Subgroups — C₈×C₂²⋊#10596833 #34003380

Order 18688.

726 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊#10596833	C₂	1	2	yes	maximal
2	C₂×C₈×#10596833	C₂	1	2	yes	maximal, complemented
3	#22624753	C₂	1	2	yes	maximal, complemented
4	C₂²×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂²⋊#10596833	C₄	1	4	yes	—
6	C₂×C₄×#10596833	C₂²	1	4	yes	—
7	#17989453	C₄	1	4	yes	—
8	C₈×#10596833	—	2	4	no	—
9	C₂³×#10596831	C₄	1	1	yes	characteristic
10	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	—	2	4	no	—
12	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₇₃	C₄	1	1	yes	characteristic
17	C₂²×#10596833	C₂×C₄	1	8	yes	—
18	C₂²⋊#10596833	C₈	1	16	yes	direct product factor
19	C₄×#10596833	—	2	4	no	—
20	C₄×#10596833	—	2	4	no	—
21	C₂³×Dic₇₃	C₂×C₄	1	2	yes	—
22	C₂²×#10596831	—	2	2	no	—
23	C₂²×#10596831	—	2	2	no	—
24	C₂²×#10596831	C₈	1	4	yes	complemented
25	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₇₃	—	2	2	no	—
28	C₂×C₄×Dic₇₃	—	2	2	no	—
29	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
32	C₈×Dic₇₃	—	2	2	no	—
33	C₈×Dic₇₃	—	2	2	no	—
34	C₈×Dic₇₃	—	2	2	no	—
35	C₈×Dic₇₃	D₄	1	4	yes	—
36	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₃	—	2	2	no	—
38	C₂²×C₈×C₇₃	—	2	2	no	—
39	C₂²×C₈×C₇₃	C₈	1	4	yes	complemented
40	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂×#10596833	—	2	8	no	—
43	C₂×#10596833	—	2	8	no	—
44	C₂×#10596833	C₂×C₈	1	16	yes	complemented
45	C₂²×Dic₇₃	—	2	4	no	—
46	C₂²×Dic₇₃	—	2	4	no	—
47	C₂²×Dic₇₃	C₄²	1	2	yes	—
48	C₂²×Dic₇₃	C₂×C₈	1	8	yes	—
49	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
50	C₂×#10596831	—	2	1	no	—
51	C₂×#10596831	—	2	1	no	—
52	C₂×#10596831	—	2	8	no	—
53	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
54	C₂×#10596831	C₂×C₈	1	2	yes	—
55	C₂×#10596831	C₂×C₈	1	2	yes	—
56	C₂×#10596831	M₁₆	1	2	yes	—
57	C₄×Dic₇₃	—	2	1	no	—
58	C₄×Dic₇₃	—	2	1	no	—
59	C₄×Dic₇₃	—	2	4	no	—
60	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
61	C₄×Dic₇₃	#41	1	2	yes	—
62	C₂⁴×C₇₃	C₄²	1	1	yes	characteristic
63	C₂²×C₄×C₇₃	—	2	1	no	—
64	C₂²×C₄×C₇₃	—	2	1	no	—
65	C₂²×C₄×C₇₃	—	2	1	no	—
66	C₂²×C₄×C₇₃	—	2	1	no	—
67	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
68	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
69	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₃	—	2	1	no	—
72	C₂×C₈×C₇₃	—	2	1	no	—
73	C₂×C₈×C₇₃	—	2	8	no	—
74	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
77	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
78	C₂³×C₇₃	—	2	1	no	—
79	C₂³×C₇₃	—	2	1	no	—
80	C₂³×C₇₃	—	2	2	no	—
81	C₂³×C₇₃	C₄×C₈	1	4	yes	—
82	C₂³×C₇₃	C₄×C₈	1	2	yes	—
83	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₇₃	—	2	1	no	—
85	C₂×C₄×C₇₃	—	2	1	no	—
86	C₂×C₄×C₇₃	—	2	2	no	—
87	C₂×C₄×C₇₃	—	2	2	no	—
88	C₂×C₄×C₇₃	—	2	2	no	—
89	C₂×C₄×C₇₃	—	2	2	no	—
90	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
91	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
93	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
97	C₂×Dic₇₃	—	2	2	no	—
98	C₂×Dic₇₃	—	2	2	no	—
99	C₂×Dic₇₃	—	2	16	no	—
100	C₂×Dic₇₃	C₄×C₈	1	4	yes	—
101	C₂×Dic₇₃	C₄×D₄	1	4	yes	—
102	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
103	C₂×Dic₇₃	#131	1	4	yes	—
104	#10596833	—	2	16	no	—
105	C₈×C₇₃	—	2	2	no	—
106	C₈×C₇₃	C₂²⋊C₈	1	4	yes	complemented
107	#10596831	—	2	2	no	—
108	#10596831	C₂²⋊C₈	1	4	yes	complemented
109	C₂²×C₇₃	—	2	4	no	—
110	C₂²×C₇₃	—	2	2	no	—
111	C₂²×C₇₃	—	2	2	no	—
112	C₂²×C₇₃	—	2	4	no	—
113	C₂²×C₇₃	C₈²	1	4	yes	complemented
114	C₂²×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
116	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
117	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
118	C₂²×C₇₃	#405	1	2	yes	—
119	C₄×C₇₃	—	2	2	no	—
120	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₄×C₇₃	#431	1	1	yes	characteristic
123	C₄×C₇₃	#431	1	1	yes	characteristic
124	Dic₇₃	—	2	4	no	—
125	Dic₇₃	C₈×D₄	1	8	yes	—
126	C₈×C₂²⋊C₈	—	73	1	no	maximal, Sylow
127	C₂×C₇₃	—	2	4	no	—
128	C₂×C₇₃	C₂×C₈²	1	1	yes	characteristic
129	C₂×C₇₃	C₈×M₁₆	1	2	yes	—
130	C₂×C₇₃	#1450	1	1	yes	characteristic
131	C₂×C₇₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
132	C₂×C₇₃	C₈×C₂²⋊C₄	1	2	yes	—
133	C₂²×C₄×C₈	—	73	1	no	—
134	C₄×C₂²⋊C₈	—	73	2	no	—
135	C₂×C₈²	—	73	2	no	—
136	#1273	—	73	2	no	—
137	C₇₃	C₈×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₈	—	73	1	no	—
139	C₂³×C₈	#64408	1	1	yes	characteristic
140	C₂²×C₄²	—	73	1	no	—
141	C₂×C₂²⋊C₈	—	73	4	no	—
142	C₂×C₄×C₈	—	73	1	no	—
143	C₂×C₄×C₈	—	73	4	no	—
144	C₂×C₄×C₈	—	73	1	no	—
145	C₂×C₄×C₈	—	73	1	no	—
146	C₂×C₄×C₈	—	73	1	no	—
147	C₂×C₄×C₈	—	146	4	no	—
148	#336	—	73	4	no	—
149	C₈²	—	146	4	no	—
150	C₂³×C₄	—	73	2	no	—
151	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	73	1	no	—
155	C₂²×C₈	—	73	8	no	—
156	C₂²×C₈	—	73	4	no	—
157	C₂²×C₈	—	73	1	no	—
158	C₂²×C₈	—	146	2	no	—
159	C₂²×C₈	—	146	2	no	—
160	C₂²×C₈	#10596833	1	4	yes	complemented
161	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
162	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
163	C₂×C₄²	—	73	1	no	—
164	C₂×C₄²	—	73	2	no	—
165	C₂×C₄²	—	73	1	no	—
166	C₂×C₄²	—	146	2	no	—
167	C₂×C₄²	—	146	2	no	—
168	C₂²⋊C₈	—	73	16	no	—
169	C₄×C₈	—	73	4	no	—
170	C₄×C₈	—	146	2	no	—
171	C₄×C₈	—	146	4	no	—
172	C₄×C₈	—	146	4	no	—
173	C₄×C₈	—	146	2	no	—
174	C₄×C₈	—	146	2	no	—
175	C₂⁴	C₄×#64408	1	1	yes	characteristic
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	73	2	no	—
181	C₂²×C₄	—	73	8	no	—
182	C₂²×C₄	—	73	2	no	—
183	C₂²×C₄	—	146	4	no	—
184	C₂²×C₄	—	146	4	no	—
185	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
186	C₂²×C₄	C₂×#10596833	1	2	yes	—
187	C₂²×C₄	#11924036	1	2	yes	—
188	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	—	73	2	no	—
193	C₂×C₈	—	73	2	no	—
194	C₂×C₈	—	73	2	no	—
195	C₂×C₈	—	73	16	no	—
196	C₂×C₈	—	73	2	no	—
197	C₂×C₈	—	146	8	no	—
198	C₂×C₈	—	146	8	no	—
199	C₂×C₈	—	146	8	no	—
200	C₂×C₈	—	146	1	no	—
201	C₂×C₈	—	146	1	no	—
202	C₂×C₈	C₂²⋊#64408	1	2	yes	—
203	C₂×C₈	C₂×#10596833	1	2	yes	—
204	C₂×C₈	C₂×#10596833	1	2	yes	—
205	C₂×C₈	#11924037	1	2	yes	—
206	C₄²	—	73	2	no	—
207	C₄²	—	73	2	no	—
208	C₄²	—	146	4	no	—
209	C₄²	—	146	1	no	—
210	C₄²	—	146	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₈×#64408	1	4	yes	—
215	C₂³	C₄×#10596833	1	2	yes	—
216	C₂³	C₂×C₄×#64408	1	1	yes	characteristic
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	73	4	no	—
224	C₂×C₄	—	73	4	no	—
225	C₂×C₄	—	73	4	no	—
226	C₂×C₄	—	73	4	no	—
227	C₂×C₄	—	146	16	no	—
228	C₂×C₄	—	146	2	no	—
229	C₂×C₄	—	146	2	no	—
230	C₂×C₄	C₄×#10596833	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
232	C₂×C₄	#14556670	1	1	yes	characteristic
233	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
234	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
235	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
236	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	73	4	no	—
239	C₈	—	146	16	no	—
240	C₈	—	146	2	no	—
241	C₈	C₂²⋊#10596833	1	4	yes	direct product factor
242	C₂²	—	2	4	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	C₈×#10596833	1	4	yes	complemented
247	C₂²	C₄×C₂²⋊#64408	1	1	yes	characteristic
248	C₂²	C₂×C₈×#64408	1	2	yes	—
249	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
250	C₂²	C₄×#11924037	1	1	yes	characteristic
251	C₂²	#17989550	1	2	yes	—
252	C₄	—	2	2	no	—
253	C₄	—	73	8	no	—
254	C₄	—	146	4	no	—
255	C₄	C₂×C₂²⋊#10596833	1	1	yes	characteristic
256	C₄	#17989555	1	1	yes	characteristic
257	C₄	#17989603	1	1	yes	characteristic
258	C₄	#17989604	1	1	yes	characteristic
259	C₂	—	2	4	no	—
260	C₂	C₂×C₈×#10596833	1	1	yes	characteristic
261	C₂	C₈×#11924037	1	2	yes	—
262	C₂	#22625134	1	1	yes	characteristic
263	C₂	C₄×C₂²⋊#10596833	1	1	yes	characteristic
264	C₂	C₈×C₂²⋊#64408	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×#64408
commutator	C₂×C₇₃	C₂×C₈²
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂³×C₈×C₇₃	C₄
socle	C₂³×C₇₃	C₂×C₄²
prime radical	C₂⁴×C₇₃	C₄²
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂³×C₈	#64408
hypocenter	C₇₃	C₈×C₂²⋊C₈
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×#10596833	C₂	semidirect
3. #22624753	C₂	semidirect
18. C₂²⋊#10596833	C₈	direct (N × H)
24. C₂²×#10596831	C₈	semidirect
39. C₂²×C₈×C₇₃	C₈	semidirect
44. C₂×#10596833	C₂×C₈	semidirect
106. C₈×C₇₃	C₂²⋊C₈	semidirect
108. #10596831	C₂²⋊C₈	semidirect
113. C₂²×C₇₃	C₈²	semidirect
137. C₇₃	C₈×C₂²⋊C₈	semidirect
160. C₂²×C₈	#10596833	semidirect
241. C₈	C₂²⋊#10596833	direct (N × H)
246. C₂²	C₈×#10596833	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊#10596833	2
C₂×C₈×#10596833	2
#22624753	2
C₂²×C₈×Dic₇₃	2
C₈×C₂²⋊C₈	73

Subgroups — C8×C22⋊#10596833 #34003380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₂²⋊#10596833 #34003380