Subgroups — C₂×#22611977 #33240215

Order 18688.

766 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556536	C₂	1	2	yes	maximal
2	C₂×#17988967	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₅₈₄	C₂	1	1	yes	maximal, characteristic
4	#22611977	C₂	1	8	yes	maximal, direct product factor
5	C₂×#17988392	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₂₉₂	—	2	1	no	—
8	C₂²×Dic₂₉₂	C₂²	1	2	yes	—
9	C₂²×D₅₈₄	C₂²	1	1	yes	characteristic
10	C₂×#14556591	—	2	1	no	—
11	C₂×C₄×D₂₉₂	—	2	1	no	—
12	C₂×#14556536	—	2	8	no	—
13	C₂×#14556536	C₂²	1	8	yes	—
14	#17988967	—	2	2	no	—
15	#17988967	C₂²	1	4	yes	complemented
16	C₄×D₅₈₄	—	2	2	no	—
17	C₄×D₅₈₄	C₂²	1	4	yes	—
18	C₂×#14556525	C₂²	1	1	yes	characteristic
19	C₂×#14556513	—	2	1	no	—
20	#17988392	C₂²	1	8	yes	complemented
21	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₁₆×C₇₃	C₂²	1	4	yes	complemented
24	C₂²×Dic₁₄₆	—	2	2	no	—
25	C₂²×D₂₉₂	—	2	1	no	—
26	C₂²×C₄×D₇₃	—	4	1	no	—
27	C₂×Dic₂₉₂	—	2	4	no	—
28	C₂×Dic₂₉₂	—	2	2	no	—
29	C₂×Dic₂₉₂	—	2	4	no	—
30	C₂×Dic₂₉₂	—	2	4	no	—
31	C₂×Dic₂₉₂	—	4	4	no	—
32	C₂×Dic₂₉₂	C₂³	1	4	yes	—
33	C₂×D₅₈₄	—	2	1	no	—
34	C₂×D₅₈₄	—	2	2	no	—
35	C₂×D₅₈₄	D₄	1	4	yes	—
36	C₂×D₅₈₄	C₂³	1	2	yes	—
37	#14556591	—	2	4	no	—
38	#14556591	—	4	2	no	—
39	C₄×D₂₉₂	—	2	4	no	—
40	C₄×D₂₉₂	—	4	2	no	—
41	C₂×C₄⋊Dic₇₃	—	2	1	no	—
42	C₂×C₄⋊Dic₇₃	—	4	1	no	—
43	C₂×C₄⋊Dic₇₃	—	4	1	no	—
44	C₂×#11924005	—	4	1	no	—
45	#14556536	—	2	16	no	—
46	#14556525	—	2	1	no	—
47	#14556525	C₂³	1	2	yes	—
48	#14556513	—	2	4	no	—
49	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
52	C₂×C₁₆×C₇₃	—	2	4	no	—
53	C₂×C₁₆×C₇₃	C₂³	1	4	yes	—
54	C₄×C₈×C₇₃	D₄	1	2	yes	—
55	C₄×C₈×C₇₃	C₂³	1	2	yes	—
56	C₂³×D₇₃	—	4	1	no	—
57	C₂×Dic₁₄₆	—	2	4	no	—
58	C₂×Dic₁₄₆	—	4	4	no	—
59	C₂×Dic₁₄₆	—	4	4	no	—
60	C₂×Dic₁₄₆	—	4	2	no	—
61	C₂×D₂₉₂	—	2	4	no	—
62	C₂×D₂₉₂	—	2	2	no	—
63	C₂×D₂₉₂	—	4	1	no	—
64	C₂×D₂₉₂	—	4	2	no	—
65	C₂²×Dic₇₃	—	4	1	no	—
66	C₂²×Dic₇₃	—	4	2	no	—
67	C₂×C₄×D₇₃	—	4	4	no	—
68	C₂×C₄×D₇₃	—	8	4	no	—
69	Dic₂₉₂	—	2	4	no	—
70	Dic₂₉₂	—	2	4	no	—
71	Dic₂₉₂	—	4	8	no	—
72	Dic₂₉₂	—	4	4	no	—
73	D₅₈₄	—	2	4	no	—
74	D₅₈₄	—	2	2	no	—
75	D₅₈₄	C₂×D₄	1	4	yes	—
76	C₄⋊Dic₇₃	—	2	2	no	—
77	C₄⋊Dic₇₃	—	4	1	no	—
78	C₄⋊Dic₇₃	—	4	4	no	—
79	C₄⋊Dic₇₃	—	4	4	no	—
80	#11924005	—	4	4	no	—
81	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₇₃	—	2	2	no	—
85	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₇₃	#41	1	2	yes	—
89	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
90	C₄²×C₇₃	D₈	1	2	yes	—
91	C₄²×C₇₃	C₂×D₄	1	2	yes	—
92	C₁₆×C₇₃	—	2	4	no	—
93	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
94	C₂²×D₇₃	—	4	2	no	—
95	C₂²×D₇₃	—	4	4	no	—
96	C₂²×D₇₃	—	8	2	no	—
97	C₂²×D₇₃	—	8	2	no	—
98	C₂×C₄×C₇₃	—	2	2	no	—
99	C₂×C₄×C₇₃	SD₃₂	1	4	yes	complemented
100	C₂×C₄×C₇₃	#121	1	2	yes	—
101	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×Dic₇₃	—	4	2	no	—
107	C₂×Dic₇₃	—	4	4	no	—
108	C₂×Dic₇₃	—	8	4	no	—
109	C₂×Dic₇₃	—	8	2	no	—
110	D₂₉₂	—	2	4	no	—
111	D₂₉₂	—	4	2	no	—
112	D₂₉₂	—	4	4	no	—
113	C₄×D₇₃	—	8	8	no	—
114	Dic₁₄₆	—	4	4	no	—
115	Dic₁₄₆	—	4	4	no	—
116	Dic₁₄₆	—	8	4	no	—
117	C₈×C₇₃	—	2	2	no	—
118	C₈×C₇₃	#121	1	2	yes	—
119	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
121	C₂²×C₇₃	#459	1	2	yes	—
122	C₂²×C₇₃	C₂×SD₃₂	1	2	yes	—
123	C₂²×C₇₃	#510	1	2	yes	—
124	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
125	C₂×D₇₃	—	4	4	no	—
126	C₂×D₇₃	—	8	2	no	—
127	C₂×D₇₃	—	8	2	no	—
128	C₂×D₇₃	—	8	8	no	—
129	C₄×C₇₃	—	2	2	no	—
130	C₄×C₇₃	#459	1	2	yes	—
131	C₄×C₇₃	C₂×SD₃₂	1	4	yes	complemented
132	C₄×C₇₃	C₂×#121	1	1	yes	characteristic
133	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
134	Dic₇₃	—	8	2	no	—
135	Dic₇₃	—	8	4	no	—
136	C₂×#2208	—	73	1	no	maximal, Sylow
137	C₂×C₇₃	#2208	1	4	yes	complemented
138	C₂×C₇₃	C₂×#459	1	1	yes	characteristic
139	C₂×C₇₃	C₂²×SD₃₂	1	1	yes	characteristic
140	C₂×C₇₃	C₂×#510	1	1	yes	characteristic
141	D₇₃	—	8	4	no	—
142	C₂²×SD₃₂	—	73	2	no	—
143	C₂×#494	—	73	1	no	—
144	C₂×C₄×D₈	—	73	1	no	—
145	#2208	—	73	8	no	—
146	C₂×C₄⋊C₁₆	D₇₃	1	1	yes	characteristic, complemented
147	C₂×#358	—	73	2	no	—
148	C₇₃	C₂×#2208	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₁₆	—	73	2	no	—
150	C₂²×Q₁₆	—	146	1	no	—
151	C₂²×D₈	—	73	1	no	—
152	C₂×C₄⋊Q₈	—	146	1	no	—
153	C₂×C₄×D₄	—	146	1	no	—
154	C₂×SD₃₂	—	73	8	no	—
155	C₂×SD₃₂	—	146	8	no	—
156	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
157	#494	—	73	4	no	—
158	#494	—	146	2	no	—
159	C₄×D₈	—	73	4	no	—
160	C₄×D₈	—	146	2	no	—
161	C₂×#107	—	73	1	no	—
162	C₂×#102	—	146	1	no	—
163	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
164	C₄⋊C₁₆	C₂×D₇₃	1	4	yes	complemented
165	#358	—	73	8	no	—
166	C₂²×Q₈	—	146	2	no	—
167	C₂²×D₄	—	146	1	no	—
168	C₂³×C₄	—	292	1	no	—
169	C₂×Q₁₆	—	292	4	no	—
170	C₂×Q₁₆	—	73	4	no	—
171	C₂×Q₁₆	—	146	4	no	—
172	C₂×Q₁₆	—	146	2	no	—
173	C₂×Q₁₆	—	146	4	no	—
174	C₂×Q₁₆	—	146	4	no	—
175	C₂×D₈	—	73	4	no	—
176	C₂×D₈	—	73	2	no	—
177	C₂×D₈	—	146	1	no	—
178	C₂×D₈	—	146	2	no	—
179	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
181	C₄⋊Q₈	—	292	2	no	—
182	C₄⋊Q₈	—	146	4	no	—
183	C₄×D₄	—	292	2	no	—
184	C₄×D₄	—	146	4	no	—
185	C₂×C₄⋊C₄	—	292	1	no	—
186	C₂×C₄⋊C₄	—	292	1	no	—
187	C₂×C₄⋊C₄	—	146	1	no	—
188	C₂×C₂²⋊C₄	—	292	1	no	—
189	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
190	SD₃₂	—	146	16	no	—
191	C₂×C₁₆	—	2	4	no	—
192	C₂×C₁₆	C₂²×D₇₃	1	4	yes	—
193	#107	—	73	2	no	—
194	#107	—	146	1	no	—
195	#102	—	146	4	no	—
196	C₄×C₈	D₂₉₂	1	2	yes	—
197	C₄×C₈	C₂²×D₇₃	1	2	yes	—
198	C₂⁴	—	292	1	no	—
199	C₂×Q₈	—	292	2	no	—
200	C₂×Q₈	—	292	4	no	—
201	C₂×Q₈	—	292	4	no	—
202	C₂×Q₈	—	146	4	no	—
203	C₂×D₄	—	292	1	no	—
204	C₂×D₄	—	292	2	no	—
205	C₂×D₄	—	146	4	no	—
206	C₂×D₄	—	146	2	no	—
207	C₂²×C₄	—	292	1	no	—
208	C₂²×C₄	—	292	2	no	—
209	C₂²×C₄	—	292	4	no	—
210	C₂²×C₄	—	584	4	no	—
211	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
212	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
214	Q₁₆	—	292	8	no	—
215	Q₁₆	—	292	4	no	—
216	Q₁₆	—	146	4	no	—
217	Q₁₆	—	146	4	no	—
218	D₈	—	73	4	no	—
219	D₈	—	146	4	no	—
220	D₈	—	146	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	D₄×D₇₃	1	2	yes	—
223	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
224	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
225	C₂×C₈	#11924029	1	2	yes	—
226	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
227	C₄⋊C₄	—	292	4	no	—
228	C₄⋊C₄	—	292	4	no	—
229	C₄⋊C₄	—	292	1	no	—
230	C₄⋊C₄	—	146	2	no	—
231	C₂²⋊C₄	—	292	4	no	—
232	C₄²	D₅₈₄	1	2	yes	—
233	C₄²	C₂×D₂₉₂	1	2	yes	—
234	C₁₆	—	2	4	no	—
235	C₂³	—	292	4	no	—
236	C₂³	—	292	2	no	—
237	C₂³	—	584	2	no	—
238	C₂³	—	584	2	no	—
239	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
240	Q₈	—	292	4	no	—
241	Q₈	—	292	4	no	—
242	Q₈	—	584	4	no	—
243	D₄	—	292	2	no	—
244	D₄	—	292	4	no	—
245	D₄	—	146	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	292	2	no	—
248	C₂×C₄	—	292	4	no	—
249	C₂×C₄	—	584	2	no	—
250	C₂×C₄	—	584	4	no	—
251	C₂×C₄	—	584	8	no	—
252	C₂×C₄	#14556536	1	4	yes	complemented
253	C₂×C₄	#14556566	1	2	yes	—
254	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
255	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
256	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
257	C₂×C₄	#14556619	1	2	yes	—
258	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	#14556566	1	2	yes	—
261	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
262	C₈	C₂×#11924029	1	1	yes	characteristic
263	C₂²	—	292	4	no	—
264	C₂²	—	584	8	no	—
265	C₂²	—	584	2	no	—
266	C₂²	—	584	2	no	—
267	C₂²	#17988783	1	2	yes	—
268	C₂²	C₂×#14556536	1	2	yes	—
269	C₂²	#17989030	1	2	yes	—
270	C₂²	C₂²×D₅₈₄	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	584	2	no	—
273	C₄	—	584	4	no	—
274	C₄	#17988783	1	2	yes	—
275	C₄	C₂×#14556536	1	4	yes	complemented
276	C₄	C₂×#14556566	1	1	yes	characteristic
277	C₄	C₂×#14556619	1	1	yes	characteristic
278	C₂	—	584	4	no	—
279	C₂	#22611977	1	4	yes	direct product factor
280	C₂	C₂×#17988783	1	1	yes	characteristic
281	C₂	C₂²×#14556536	1	1	yes	characteristic
282	C₂	C₂×#17989030	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₈₄
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×C₄⋊C₁₆×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₈
prime radical	C₂²×D₅₈₄	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₇₃
hypocenter	C₇₃	C₂×#2208
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#17988967	C₂	semidirect
4. #22611977	C₂	direct (N × H)
5. C₂×#17988392	C₂	semidirect
6. C₂×C₄⋊C₁₆×C₇₃	C₂	semidirect
15. #17988967	C₂²	semidirect
20. #17988392	C₂²	semidirect
23. C₄⋊C₁₆×C₇₃	C₂²	semidirect
99. C₂×C₄×C₇₃	SD₃₂	semidirect
131. C₄×C₇₃	C₂×SD₃₂	semidirect
137. C₂×C₇₃	#2208	semidirect
146. C₂×C₄⋊C₁₆	D₇₃	semidirect
148. C₇₃	C₂×#2208	semidirect
164. C₄⋊C₁₆	C₂×D₇₃	semidirect
252. C₂×C₄	#14556536	semidirect
275. C₄	C₂×#14556536	semidirect
279. C₂	#22611977	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×#14556536	2
C₂×#17988967	2
C₂×C₄×D₅₈₄	2
#22611977	2
C₂×#17988392	2
C₂×C₄⋊C₁₆×C₇₃	2
C₂×#2208	73

Subgroups — C2×#22611977 #33240215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22611977 #33240215