Subgroups — C₂×#22609301 #33068318

Order 18688.

702 conjugacy classes of subgroups.

The subgroup lattice has 292 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
2	C₂×#17988872	C₂	1	2	yes	maximal
3	#22609301	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988340	C₂	1	1	yes	maximal, characteristic
5	C₂×#17988334	C₂	1	2	yes	maximal
6	C₂×C₈×Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₉₂	—	2	1	no	—
8	C₂²×Dic₂₉₂	—	2	1	no	—
9	C₂×#14556591	C₂²	1	2	yes	—
10	#17988967	—	2	2	no	—
11	#17988967	C₂²	1	4	yes	—
12	#17988872	—	2	4	no	—
13	#17988872	C₂²	1	8	yes	—
14	C₂×#14556522	—	2	2	no	—
15	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
16	C₂×#14556515	—	2	2	no	—
17	#17988340	C₂²	1	4	yes	—
18	#17988334	C₂²	1	8	yes	—
19	C₂×C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₈×Q₈×C₇₃	—	2	2	no	—
21	C₈×Q₈×C₇₃	C₂²	1	4	yes	—
22	C₂×C₄⋊C₈×C₇₃	—	2	1	no	—
23	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₇₃	—	2	1	no	—
25	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₄₆	—	2	2	no	—
27	C₂×Dic₂₉₂	—	2	4	no	—
28	C₂×Dic₂₉₂	—	2	4	no	—
29	C₂×Dic₂₉₂	—	4	4	no	—
30	C₂×Dic₂₉₂	—	4	4	no	—
31	C₂²×#10596831	—	2	2	no	—
32	#14556591	—	2	2	no	—
33	#14556591	—	2	2	no	—
34	#14556591	—	2	2	no	—
35	#14556591	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₇₃	—	2	2	no	—
37	C₂×C₄⋊Dic₇₃	—	2	2	no	—
38	#14556522	—	2	8	no	—
39	C₄⋊#10596831	—	2	2	no	—
40	C₄⋊#10596831	C₂³	1	4	yes	—
41	#14556515	—	2	8	no	—
42	C₂²×Q₈×C₇₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₃	—	2	1	no	—
44	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
46	C₄×Q₈×C₇₃	—	2	1	no	—
47	C₄×Q₈×C₇₃	—	2	1	no	—
48	C₄×Q₈×C₇₃	D₄	1	2	yes	—
49	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₇₃	—	2	1	no	—
53	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₇₃	—	2	4	no	—
55	C₄⋊C₈×C₇₃	—	2	1	no	—
56	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
57	C₄×C₈×C₇₃	—	2	4	no	—
58	C₄×C₈×C₇₃	D₄	1	2	yes	—
59	C₄×C₈×C₇₃	C₂³	1	2	yes	—
60	C₂×Dic₁₄₆	—	2	4	no	—
61	C₂×Dic₁₄₆	—	4	4	no	—
62	C₂×Dic₁₄₆	—	4	4	no	—
63	C₂×Dic₁₄₆	—	4	2	no	—
64	C₂²×Dic₇₃	—	4	2	no	—
65	Dic₂₉₂	—	4	8	no	—
66	Dic₂₉₂	—	4	8	no	—
67	C₂×#10596831	—	2	4	no	—
68	C₂×#10596831	—	4	4	no	—
69	C₄⋊Dic₇₃	—	2	4	no	—
70	C₄⋊Dic₇₃	—	2	4	no	—
71	C₄⋊Dic₇₃	—	4	2	no	—
72	C₄⋊Dic₇₃	—	4	2	no	—
73	C₂×Q₈×C₇₃	—	2	1	no	—
74	C₂×Q₈×C₇₃	—	2	2	no	—
75	C₂×Q₈×C₇₃	Q₁₆	1	4	yes	complemented
76	C₂×Q₈×C₇₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₇₃	—	2	1	no	—
78	C₂²×C₄×C₇₃	—	2	1	no	—
79	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
82	C₂×C₈×C₇₃	—	2	2	no	—
83	C₂×C₈×C₇₃	—	2	2	no	—
84	C₂×C₈×C₇₃	—	4	2	no	—
85	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
86	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
88	C₄⋊C₄×C₇₃	—	2	2	no	—
89	C₄⋊C₄×C₇₃	—	2	1	no	—
90	C₄⋊C₄×C₇₃	—	2	2	no	—
91	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
92	C₄²×C₇₃	—	2	2	no	—
93	C₄²×C₇₃	—	2	2	no	—
94	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₇₃	#41	1	1	yes	characteristic
97	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
98	C₂³×C₇₃	#121	1	1	yes	characteristic
99	Q₈×C₇₃	—	2	2	no	—
100	Q₈×C₇₃	—	2	4	no	—
101	Q₈×C₇₃	C₂×Q₁₆	1	4	yes	complemented
102	C₂×C₄×C₇₃	—	2	2	no	—
103	C₂×C₄×C₇₃	—	2	4	no	—
104	C₂×C₄×C₇₃	—	2	2	no	—
105	C₂×C₄×C₇₃	—	2	2	no	—
106	C₂×C₄×C₇₃	—	4	2	no	—
107	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
108	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
109	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
111	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
112	C₂×C₄×C₇₃	#135	1	2	yes	—
113	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
114	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
117	C₂×Dic₇₃	—	4	4	no	—
118	C₂×Dic₇₃	—	8	4	no	—
119	Dic₁₄₆	—	4	4	no	—
120	Dic₁₄₆	—	4	4	no	—
121	Dic₁₄₆	—	8	4	no	—
122	C₈×C₇₃	—	2	2	no	—
123	C₈×C₇₃	—	4	2	no	—
124	C₈×C₇₃	C₂×SD₁₆	1	4	yes	—
125	#10596831	—	4	4	no	—
126	C₂²×C₇₃	#460	1	2	yes	—
127	C₂²×C₇₃	#467	1	2	yes	—
128	C₂²×C₇₃	#473	1	2	yes	—
129	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
130	C₄×C₇₃	—	2	4	no	—
131	C₄×C₇₃	—	2	1	no	—
132	C₄×C₇₃	—	2	1	no	—
133	C₄×C₇₃	—	4	2	no	—
134	C₄×C₇₃	#460	1	2	yes	—
135	C₄×C₇₃	#467	1	2	yes	—
136	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
137	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
138	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
139	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
140	Dic₇₃	—	8	4	no	—
141	C₂×#1670	—	73	1	no	maximal, Sylow
142	C₂×C₇₃	#1670	1	4	yes	complemented
143	C₂×C₇₃	C₂×#460	1	1	yes	characteristic
144	C₂×C₇₃	C₂×#467	1	1	yes	characteristic
145	C₂×C₇₃	C₂×#473	1	1	yes	characteristic
146	C₂×#494	—	73	1	no	—
147	C₂×#475	—	73	2	no	—
148	C₂×C₈×Q₈	D₇₃	1	1	yes	characteristic
149	#1670	—	73	8	no	—
150	C₂×#335	—	73	1	no	—
151	C₂×#332	—	73	2	no	—
152	C₇₃	C₂×#1670	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₁₆	—	146	1	no	—
154	C₂²×Q₁₆	—	146	1	no	—
155	C₂×C₄⋊Q₈	—	73	2	no	—
156	C₂×C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
157	#494	—	73	4	no	—
158	#494	—	146	2	no	—
159	#475	—	73	8	no	—
160	#475	—	146	4	no	—
161	C₈×Q₈	—	2	2	no	—
162	C₈×Q₈	C₂×D₇₃	1	4	yes	—
163	C₂×#106	—	146	2	no	—
164	C₂×C₄⋊C₈	—	2	1	no	—
165	C₂×C₄⋊C₈	—	73	2	no	—
166	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
167	C₂×#103	—	146	2	no	—
168	C₂×C₄×C₈	—	2	1	no	—
169	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
170	#335	—	73	4	no	—
171	#332	—	73	8	no	—
172	C₂²×Q₈	—	146	2	no	—
173	C₂²×Q₈	D₂₉₂	1	1	yes	characteristic
174	C₂×Q₁₆	—	292	4	no	—
175	C₂×Q₁₆	—	292	4	no	—
176	C₂×Q₁₆	—	146	4	no	—
177	C₂×Q₁₆	—	146	4	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	146	2	no	—
180	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
181	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
182	C₄⋊Q₈	—	73	4	no	—
183	C₄⋊Q₈	—	146	2	no	—
184	C₄⋊Q₈	—	146	2	no	—
185	C₄⋊Q₈	—	146	2	no	—
186	C₄×Q₈	—	2	1	no	—
187	C₄×Q₈	—	2	1	no	—
188	C₄×Q₈	D₂₉₂	1	2	yes	—
189	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	146	2	no	—
192	C₂×C₄⋊C₄	—	146	2	no	—
193	C₂×C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
196	#106	—	146	8	no	—
197	C₄⋊C₈	—	2	4	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	73	4	no	—
200	C₄⋊C₈	—	146	2	no	—
201	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
202	#103	—	146	8	no	—
203	C₄×C₈	—	2	4	no	—
204	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
205	C₄×C₈	C₂²×D₇₃	1	2	yes	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	292	2	no	—
209	C₂×Q₈	—	292	4	no	—
210	C₂×Q₈	—	292	4	no	—
211	C₂×Q₈	—	146	4	no	—
212	C₂×Q₈	Dic₂₉₂	1	4	yes	complemented
213	C₂×Q₈	C₂×D₂₉₂	1	2	yes	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	292	2	no	—
217	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
219	C₂²×C₄	#11924028	1	1	yes	characteristic
220	Q₁₆	—	292	8	no	—
221	Q₁₆	—	292	8	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	292	4	no	—
226	C₂×C₈	—	146	4	no	—
227	C₂×C₈	#11924015	1	4	yes	—
228	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
229	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	292	2	no	—
234	C₄⋊C₄	—	292	2	no	—
235	C₄⋊C₄	—	146	4	no	—
236	C₄⋊C₄	—	146	4	no	—
237	C₄⋊C₄	C₂×D₂₉₂	1	2	yes	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
241	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
242	C₄²	#11924028	1	1	yes	characteristic
243	C₄²	C₂³×D₇₃	1	1	yes	characteristic
244	C₂³	#14556565	1	1	yes	characteristic
245	Q₈	—	2	4	no	—
246	Q₈	—	2	2	no	—
247	Q₈	—	292	4	no	—
248	Q₈	—	292	4	no	—
249	Q₈	—	584	4	no	—
250	Q₈	C₂×Dic₂₉₂	1	4	yes	complemented
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	292	4	no	—
257	C₂×C₄	—	584	4	no	—
258	C₂×C₄	#14556565	1	1	yes	characteristic
259	C₂×C₄	#14556565	1	1	yes	characteristic
260	C₂×C₄	#14556565	1	1	yes	characteristic
261	C₂×C₄	C₂×#11924015	1	2	yes	—
262	C₂×C₄	C₂×Dic₂₉₂	1	2	yes	—
263	C₂×C₄	#14556613	1	2	yes	—
264	C₂×C₄	#14556616	1	2	yes	—
265	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
267	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	4	2	no	—
270	C₈	—	292	4	no	—
271	C₈	C₂×#11924015	1	4	yes	—
272	C₂²	#17988787	1	2	yes	—
273	C₂²	#17988828	1	2	yes	—
274	C₂²	#17988862	1	2	yes	—
275	C₂²	C₂×#14556565	1	1	yes	characteristic
276	C₄	—	2	4	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	4	2	no	—
280	C₄	—	584	4	no	—
281	C₄	#17988787	1	2	yes	—
282	C₄	#17988828	1	2	yes	—
283	C₄	C₂²×#11924015	1	1	yes	characteristic
284	C₄	C₂²×Dic₂₉₂	1	1	yes	characteristic
285	C₄	C₂×#14556613	1	1	yes	characteristic
286	C₄	C₂×#14556616	1	1	yes	characteristic
287	C₂	#22609301	1	4	yes	direct product factor
288	C₂	C₂×#17988787	1	1	yes	characteristic
289	C₂	C₂×#17988828	1	1	yes	characteristic
290	C₂	C₂×#17988862	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556565
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×C₈×Q₈×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	#121
hypercenter	C₂×C₈×Q₈	D₇₃
hypocenter	C₇₃	C₂×#1670
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #22609301	C₂	direct (N × H)
75. C₂×Q₈×C₇₃	Q₁₆	semidirect
101. Q₈×C₇₃	C₂×Q₁₆	semidirect
142. C₂×C₇₃	#1670	semidirect
152. C₇₃	C₂×#1670	semidirect
212. C₂×Q₈	Dic₂₉₂	semidirect
250. Q₈	C₂×Dic₂₉₂	semidirect
287. C₂	#22609301	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×#17988872	2
#22609301	2
C₂×#17988340	2
C₂×#17988334	2
C₂×C₈×Q₈×C₇₃	2
C₂×#1670	73

Subgroups — C2×#22609301 #33068318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22609301 #33068318