331	C₂	1	8	yes	maximal, direct product factor
5	C₂×#17988335	C₂	1	2	yes	maximal, complemented
6	C₂×#335×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₇₃	—	2	1	no	—
8	C₂×#14556591	C₂²	1	2	yes	—
9	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
10	#17988992	—	2	2	no	—
11	#17988992	C₂²	1	4	yes	complemented
12	#17988878	—	2	4	no	—
13	#17988878	C₂²	1	8	yes	—
14	C₈×D₂₉₂	—	2	2	no	—
15	C₈×D₂₉₂	C₂²	1	4	yes	—
16	C₂×#14556525	—	2	1	no	—
17	C₂×#14556525	—	2	1	no	—
18	C₂×#14556523	—	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	#17988335	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	C₂²×#10596831	—	2	1	no	—
30	C₂×C₈×D₇₃	—	2	4	no	—
31	C₂×C₈×D₇₃	—	4	4	no	—
32	#14556591	—	2	2	no	—
33	#14556591	—	2	2	no	—
34	#14556591	—	2	2	no	—
35	#14556591	C₂³	1	4	yes	—
36	C₄×D₂₉₂	—	2	1	no	—
37	C₄×D₂₉₂	—	2	1	no	—
38	C₄×D₂₉₂	D₄	1	2	yes	—
39	C₄×D₂₉₂	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₇₃	—	2	2	no	—
41	C₂×C₄⋊Dic₇₃	—	2	2	no	—
42	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
43	C₂×#11924005	—	2	1	no	—
44	#14556525	—	2	4	no	—
45	#14556525	—	2	4	no	—
46	#14556523	—	2	8	no	—
47	C₄⋊#10596831	—	2	1	no	—
48	C₄⋊#10596831	C₂³	1	2	yes	—
49	#14556513	—	2	8	no	—
50	#14556503	—	2	4	no	—
51	C₂²×C₈×C₇₃	—	2	2	no	—
52	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₇₃	—	2	2	no	—
56	C₄⋊C₈×C₇₃	C₂³	1	4	yes	—
57	C₄×C₈×C₇₃	D₄	1	2	yes	—
58	C₄×C₈×C₇₃	C₂³	1	2	yes	—
59	C₂³×D₇₃	—	2	1	no	—
60	C₂×Dic₁₄₆	—	2	4	no	—
61	C₂×Dic₁₄₆	—	4	2	no	—
62	C₂×Dic₁₄₆	—	8	4	no	—
63	C₂×D₂₉₂	—	2	1	no	—
64	C₂×D₂₉₂	—	2	2	no	—
65	C₂×D₂₉₂	Q₁₆	1	4	yes	—
66	C₂×D₂₉₂	C₂×D₄	1	2	yes	—
67	C₂²×Dic₇₃	—	2	1	no	—
68	C₂²×Dic₇₃	—	4	2	no	—
69	C₂×C₄×D₇₃	—	2	2	no	—
70	C₂×C₄×D₇₃	—	2	2	no	—
71	C₂×C₄×D₇₃	—	4	2	no	—
72	C₂×C₄×D₇₃	—	4	2	no	—
73	C₂×#10596831	—	2	2	no	—
74	C₂×#10596831	—	4	2	no	—
75	C₈×D₇₃	—	4	8	no	—
76	C₄⋊Dic₇₃	—	2	1	no	—
77	C₄⋊Dic₇₃	—	2	4	no	—
78	C₄⋊Dic₇₃	—	2	4	no	—
79	C₄⋊Dic₇₃	—	4	2	no	—
80	C₄⋊Dic₇₃	—	4	2	no	—
81	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
82	#11924005	—	2	2	no	—
83	#11924005	—	2	2	no	—
84	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
87	C₂×C₈×C₇₃	—	2	2	no	—
88	C₂×C₈×C₇₃	—	2	4	no	—
89	C₂×C₈×C₇₃	—	4	4	no	—
90	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
91	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
93	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₇₃	#41	1	1	yes	characteristic
96	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
97	C₂³×C₇₃	#121	1	1	yes	characteristic
98	C₂²×D₇₃	—	2	2	no	—
99	C₂²×D₇₃	—	2	4	no	—
100	C₂²×D₇₃	—	4	2	no	—
101	C₂²×D₇₃	—	4	2	no	—
102	C₂×C₄×C₇₃	—	2	2	no	—
103	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
104	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
105	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
107	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
108	C₂×C₄×C₇₃	#135	1	2	yes	—
109	C₂×C₄×C₇₃	#137	1	2	yes	—
110	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
113	C₂×Dic₇₃	—	2	2	no	—
114	C₂×Dic₇₃	—	4	2	no	—
115	C₂×Dic₇₃	—	4	4	no	—
116	C₂×Dic₇₃	—	8	4	no	—
117	D₂₉₂	—	2	2	no	—
118	D₂₉₂	—	2	4	no	—
119	D₂₉₂	C₂×Q₁₆	1	4	yes	—
120	C₄×D₇₃	—	4	2	no	—
121	C₄×D₇₃	—	4	2	no	—
122	C₄×D₇₃	—	4	4	no	—
123	Dic₁₄₆	—	8	4	no	—
124	Dic₁₄₆	—	8	4	no	—
125	C₈×C₇₃	—	2	2	no	—
126	C₈×C₇₃	—	4	4	no	—
127	C₈×C₇₃	C₂×SD₁₆	1	4	yes	—
128	#10596831	—	4	2	no	—
129	C₂²×C₇₃	#461	1	2	yes	—
130	C₂²×C₇₃	#467	1	2	yes	—
131	C₂²×C₇₃	#472	1	2	yes	—
132	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
133	C₂×D₇₃	—	2	4	no	—
134	C₂×D₇₃	—	4	8	no	—
135	C₂×D₇₃	—	4	2	no	—
136	C₂×D₇₃	—	4	2	no	—
137	C₄×C₇₃	—	2	1	no	—
138	C₄×C₇₃	—	2	1	no	—
139	C₄×C₇₃	#461	1	2	yes	—
140	C₄×C₇₃	#467	1	2	yes	—
141	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
142	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
143	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
144	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
145	Dic₇₃	—	4	2	no	—
146	Dic₇₃	—	8	4	no	—
147	C₂×#1675	—	73	1	no	maximal, Sylow
148	C₂×C₇₃	#1675	1	4	yes	complemented
149	C₂×C₇₃	C₂×#461	1	1	yes	characteristic
150	C₂×C₇₃	C₂×#467	1	1	yes	characteristic
151	C₂×C₇₃	C₂×#472	1	1	yes	characteristic
152	D₇₃	—	4	4	no	—
153	C₂×#500	—	73	1	no	—
154	C₂×#476	—	73	2	no	—
155	C₂×C₈×D₄	—	73	1	no	—
156	#1675	—	73	8	no	—
157	C₂×#335	D₇₃	1	1	yes	characteristic, complemented
158	C₂×#333	—	73	2	no	—
159	C₇₃	C₂×#1675	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₈	—	146	1	no	—
161	C₂×C₄⋊Q₈	—	73	2	no	—
162	C₂×C₄×D₄	—	73	1	no	—
163	#500	—	73	4	no	—
164	#500	—	146	2	no	—
165	#476	—	73	8	no	—
166	#476	—	146	4	no	—
167	C₈×D₄	—	73	4	no	—
168	C₈×D₄	—	146	2	no	—
169	C₂×#107	—	146	1	no	—
170	C₂×#107	—	146	1	no	—
171	C₂×#106	—	146	2	no	—
172	C₂×C₄⋊C₈	—	73	1	no	—
173	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
174	C₂×#102	—	146	2	no	—
175	C₂×C₂²⋊C₈	—	146	1	no	—
176	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
177	#335	C₂×D₇₃	1	4	yes	complemented
178	#333	—	73	8	no	—
179	C₂²×Q₈	—	146	2	no	—
180	C₂²×D₄	—	73	1	no	—
181	C₂³×C₄	—	146	1	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	292	4	no	—
184	C₂²×C₈	—	146	1	no	—
185	C₂²×C₈	—	146	4	no	—
186	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
187	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
188	C₄⋊Q₈	—	73	4	no	—
189	C₄⋊Q₈	—	146	2	no	—
190	C₄⋊Q₈	—	146	2	no	—
191	C₄⋊Q₈	—	146	2	no	—
192	C₄×D₄	—	73	2	no	—
193	C₄×D₄	—	73	2	no	—
194	C₄×D₄	—	146	1	no	—
195	C₄×D₄	—	146	1	no	—
196	C₂×C₄⋊C₄	—	73	1	no	—
197	C₂×C₄⋊C₄	—	146	2	no	—
198	C₂×C₄⋊C₄	—	146	2	no	—
199	C₂×C₂²⋊C₄	—	146	1	no	—
200	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
201	#107	—	146	4	no	—
202	#107	—	146	4	no	—
203	#106	—	146	8	no	—
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	73	2	no	—
206	C₄⋊C₈	—	146	1	no	—
207	C₄⋊C₈	C₂²×D₇₃	1	4	yes	—
208	#102	—	146	8	no	—
209	C₂²⋊C₈	—	146	4	no	—
210	C₄×C₈	D₂₉₂	1	2	yes	—
211	C₄×C₈	C₂²×D₇₃	1	2	yes	—
212	C₂⁴	—	146	1	no	—
213	C₂×Q₈	—	292	2	no	—
214	C₂×Q₈	—	584	4	no	—
215	C₂×Q₈	—	146	4	no	—
216	C₂×D₄	—	73	4	no	—
217	C₂×D₄	—	73	2	no	—
218	C₂×D₄	—	146	1	no	—
219	C₂×D₄	—	146	2	no	—
220	C₂²×C₄	—	292	2	no	—
221	C₂²×C₄	—	292	2	no	—
222	C₂²×C₄	—	292	2	no	—
223	C₂²×C₄	—	146	1	no	—
224	C₂²×C₄	—	146	2	no	—
225	C₂²×C₄	—	146	2	no	—
226	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
228	C₂²×C₄	#11924029	1	1	yes	characteristic
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	4	4	no	—
232	C₂×C₈	—	292	2	no	—
233	C₂×C₈	—	292	8	no	—
234	C₂×C₈	—	146	2	no	—
235	C₂×C₈	#11924017	1	4	yes	—
236	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
237	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
238	C₄⋊C₄	—	292	2	no	—
239	C₄⋊C₄	—	292	2	no	—
240	C₄⋊C₄	—	73	2	no	—
241	C₄⋊C₄	—	146	4	no	—
242	C₄⋊C₄	—	146	4	no	—
243	C₄⋊C₄	—	146	1	no	—
244	C₂²⋊C₄	—	146	2	no	—
245	C₂²⋊C₄	—	146	2	no	—
246	C₄²	D₄×D₇₃	1	1	yes	characteristic
247	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
248	C₄²	#11924029	1	1	yes	characteristic
249	C₄²	C₂³×D₇₃	1	1	yes	characteristic
250	C₂³	—	292	2	no	—
251	C₂³	—	292	2	no	—
252	C₂³	—	146	2	no	—
253	C₂³	—	146	4	no	—
254	C₂³	#14556566	1	1	yes	characteristic
255	Q₈	—	584	4	no	—
256	Q₈	—	584	4	no	—
257	D₄	—	73	4	no	—
258	D₄	—	146	4	no	—
259	D₄	—	146	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	292	2	no	—
262	C₂×C₄	—	292	2	no	—
263	C₂×C₄	—	292	4	no	—
264	C₂×C₄	—	292	2	no	—
265	C₂×C₄	—	292	4	no	—
266	C₂×C₄	—	584	4	no	—
267	C₂×C₄	—	146	2	no	—
268	C₂×C₄	#14556566	1	1	yes	characteristic
269	C₂×C₄	#14556566	1	1	yes	characteristic
270	C₂×C₄	#14556566	1	1	yes	characteristic
271	C₂×C₄	C₂×#11924017	1	2	yes	—
272	C₂×C₄	Q₁₆×D₇₃	1	2	yes	—
273	C₂×C₄	#14556610	1	2	yes	—
274	C₂×C₄	#14556624	1	2	yes	—
275	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
277	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	4	4	no	—
280	C₈	—	292	2	no	—
281	C₈	C₂×#11924017	1	4	yes	—
282	C₂²	—	292	8	no	—
283	C₂²	—	292	2	no	—
284	C₂²	—	292	2	no	—
285	C₂²	—	146	4	no	—
286	C₂²	#17988793	1	2	yes	—
287	C₂²	#17988829	1	2	yes	—
288	C₂²	#17988858	1	2	yes	—
289	C₂²	C₂×#14556566	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	292	2	no	—
293	C₄	—	584	4	no	—
294	C₄	#17988793	1	2	yes	—
295	C₄	#17988829	1	2	yes	—
296	C₄	C₂²×#11924017	1	1	yes	characteristic
297	C₄	C₂×Q₁₆×D₇₃	1	1	yes	characteristic
298	C₄	C₂×#14556610	1	1	yes	characteristic
299	C₄	C₂×#14556624	1	1	yes	characteristic
300	C₂	—	292	4	no	—
301	C₂	#22609331	1	4	yes	direct product factor
302	C₂	C₂×#17988793	1	1	yes	characteristic
303	C₂	C₂×#17988829	1	1	yes	characteristic
304	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	C₂²×D₂₉₂	D₄
hypercenter	C₂×#335	D₇₃
hypocenter	C₇₃	C₂×#1675
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×#17988992	C₂	semidirect
4. #22609331	C₂	direct (N × H)
5. C₂×#17988335	C₂	semidirect
6. C₂×#335×C₇₃	C₂	semidirect
11. #17988992	C₂²	semidirect
22. #17988335	C₂²	semidirect
25. #335×C₇₃	C₂²	semidirect
148. C₂×C₇₃	#1675	semidirect
157. C₂×#335	D₇₃	semidirect
159. C₇₃	C₂×#1675	semidirect
177. #335	C₂×D₇₃	semidirect
301. C₂	#22609331	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×#17988992	2
C₂×#17988878	2
C₂×C₈×D₂₉₂	2
#22609331	2
C₂×#17988335	2
C₂×#335×C₇₃	2
C₂×#1675	73

Subgroups — C2×#22609331 #33068248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22609331 #33068248