Subgroups — C₂×#22609528 #33073633

Order 18688.

814 conjugacy classes of subgroups.

The subgroup lattice has 244 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
2	C₂×#17988985	C₂	1	2	yes	maximal
3	C₂×#17988967	C₂	1	1	yes	maximal, characteristic
4	C₂×#17988938	C₂	1	2	yes	maximal
5	#22609528	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₈²×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₉₂	—	2	2	no	—
8	C₂×#14556591	C₂²	1	2	yes	—
9	C₂×#14556591	C₂²	1	2	yes	—
10	#17988992	—	2	2	no	—
11	#17988992	C₂²	1	4	yes	—
12	#17988985	—	2	4	no	—
13	#17988985	C₂²	1	8	yes	—
14	#17988967	—	2	2	no	—
15	#17988967	C₂²	1	4	yes	—
16	#17988938	—	2	4	no	—
17	#17988938	C₂²	1	8	yes	—
18	C₂×#14556525	—	2	2	no	—
19	C₂×#14556523	—	2	4	no	—
20	C₂×#14556516	—	2	4	no	—
21	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₈²×C₇₃	C₂²	1	4	yes	—
25	C₂²×Dic₁₄₆	—	2	2	no	—
26	C₂²×Dic₁₄₆	—	2	2	no	—
27	C₂×Dic₂₉₂	—	2	8	no	—
28	C₂×Dic₂₉₂	—	4	8	no	—
29	#14556591	—	2	2	no	—
30	#14556591	—	2	2	no	—
31	#14556591	—	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	#14556591	C₂³	1	4	yes	—
37	C₂×C₄⋊Dic₇₃	—	2	2	no	—
38	C₂×C₄⋊Dic₇₃	—	2	2	no	—
39	C₂×C₄⋊Dic₇₃	—	2	2	no	—
40	C₂×C₄⋊Dic₇₃	—	2	2	no	—
41	#14556525	—	2	8	no	—
42	#14556523	—	2	16	no	—
43	#14556516	—	2	16	no	—
44	C₂²×C₈×C₇₃	D₄	1	2	yes	—
45	C₂²×C₈×C₇₃	D₄	1	2	yes	—
46	C₂²×C₈×C₇₃	D₄	1	2	yes	—
47	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₇₃	D₄	1	2	yes	—
49	C₄×C₈×C₇₃	D₄	1	2	yes	—
50	C₄×C₈×C₇₃	D₄	1	2	yes	—
51	C₄×C₈×C₇₃	C₂³	1	2	yes	—
52	C₄×C₈×C₇₃	C₂³	1	2	yes	—
53	C₄×C₈×C₇₃	C₂³	1	2	yes	—
54	C₂×Dic₁₄₆	—	2	4	no	—
55	C₂×Dic₁₄₆	—	2	4	no	—
56	C₂×Dic₁₄₆	—	4	8	no	—
57	C₂×Dic₁₄₆	—	4	8	no	—
58	C₂×Dic₁₄₆	—	4	2	no	—
59	C₂×Dic₁₄₆	—	4	2	no	—
60	C₂²×Dic₇₃	—	4	2	no	—
61	C₂²×Dic₇₃	—	4	2	no	—
62	Dic₂₉₂	—	4	16	no	—
63	C₄⋊Dic₇₃	—	2	4	no	—
64	C₄⋊Dic₇₃	—	2	4	no	—
65	C₄⋊Dic₇₃	—	2	4	no	—
66	C₄⋊Dic₇₃	—	2	4	no	—
67	C₄⋊Dic₇₃	—	4	2	no	—
68	C₄⋊Dic₇₃	—	4	2	no	—
69	C₄⋊Dic₇₃	—	4	2	no	—
70	C₄⋊Dic₇₃	—	4	2	no	—
71	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₇₃	—	2	4	no	—
75	C₂×C₈×C₇₃	SD₁₆	1	8	yes	—
76	C₂×C₈×C₇₃	Q₁₆	1	8	yes	complemented
77	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
80	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
84	C₂³×C₇₃	#127	1	1	yes	characteristic
85	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
86	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
87	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
88	C₂×C₄×C₇₃	C₂×SD₁₆	1	4	yes	—
89	C₂×C₄×C₇₃	C₂×Q₁₆	1	4	yes	—
90	C₂×C₄×C₇₃	#135	1	4	yes	—
91	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×Dic₇₃	—	4	4	no	—
95	C₂×Dic₇₃	—	4	4	no	—
96	C₂×Dic₇₃	—	8	4	no	—
97	C₂×Dic₇₃	—	8	4	no	—
98	Dic₁₄₆	—	4	8	no	—
99	Dic₁₄₆	—	4	8	no	—
100	Dic₁₄₆	—	8	4	no	—
101	Dic₁₄₆	—	8	4	no	—
102	C₈×C₇₃	—	2	4	no	—
103	C₈×C₇₃	C₂×SD₁₆	1	8	yes	—
104	C₈×C₇₃	C₂×Q₁₆	1	8	yes	complemented
105	C₂²×C₇₃	#492	1	2	yes	—
106	C₂²×C₇₃	#494	1	2	yes	—
107	C₂²×C₇₃	#495	1	2	yes	—
108	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
109	C₄×C₇₃	#492	1	2	yes	—
110	C₄×C₇₃	#494	1	2	yes	—
111	C₄×C₇₃	#495	1	2	yes	—
112	C₄×C₇₃	C₂²×SD₁₆	1	2	yes	—
113	C₄×C₇₃	C₂²×Q₁₆	1	2	yes	—
114	C₄×C₇₃	C₂×#135	1	2	yes	—
115	Dic₇₃	—	8	4	no	—
116	Dic₇₃	—	8	4	no	—
117	C₂×#1707	—	73	1	no	maximal, Sylow
118	C₂×C₇₃	#1707	1	4	yes	complemented
119	C₂×C₇₃	C₂×#492	1	1	yes	characteristic
120	C₂×C₇₃	C₂×#494	1	1	yes	characteristic
121	C₂×C₇₃	C₂×#495	1	1	yes	characteristic
122	C₂×#500	—	73	1	no	—
123	C₂×#498	—	73	2	no	—
124	C₂×#494	—	73	1	no	—
125	C₂×#487	—	73	2	no	—
126	#1707	—	73	8	no	—
127	C₂×C₈²	D₇₃	1	1	yes	characteristic
128	C₇₃	C₂×#1707	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₁₆	—	146	2	no	—
130	C₂×C₄⋊Q₈	—	73	2	no	—
131	C₂×C₄⋊Q₈	—	73	2	no	—
132	#500	—	73	4	no	—
133	#500	—	146	2	no	—
134	#498	—	73	8	no	—
135	#498	—	146	4	no	—
136	#494	—	73	4	no	—
137	#494	—	146	2	no	—
138	#487	—	73	8	no	—
139	#487	—	146	4	no	—
140	C₂×#107	—	146	2	no	—
141	C₂×#106	—	146	4	no	—
142	C₂×#103	—	146	4	no	—
143	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
146	C₈²	C₂×D₇₃	1	4	yes	—
147	C₂²×Q₈	—	146	2	no	—
148	C₂²×Q₈	—	146	2	no	—
149	C₂×Q₁₆	—	292	8	no	—
150	C₂×Q₁₆	—	146	8	no	—
151	C₂²×C₈	D₂₉₂	1	2	yes	—
152	C₂²×C₈	D₂₉₂	1	2	yes	—
153	C₂²×C₈	D₂₉₂	1	2	yes	—
154	C₄⋊Q₈	—	73	4	no	—
155	C₄⋊Q₈	—	73	4	no	—
156	C₄⋊Q₈	—	146	2	no	—
157	C₄⋊Q₈	—	146	2	no	—
158	C₄⋊Q₈	—	146	2	no	—
159	C₄⋊Q₈	—	146	2	no	—
160	C₄⋊Q₈	—	146	2	no	—
161	C₄⋊Q₈	—	146	2	no	—
162	C₂×C₄⋊C₄	—	146	2	no	—
163	C₂×C₄⋊C₄	—	146	2	no	—
164	C₂×C₄⋊C₄	—	146	2	no	—
165	C₂×C₄⋊C₄	—	146	2	no	—
166	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
167	#107	—	146	8	no	—
168	#106	—	146	16	no	—
169	#103	—	146	16	no	—
170	C₄×C₈	D₂₉₂	1	2	yes	—
171	C₄×C₈	D₂₉₂	1	2	yes	—
172	C₄×C₈	D₂₉₂	1	2	yes	—
173	C₄×C₈	C₂²×D₇₃	1	2	yes	—
174	C₄×C₈	C₂²×D₇₃	1	2	yes	—
175	C₄×C₈	C₂²×D₇₃	1	2	yes	—
176	C₂×Q₈	—	292	2	no	—
177	C₂×Q₈	—	292	2	no	—
178	C₂×Q₈	—	292	8	no	—
179	C₂×Q₈	—	292	8	no	—
180	C₂×Q₈	—	146	4	no	—
181	C₂×Q₈	—	146	4	no	—
182	C₂²×C₄	—	292	2	no	—
183	C₂²×C₄	—	292	2	no	—
184	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
187	Q₁₆	—	292	16	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	#11924017	1	8	yes	—
190	C₂×C₈	Dic₂₉₂	1	8	yes	complemented
191	C₂×C₈	C₂×D₂₉₂	1	4	yes	—
192	C₂×C₈	C₂×D₂₉₂	1	4	yes	—
193	C₂×C₈	C₂×D₂₉₂	1	4	yes	—
194	C₄⋊C₄	—	292	2	no	—
195	C₄⋊C₄	—	292	2	no	—
196	C₄⋊C₄	—	292	2	no	—
197	C₄⋊C₄	—	292	2	no	—
198	C₄⋊C₄	—	146	4	no	—
199	C₄⋊C₄	—	146	4	no	—
200	C₄⋊C₄	—	146	4	no	—
201	C₄⋊C₄	—	146	4	no	—
202	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
203	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
204	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
205	C₄²	C₂³×D₇₃	1	1	yes	characteristic
206	C₂³	#14556588	1	1	yes	characteristic
207	Q₈	—	292	8	no	—
208	Q₈	—	292	8	no	—
209	Q₈	—	584	4	no	—
210	Q₈	—	584	4	no	—
211	C₂×C₄	—	292	4	no	—
212	C₂×C₄	—	292	4	no	—
213	C₂×C₄	—	584	4	no	—
214	C₂×C₄	—	584	4	no	—
215	C₂×C₄	#14556588	1	1	yes	characteristic
216	C₂×C₄	#14556588	1	1	yes	characteristic
217	C₂×C₄	#14556588	1	1	yes	characteristic
218	C₂×C₄	C₂×#11924017	1	4	yes	—
219	C₂×C₄	C₂×Dic₂₉₂	1	4	yes	—
220	C₂×C₄	#14556613	1	4	yes	—
221	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
223	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
224	C₈	—	2	4	no	—
225	C₈	C₂×#11924017	1	8	yes	—
226	C₈	C₂×Dic₂₉₂	1	8	yes	complemented
227	C₂²	#17988960	1	2	yes	—
228	C₂²	#17988967	1	2	yes	—
229	C₂²	#17988971	1	2	yes	—
230	C₂²	C₂×#14556588	1	1	yes	characteristic
231	C₄	—	584	4	no	—
232	C₄	—	584	4	no	—
233	C₄	#17988960	1	2	yes	—
234	C₄	#17988967	1	2	yes	—
235	C₄	#17988971	1	2	yes	—
236	C₄	C₂²×#11924017	1	2	yes	—
237	C₄	C₂²×Dic₂₉₂	1	2	yes	—
238	C₄	C₂×#14556613	1	2	yes	—
239	C₂	#22609528	1	4	yes	direct product factor
240	C₂	C₂×#17988960	1	1	yes	characteristic
241	C₂	C₂×#17988967	1	1	yes	characteristic
242	C₂	C₂×#17988971	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556588
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×C₈²×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	#127
hypercenter	C₂×C₈²	D₇₃
hypocenter	C₇₃	C₂×#1707
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
5. #22609528	C₂	direct (N × H)
76. C₂×C₈×C₇₃	Q₁₆	semidirect
104. C₈×C₇₃	C₂×Q₁₆	semidirect
118. C₂×C₇₃	#1707	semidirect
128. C₇₃	C₂×#1707	semidirect
190. C₂×C₈	Dic₂₉₂	semidirect
226. C₈	C₂×Dic₂₉₂	semidirect
239. C₂	#22609528	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×#17988985	2
C₂×#17988967	2
C₂×#17988938	2
#22609528	2
C₂×C₈²×C₇₃	2
C₂×#1707	73

Subgroups — C2×#22609528 #33073633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22609528 #33073633