Subgroups — C₂×#22627001 #34034037

Order 18688.

726 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11924040	C₂	1	2	yes	maximal
2	#22627001	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₄×#10596833	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924040	—	2	4	no	—
5	C₂²×#11924040	—	2	4	no	—
6	C₂²×#11924040	C₄	1	8	yes	complemented
7	C₂²×#11924040	C₂²	1	4	yes	—
8	C₂³×#10596833	C₄	1	1	yes	characteristic
9	C₂³×#10596833	C₂²	1	1	yes	characteristic
10	C₂×C₄×#10596833	—	2	4	no	—
11	C₂×C₄×#10596833	C₄	1	2	yes	—
12	C₂×C₄×#10596833	C₂²	1	2	yes	—
13	C₂²×C₄×Dic₇₃	C₄	1	1	yes	characteristic
14	C₂×#11924040	—	2	16	no	—
15	C₂×#11924040	—	2	4	no	—
16	C₂×#11924040	—	2	4	no	—
17	C₂×#11924040	C₂×C₄	1	8	yes	complemented
18	C₂²×#10596833	—	2	2	no	—
19	C₂²×#10596833	—	2	2	no	—
20	C₂²×#10596833	C₈	1	4	yes	—
21	C₂²×#10596833	C₂×C₄	1	4	yes	—
22	C₂²×#10596833	C₂×C₄	1	2	yes	—
23	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
24	C₂²×#10596833	D₄	1	4	yes	—
25	C₂²×#10596833	D₄	1	2	yes	—
26	C₂²×#10596833	Q₈	1	2	yes	—
27	C₂²×#10596833	C₂³	1	1	yes	characteristic
28	C₄×#10596833	—	2	4	no	—
29	C₄×#10596833	—	2	4	no	—
30	C₂³×Dic₇₃	C₈	1	1	yes	characteristic
31	C₂³×Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₇₃	—	2	2	no	—
33	C₂×C₄×Dic₇₃	—	2	2	no	—
34	C₂×C₄×Dic₇₃	C₈	1	2	yes	—
35	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₇₃	C₈	1	1	yes	characteristic
37	#11924040	—	2	8	no	—
38	C₂×#10596833	—	2	8	no	—
39	C₂×#10596833	—	2	2	no	—
40	C₂×#10596833	—	2	2	no	—
41	C₂×#10596833	C₄²	1	2	yes	—
42	C₂×#10596833	C₂²⋊C₄	1	2	yes	—
43	C₂×#10596833	C₂²⋊C₄	1	8	yes	—
44	C₂×#10596833	C₄⋊C₄	1	2	yes	—
45	C₂×#10596833	C₄⋊C₄	1	8	yes	—
46	C₂×#10596833	C₂×C₈	1	4	yes	—
47	C₂×#10596833	C₂²×C₄	1	2	yes	—
48	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
49	C₂×#10596833	C₂×D₄	1	2	yes	—
50	C₂×#10596833	C₂×Q₈	1	1	yes	characteristic
51	C₂²×Dic₇₃	—	2	2	no	—
52	C₂²×Dic₇₃	—	2	2	no	—
53	C₂²×Dic₇₃	C₁₆	1	4	yes	complemented
54	C₂²×Dic₇₃	C₄²	1	2	yes	—
55	C₂²×Dic₇₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
57	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
58	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
60	C₂²×Dic₇₃	M₁₆	1	2	yes	—
61	C₂²×Dic₇₃	M₁₆	1	2	yes	—
62	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₄×Dic₇₃	—	2	4	no	—
64	C₄×Dic₇₃	—	2	2	no	—
65	C₄×Dic₇₃	—	2	2	no	—
66	C₂⁴×C₇₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	—	2	2	no	—
68	C₂²×C₄×C₇₃	—	2	2	no	—
69	C₂²×C₄×C₇₃	C₁₆	1	4	yes	complemented
70	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
71	C₂³×C₇₃	C₄×C₈	1	2	yes	—
72	C₂³×C₇₃	C₂²⋊C₈	1	2	yes	—
73	C₂³×C₇₃	C₄⋊C₈	1	2	yes	—
74	C₂³×C₇₃	C₂×C₁₆	1	2	yes	—
75	C₂³×C₇₃	C₂×C₁₆	1	2	yes	—
76	C₂³×C₇₃	M₃₂	1	2	yes	—
77	C₂³×C₇₃	M₃₂	1	2	yes	—
78	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₇₃	—	2	8	no	—
80	C₂×C₄×C₇₃	—	2	2	no	—
81	C₂×C₄×C₇₃	—	2	2	no	—
82	C₂×C₄×C₇₃	C₂×C₁₆	1	4	yes	complemented
83	C₂×Dic₇₃	—	2	8	no	—
84	C₂×Dic₇₃	—	2	2	no	—
85	C₂×Dic₇₃	—	2	2	no	—
86	C₂×Dic₇₃	#95	1	4	yes	—
87	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
88	C₂×Dic₇₃	#97	1	4	yes	—
89	C₂×Dic₇₃	C₂²⋊C₈	1	2	yes	—
90	C₂×Dic₇₃	C₂²⋊C₈	1	4	yes	—
91	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
92	C₂×Dic₇₃	C₄⋊C₈	1	4	yes	—
93	C₂×Dic₇₃	C₂×C₁₆	1	4	yes	complemented
94	C₂×Dic₇₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×Dic₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
100	#10596833	—	2	4	no	—
101	#10596833	#95	1	4	yes	—
102	#10596833	C₂×C₂²⋊C₄	1	2	yes	—
103	#10596833	C₂×C₄⋊C₄	1	2	yes	—
104	C₂²×C₇₃	#336	1	4	yes	—
105	C₂²×C₇₃	C₄×C₁₆	1	4	yes	—
106	C₂²×C₇₃	#346	1	4	yes	—
107	C₂²×C₇₃	C₂²⋊C₁₆	1	4	yes	—
108	C₂²×C₇₃	C₂²⋊C₁₆	1	4	yes	—
109	C₂²×C₇₃	C₄⋊C₁₆	1	4	yes	—
110	C₂²×C₇₃	C₄⋊C₁₆	1	4	yes	—
111	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
112	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
113	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
115	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
116	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
117	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
118	C₄×C₇₃	—	2	4	no	—
119	Dic₇₃	—	2	4	no	—
120	Dic₇₃	#336	1	4	yes	—
121	Dic₇₃	C₂×#95	1	1	yes	characteristic
122	Dic₇₃	C₂×#97	1	1	yes	characteristic
123	Dic₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	Dic₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
125	C₂×#1374	—	73	1	no	maximal, Sylow
126	C₂×C₇₃	#1374	1	8	yes	complemented
127	C₂×C₇₃	C₂×#336	1	1	yes	characteristic
128	C₂×C₇₃	C₂×C₄×C₁₆	1	1	yes	characteristic
129	C₂×C₇₃	C₂×#346	1	1	yes	characteristic
130	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
131	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
132	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
133	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
134	C₂³×C₁₆	—	73	2	no	—
135	C₂²×C₄×C₈	—	73	1	no	—
136	#1374	—	73	4	no	—
137	C₇₃	C₂×#1374	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₈	—	73	1	no	—
139	C₂³×C₈	—	73	1	no	—
140	C₂²×C₄²	—	73	1	no	—
141	C₂²×C₁₆	—	73	8	no	—
142	C₂²×C₁₆	—	73	4	no	—
143	C₂²×C₁₆	—	146	4	no	—
144	C₂²×C₁₆	—	146	4	no	—
145	C₂×C₄×C₈	—	73	2	no	—
146	C₂×C₄×C₈	—	73	2	no	—
147	C₂×C₄×C₈	—	146	4	no	—
148	C₂³×C₄	—	73	1	no	—
149	C₂³×C₄	—	73	1	no	—
150	C₂³×C₄	#10596842	1	1	yes	characteristic
151	C₂²×C₈	—	73	1	no	—
152	C₂²×C₈	—	73	4	no	—
153	C₂²×C₈	—	73	4	no	—
154	C₂²×C₈	—	73	2	no	—
155	C₂²×C₈	—	73	2	no	—
156	C₂²×C₈	—	73	1	no	—
157	C₂²×C₈	—	73	4	no	—
158	C₂²×C₈	—	73	2	no	—
159	C₂²×C₈	—	146	2	no	—
160	C₂²×C₈	—	146	2	no	—
161	C₂×C₄²	—	73	2	no	—
162	C₂×C₄²	—	73	2	no	—
163	C₂×C₄²	—	146	2	no	—
164	C₂×C₄²	—	146	2	no	—
165	C₂×C₁₆	—	73	8	no	—
166	C₂×C₁₆	—	146	16	no	—
167	C₂×C₁₆	—	146	4	no	—
168	C₂×C₁₆	—	146	4	no	—
169	C₄×C₈	—	146	4	no	—
170	C₄×C₈	—	146	4	no	—
171	C₂⁴	C₂×#10596842	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	73	2	no	—
175	C₂²×C₄	—	73	2	no	—
176	C₂²×C₄	—	73	2	no	—
177	C₂²×C₄	—	73	2	no	—
178	C₂²×C₄	—	73	2	no	—
179	C₂²×C₄	—	73	2	no	—
180	C₂²×C₄	—	73	1	no	—
181	C₂²×C₄	—	73	1	no	—
182	C₂²×C₄	—	73	4	no	—
183	C₂²×C₄	—	73	2	no	—
184	C₂²×C₄	—	146	2	no	—
185	C₂²×C₄	—	146	2	no	—
186	C₂²×C₄	#11924040	1	4	yes	complemented
187	C₂²×C₄	C₂×#10596842	1	2	yes	—
188	C₂×C₈	—	73	2	no	—
189	C₂×C₈	—	73	2	no	—
190	C₂×C₈	—	73	8	no	—
191	C₂×C₈	—	73	2	no	—
192	C₂×C₈	—	73	8	no	—
193	C₂×C₈	—	73	2	no	—
194	C₂×C₈	—	73	2	no	—
195	C₂×C₈	—	73	1	no	—
196	C₂×C₈	—	73	1	no	—
197	C₂×C₈	—	73	4	no	—
198	C₂×C₈	—	146	2	no	—
199	C₂×C₈	—	146	8	no	—
200	C₂×C₈	—	146	2	no	—
201	C₄²	—	146	2	no	—
202	C₄²	—	146	4	no	—
203	C₄²	—	146	2	no	—
204	C₁₆	—	146	8	no	—
205	C₂³	C₄×#10596842	1	2	yes	—
206	C₂³	C₂²⋊#10596842	1	2	yes	—
207	C₂³	C₄⋊#10596842	1	2	yes	—
208	C₂³	C₂×#11924040	1	2	yes	—
209	C₂³	#14556686	1	2	yes	—
210	C₂³	#14556687	1	2	yes	—
211	C₂³	#14556688	1	2	yes	—
212	C₂³	C₂²×#10596842	1	1	yes	characteristic
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	73	4	no	—
217	C₂×C₄	—	73	4	no	—
218	C₂×C₄	—	73	4	no	—
219	C₂×C₄	—	73	4	no	—
220	C₂×C₄	—	73	1	no	—
221	C₂×C₄	—	73	1	no	—
222	C₂×C₄	—	73	2	no	—
223	C₂×C₄	—	73	1	no	—
224	C₂×C₄	—	73	2	no	—
225	C₂×C₄	—	73	1	no	—
226	C₂×C₄	—	73	1	no	—
227	C₂×C₄	—	73	1	no	—
228	C₂×C₄	—	73	2	no	—
229	C₂×C₄	—	73	4	no	—
230	C₂×C₄	—	146	8	no	—
231	C₂×C₄	—	146	2	no	—
232	C₂×C₄	—	146	2	no	—
233	C₂×C₄	C₂×#11924040	1	4	yes	complemented
234	C₈	—	73	2	no	—
235	C₈	—	73	4	no	—
236	C₈	—	73	2	no	—
237	C₈	—	146	4	no	—
238	C₂²	(C₂×#10596842)⋊C₄	1	4	yes	—
239	C₂²	C₄×#11924040	1	4	yes	—
240	C₂²	#17989681	1	4	yes	—
241	C₂²	C₂²⋊#11924040	1	4	yes	—
242	C₂²	#17989684	1	4	yes	—
243	C₂²	C₄⋊#11924040	1	4	yes	—
244	C₂²	#17989690	1	4	yes	—
245	C₂²	C₂×C₄×#10596842	1	1	yes	characteristic
246	C₂²	C₂×C₂²⋊#10596842	1	1	yes	characteristic
247	C₂²	C₂×C₄⋊#10596842	1	1	yes	characteristic
248	C₂²	C₂²×#11924040	1	1	yes	characteristic
249	C₂²	C₂×#14556686	1	1	yes	characteristic
250	C₂²	C₂×#14556687	1	1	yes	characteristic
251	C₂²	C₂×#14556688	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	73	1	no	—
254	C₄	—	73	1	no	—
255	C₄	—	73	4	no	—
256	C₄	—	73	1	no	—
257	C₄	—	73	1	no	—
258	C₄	—	146	4	no	—
259	C₂	#22627001	1	8	yes	direct product factor
260	C₂	C₂×(C₂×#10596842)⋊C₄	1	1	yes	characteristic
261	C₂	C₂×C₄×#11924040	1	1	yes	characteristic
262	C₂	C₂×#17989681	1	1	yes	characteristic
263	C₂	C₂×C₂²⋊#11924040	1	1	yes	characteristic
264	C₂	C₂×#17989684	1	1	yes	characteristic
265	C₂	C₂×C₄⋊#11924040	1	1	yes	characteristic
266	C₂	C₂×#17989690	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×#10596842
commutator	C₂×C₇₃	C₂×C₄×C₁₆
Frattini	C₂³	C₂²×#10596842
Fitting	C₂³×C₄×C₇₃	C₈
socle, prime radical	C₂⁴×C₇₃	C₂×C₈
cosocle	C₂²×#10596833	C₂³
hypercenter	C₂³×C₄	#10596842
hypocenter	C₇₃	C₂×#1374
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. #22627001	C₂	direct (N × H)
6. C₂²×#11924040	C₄	semidirect
17. C₂×#11924040	C₂×C₄	semidirect
53. C₂²×Dic₇₃	C₁₆	semidirect
69. C₂²×C₄×C₇₃	C₁₆	semidirect
82. C₂×C₄×C₇₃	C₂×C₁₆	semidirect
93. C₂×Dic₇₃	C₂×C₁₆	semidirect
126. C₂×C₇₃	#1374	semidirect
137. C₇₃	C₂×#1374	semidirect
186. C₂²×C₄	#11924040	semidirect
233. C₂×C₄	C₂×#11924040	semidirect
259. C₂	#22627001	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#11924040	2
#22627001	2
C₂²×C₄×#10596833	2
C₂×#1374	73

Subgroups — C2×#22627001 #34034037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22627001 #34034037