Subgroups — C₂×#22608750 #33055377

Order 18688.

718 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#10596831	C₂	1	2	yes	maximal
2	#22608750	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17988338	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊#10596831	C₂	1	2	yes	maximal
5	C₂×#17988308	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
7	Q₈×#10596831	—	2	4	no	—
8	Q₈×#10596831	C₂²	1	8	yes	—
9	C₂×C₄⋊#10596831	—	2	2	no	—
10	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
11	C₂×C₄×#10596831	—	2	2	no	—
12	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
13	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
14	#17988338	C₂²	1	4	yes	—
15	Q₈⋊#10596831	C₂²	1	8	yes	—
16	#17988308	C₂²	1	4	yes	—
17	C₂²×Q₁₆×C₇₃	C₄	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₇₃	C₂²	1	2	yes	—
19	C₄×Q₁₆×C₇₃	C₄	1	4	yes	—
20	C₄×Q₁₆×C₇₃	C₂²	1	4	yes	—
21	C₂×#107×C₇₃	C₄	1	1	yes	characteristic
22	C₂×#103×C₇₃	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₂²×#10596831	—	2	2	no	—
25	C₂²×#10596831	—	2	1	no	—
26	C₂²×#10596831	D₄	1	2	yes	—
27	C₄⋊#10596831	—	2	8	no	—
28	C₄⋊#10596831	—	2	2	no	—
29	C₄⋊#10596831	C₂³	1	4	yes	—
30	C₄×#10596831	—	2	8	no	—
31	C₄×#10596831	—	2	1	no	—
32	C₄×#10596831	D₄	1	2	yes	—
33	C₄×#10596831	C₂³	1	2	yes	—
34	C₄×#10596831	C₂³	1	2	yes	—
35	C₂²×Q₈×C₇₃	C₂×C₄	1	2	yes	—
36	C₂×Q₁₆×C₇₃	C₈	1	8	yes	complemented
37	C₂×Q₁₆×C₇₃	C₂×C₄	1	4	yes	—
38	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄×Q₈×C₇₃	—	2	2	no	—
41	C₄×Q₈×C₇₃	—	2	2	no	—
42	C₄×Q₈×C₇₃	C₂×C₄	1	4	yes	—
43	C₄×Q₈×C₇₃	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₇₃	—	2	2	no	—
47	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
48	#107×C₇₃	C₂×C₄	1	4	yes	—
49	#103×C₇₃	C₂×C₄	1	8	yes	—
50	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₇₃	C₂³	1	2	yes	—
52	C₂×#10596831	—	2	4	no	—
53	C₂×#10596831	—	2	4	no	—
54	C₂×#10596831	—	2	2	no	—
55	C₂×#10596831	—	4	2	no	—
56	C₂×#10596831	—	4	4	no	—
57	C₂×#10596831	C₂×D₄	1	4	yes	—
58	C₂×Q₈×C₇₃	—	2	2	no	—
59	C₂×Q₈×C₇₃	—	2	4	no	—
60	C₂×Q₈×C₇₃	C₂×C₈	1	8	yes	—
61	C₂×Q₈×C₇₃	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₇₃	—	2	2	no	—
63	C₂²×C₄×C₇₃	—	2	2	no	—
64	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
67	Q₁₆×C₇₃	C₂×C₈	1	16	yes	complemented
68	C₂×C₈×C₇₃	—	2	2	no	—
69	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
70	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
72	C₄⋊C₄×C₇₃	—	2	4	no	—
73	C₄⋊C₄×C₇₃	—	2	2	no	—
74	C₄⋊C₄×C₇₃	—	2	4	no	—
75	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
76	C₄²×C₇₃	—	2	4	no	—
77	C₄²×C₇₃	—	2	4	no	—
78	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₇₃	#41	1	1	yes	characteristic
81	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
82	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
83	Q₈×C₇₃	—	2	4	no	—
84	Q₈×C₇₃	—	2	8	no	—
85	Q₈×C₇₃	C₂²×C₈	1	8	yes	—
86	C₂×C₄×C₇₃	—	2	4	no	—
87	C₂×C₄×C₇₃	—	2	8	no	—
88	C₂×C₄×C₇₃	—	2	2	no	—
89	C₂×C₄×C₇₃	—	2	4	no	—
90	C₂×C₄×C₇₃	—	4	4	no	—
91	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₇₃	#131	1	2	yes	—
96	C₂×C₄×C₇₃	#137	1	4	yes	—
97	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
100	C₈×C₇₃	—	2	2	no	—
101	C₈×C₇₃	C₂²×C₈	1	4	yes	—
102	#10596831	—	2	4	no	—
103	#10596831	—	4	2	no	—
104	#10596831	—	4	4	no	—
105	C₂²×C₇₃	C₈×D₄	1	2	yes	—
106	C₂²×C₇₃	#441	1	2	yes	—
107	C₂²×C₇₃	#444	1	2	yes	—
108	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
109	C₄×C₇₃	—	2	8	no	—
110	C₄×C₇₃	—	2	1	no	—
111	C₄×C₇₃	—	2	1	no	—
112	C₄×C₇₃	—	4	4	no	—
113	C₄×C₇₃	C₈×D₄	1	2	yes	—
114	C₄×C₇₃	#441	1	2	yes	—
115	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
116	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
117	C₄×C₇₃	C₂×#137	1	2	yes	—
118	C₂×#1579	—	73	1	no	maximal, Sylow
119	C₂×C₇₃	#1579	1	4	yes	complemented
120	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
121	C₂×C₇₃	C₂×#441	1	1	yes	characteristic
122	C₂×C₇₃	C₂×#444	1	1	yes	characteristic
123	C₂×C₈×Q₈	—	73	2	no	—
124	C₂×C₄×Q₁₆	D₇₃	1	1	yes	characteristic
125	#1579	—	73	8	no	—
126	C₂×#334	—	73	1	no	—
127	C₂×Q₈⋊C₈	—	73	2	no	—
128	C₂×#322	—	73	1	no	—
129	C₇₃	C₂×#1579	1	1	yes	characteristic, complemented, Sylow
130	C₂²×Q₁₆	Dic₇₃	1	1	yes	characteristic
131	C₂×C₄×Q₈	C₂×D₇₃	1	2	yes	—
132	C₈×Q₈	—	73	8	no	—
133	C₈×Q₈	—	146	4	no	—
134	C₄×Q₁₆	Dic₇₃	1	4	yes	—
135	C₄×Q₁₆	C₂×D₇₃	1	4	yes	—
136	C₂×#107	Dic₇₃	1	1	yes	characteristic
137	C₂×C₄⋊C₈	—	73	2	no	—
138	C₂×C₄⋊C₈	—	146	2	no	—
139	C₂×#103	Dic₇₃	1	2	yes	—
140	C₂×C₄×C₈	—	73	1	no	—
141	C₂×C₄×C₈	—	73	1	no	—
142	C₂×C₄×C₈	—	146	2	no	—
143	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
144	#334	—	73	4	no	—
145	Q₈⋊C₈	—	73	8	no	—
146	#322	—	73	4	no	—
147	C₂²×Q₈	C₂×Dic₇₃	1	2	yes	—
148	C₂×Q₁₆	#10596831	1	8	yes	complemented
149	C₂×Q₁₆	C₂×Dic₇₃	1	4	yes	—
150	C₂²×C₈	—	73	2	no	—
151	C₂²×C₈	—	146	2	no	—
152	C₂²×C₈	—	146	1	no	—
153	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
154	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
155	C₄×Q₈	—	2	2	no	—
156	C₄×Q₈	—	2	2	no	—
157	C₄×Q₈	C₂×Dic₇₃	1	4	yes	—
158	C₄×Q₈	C₂²×D₇₃	1	4	yes	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
161	C₂×C₄²	—	2	2	no	—
162	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
163	#107	C₂×Dic₇₃	1	4	yes	—
164	C₄⋊C₈	—	73	4	no	—
165	C₄⋊C₈	—	146	8	no	—
166	C₄⋊C₈	—	146	2	no	—
167	#103	C₂×Dic₇₃	1	8	yes	—
168	C₄×C₈	—	73	2	no	—
169	C₄×C₈	—	73	2	no	—
170	C₄×C₈	—	73	2	no	—
171	C₄×C₈	—	146	8	no	—
172	C₄×C₈	—	146	1	no	—
173	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
174	C₄×C₈	C₂²×D₇₃	1	2	yes	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	C₂×#10596831	1	8	yes	—
178	C₂×Q₈	C₂²×Dic₇₃	1	4	yes	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
182	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
183	C₂²×C₄	#11924027	1	1	yes	characteristic
184	Q₁₆	C₂×#10596831	1	16	yes	complemented
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	292	2	no	—
187	C₂×C₈	—	292	4	no	—
188	C₂×C₈	—	73	4	no	—
189	C₂×C₈	—	146	4	no	—
190	C₂×C₈	—	146	2	no	—
191	C₂×C₈	—	146	4	no	—
192	C₂×C₈	C₂×#10596831	1	4	yes	—
193	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
194	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	C₂²×Dic₇₃	1	4	yes	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	4	no	—
201	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
202	C₄²	D₄×D₇₃	1	1	yes	characteristic
203	C₄²	#11924027	1	1	yes	characteristic
204	C₄²	C₂³×D₇₃	1	1	yes	characteristic
205	C₂³	D₄×Dic₇₃	1	1	yes	characteristic
206	Q₈	—	2	8	no	—
207	Q₈	—	2	4	no	—
208	Q₈	C₂²×#10596831	1	8	yes	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
215	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
216	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
217	C₂×C₄	C₂²×#10596831	1	2	yes	—
218	C₂×C₄	#14556598	1	2	yes	—
219	C₂×C₄	#14556623	1	4	yes	—
220	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
222	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	292	2	no	—
225	C₈	—	292	4	no	—
226	C₈	—	146	4	no	—
227	C₈	C₂²×#10596831	1	4	yes	—
228	C₂²	D₄×#10596831	1	2	yes	—
229	C₂²	#17988694	1	2	yes	—
230	C₂²	#17988709	1	2	yes	—
231	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	8	no	—
235	C₄	—	4	4	no	—
236	C₄	D₄×#10596831	1	2	yes	—
237	C₄	#17988694	1	2	yes	—
238	C₄	C₂³×#10596831	1	1	yes	characteristic
239	C₄	C₂×#14556598	1	1	yes	characteristic
240	C₄	C₂×#14556623	1	2	yes	—
241	C₂	#22608750	1	4	yes	direct product factor
242	C₂	C₂×D₄×#10596831	1	1	yes	characteristic
243	C₂	C₂×#17988694	1	1	yes	characteristic
244	C₂	C₂×#17988709	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₃
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×C₄×Q₁₆×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₇₃
hypocenter	C₇₃	C₂×#1579
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #22608750	C₂	direct (N × H)
36. C₂×Q₁₆×C₇₃	C₈	semidirect
67. Q₁₆×C₇₃	C₂×C₈	semidirect
119. C₂×C₇₃	#1579	semidirect
129. C₇₃	C₂×#1579	semidirect
148. C₂×Q₁₆	#10596831	semidirect
184. Q₁₆	C₂×#10596831	semidirect
241. C₂	#22608750	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×Q₈×#10596831	2
#22608750	2
C₂×#17988338	2
C₂×Q₈⋊#10596831	2
C₂×#17988308	2
C₂×C₄×Q₁₆×C₇₃	2
C₂×#1579	73

Subgroups — C2×#22608750 #33055377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22608750 #33055377