Subgroups — #34012627

Order 18688.

414 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626261	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989474	C₂	1	2	yes	maximal
3	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
4	#22624968	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556664	C₂²	1	2	yes	—
7	C₄×#11924036	C₄	1	1	yes	characteristic
8	C₂×C₈×#64408	C₄	1	1	yes	characteristic
9	C₂×C₈×#64408	C₄	1	1	yes	characteristic
10	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
11	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
12	C₄²×#64408	C₄	1	1	yes	characteristic
13	#17989508	C₄	1	2	yes	complemented
14	C₄⋊#11924031	C₄	1	2	yes	complemented
15	#17989474	—	2	4	no	—
16	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
17	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
18	C₄⋊#11924003	C₄	1	1	yes	characteristic, complemented
19	#17988405	C₄	1	2	yes	complemented
20	C₄⋊C₁₆×C₇₃	C₄	1	1	yes	characteristic, complemented
21	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
22	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×#64408	C₈	1	2	yes	—
24	C₂×C₄×#64408	C₈	1	2	yes	—
25	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
27	#14556664	—	2	2	no	—
28	#14556664	—	2	2	no	—
29	C₂×#11924031	C₂×C₄	1	4	yes	—
30	C₄×#10596833	C₈	1	2	yes	—
31	C₄×#10596833	C₈	1	2	yes	—
32	C₈×#64408	—	2	2	no	—
33	C₈×#64408	—	2	2	no	—
34	C₈×#64408	—	2	1	no	—
35	C₈×#64408	—	2	1	no	—
36	C₈×#64408	D₄	1	2	yes	—
37	C₈×#64408	Q₈	1	2	yes	—
38	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
40	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂×#11924003	C₂×C₄	1	2	yes	—
42	C₁₆×D₇₃	—	2	2	no	—
43	C₁₆×D₇₃	—	2	2	no	—
44	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
45	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
46	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×#64408	C₂×C₈	1	2	yes	—
50	C₂²×#64408	C₂×C₈	1	2	yes	—
51	#11924036	—	2	1	no	—
52	#11924036	—	2	1	no	—
53	#11924036	C₄⋊C₄	1	2	yes	—
54	#11924036	M₁₆	1	2	yes	—
55	C₂×#10596833	C₂×C₈	1	2	yes	—
56	C₂×#10596833	C₂×C₈	1	2	yes	—
57	C₄×#64408	—	2	4	no	—
58	C₄×#64408	—	2	4	no	—
59	C₄×#64408	—	2	1	no	—
60	C₄×#64408	—	2	1	no	—
61	C₄×#64408	C₄⋊C₄	1	2	yes	—
62	C₄×#64408	M₁₆	1	2	yes	—
63	#11924031	—	2	4	no	—
64	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
65	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
66	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×#10596831	C₄²	1	1	yes	characteristic
68	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
69	C₈×D₇₃	—	2	1	no	—
70	C₈×D₇₃	—	2	1	no	—
71	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
72	C₈×D₇₃	C₂×Q₈	1	1	yes	characteristic
73	C₈×D₇₃	#41	1	1	yes	characteristic
74	C₈×D₇₃	#41	1	1	yes	characteristic
75	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
76	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
77	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
78	#11924003	—	2	2	no	—
79	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
80	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₇₃	C₄²	1	1	yes	characteristic
82	C₁₆×C₇₃	—	2	2	no	—
83	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×#64408	—	2	2	no	—
85	C₂×#64408	—	2	2	no	—
86	C₂×#64408	—	2	2	no	—
87	C₂×#64408	—	2	2	no	—
88	C₂×#64408	C₄⋊C₈	1	4	yes	—
89	C₂×#64408	M₃₂	1	4	yes	—
90	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
91	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
92	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
93	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
94	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
95	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
96	C₄×D₇₃	—	2	2	no	—
97	C₄×D₇₃	M₃₂	1	4	yes	—
98	C₄×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₄×D₇₃	#117	1	1	yes	characteristic
100	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
101	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
102	#10596833	—	2	2	no	—
103	#10596833	C₄⋊C₈	1	4	yes	—
104	C₈×C₇₃	—	2	1	no	—
105	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
106	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
107	#10596831	—	2	1	no	—
108	#10596831	C₄×D₄	1	1	yes	characteristic
109	#10596831	C₄×Q₈	1	1	yes	characteristic
110	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
111	C₂×D₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂×D₇₃	#431	1	1	yes	characteristic
113	C₂×D₇₃	C₂×M₃₂	1	2	yes	—
114	C₂×D₇₃	C₂×M₃₂	1	2	yes	—
115	#64408	—	2	4	no	—
116	#64408	—	2	4	no	—
117	C₄×C₇₃	—	2	1	no	—
118	C₄×C₇₃	#346	1	2	yes	complemented
119	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
120	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
121	Dic₇₃	—	2	1	no	—
122	Dic₇₃	#346	1	2	yes	complemented
123	Dic₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
124	Dic₇₃	#431	1	1	yes	characteristic
125	#12653	—	73	1	no	maximal, Sylow
126	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
127	C₂×C₇₃	C₂×#346	1	1	yes	characteristic
128	C₂×C₇₃	C₄×M₃₂	1	1	yes	characteristic
129	D₇₃	#2150	1	2	yes	—
130	D₇₃	#2163	1	2	yes	—
131	C₂×C₄⋊C₁₆	—	73	1	no	—
132	C₂×C₄⋊C₁₆	—	73	1	no	—
133	C₂×#346	—	73	2	no	—
134	C₄²×C₈	—	73	1	no	—
135	#1374	—	73	2	no	—
136	C₇₃	#12653	1	1	yes	characteristic, complemented, Sylow
137	C₂²×C₁₆	—	73	2	no	—
138	C₂²×C₁₆	—	73	2	no	—
139	C₂×C₄×C₈	—	73	1	no	—
140	C₂×C₄×C₈	—	73	1	no	—
141	C₂×C₄×C₈	—	73	1	no	—
142	C₂×C₄×C₈	—	73	1	no	—
143	C₂×C₄×C₈	—	73	1	no	—
144	C₂×C₄×C₈	—	73	1	no	—
145	C₄³	—	73	1	no	—
146	C₄⋊C₁₆	—	73	2	no	—
147	C₄⋊C₁₆	—	73	2	no	—
148	C₄⋊C₁₆	—	73	2	no	—
149	C₄⋊C₁₆	—	73	1	no	—
150	C₄⋊C₁₆	#64408	1	1	yes	characteristic, complemented
151	#346	—	146	4	no	—
152	C₂²×C₈	—	73	1	no	—
153	C₂²×C₈	—	73	1	no	—
154	C₂²×C₈	—	73	1	no	—
155	C₂²×C₈	—	73	1	no	—
156	C₂×C₄²	—	73	1	no	—
157	C₂×C₄²	—	73	1	no	—
158	C₂×C₄²	—	73	1	no	—
159	C₂×C₄²	—	73	2	no	—
160	C₂×C₄²	—	73	2	no	—
161	C₂×C₁₆	—	73	4	no	—
162	C₂×C₁₆	—	73	2	no	—
163	C₂×C₁₆	—	146	2	no	—
164	C₂×C₁₆	—	146	2	no	—
165	C₂×C₁₆	—	146	2	no	—
166	C₂×C₁₆	—	146	2	no	—
167	C₂×C₁₆	C₂×#64408	1	2	yes	—
168	C₄×C₈	—	73	2	no	—
169	C₄×C₈	—	73	2	no	—
170	C₄×C₈	—	73	2	no	—
171	C₄×C₈	—	73	2	no	—
172	C₄×C₈	—	73	1	no	—
173	C₄×C₈	—	73	1	no	—
174	C₄×C₈	—	73	1	no	—
175	C₄×C₈	—	146	2	no	—
176	C₄×C₈	—	146	2	no	—
177	C₄×C₈	—	146	1	no	—
178	C₄×C₈	—	146	1	no	—
179	C₄×C₈	C₂×#64408	1	1	yes	characteristic
180	C₂²×C₄	—	73	1	no	—
181	C₂²×C₄	—	73	1	no	—
182	C₂²×C₄	—	73	2	no	—
183	C₂²×C₄	—	73	2	no	—
184	C₂²×C₄	—	73	1	no	—
185	C₂×C₈	—	73	2	no	—
186	C₂×C₈	—	73	2	no	—
187	C₂×C₈	—	73	2	no	—
188	C₂×C₈	—	73	1	no	—
189	C₂×C₈	—	73	1	no	—
190	C₂×C₈	—	73	1	no	—
191	C₂×C₈	—	73	1	no	—
192	C₂×C₈	—	73	1	no	—
193	C₂×C₈	—	73	2	no	—
194	C₂×C₈	—	73	1	no	—
195	C₂×C₈	—	146	1	no	—
196	C₂×C₈	—	146	1	no	—
197	C₂×C₈	—	146	1	no	—
198	C₂×C₈	—	146	1	no	—
199	C₂×C₈	C₄×#64408	1	1	yes	characteristic
200	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
201	C₄²	—	73	2	no	—
202	C₄²	—	73	2	no	—
203	C₄²	—	73	1	no	—
204	C₄²	—	73	1	no	—
205	C₄²	—	73	1	no	—
206	C₄²	—	146	1	no	—
207	C₄²	—	146	1	no	—
208	C₄²	—	146	4	no	—
209	C₄²	—	146	4	no	—
210	C₄²	C₄×#64408	1	1	yes	characteristic
211	C₁₆	—	2	2	no	—
212	C₁₆	—	146	4	no	—
213	C₁₆	—	146	2	no	—
214	C₂³	—	73	1	no	—
215	C₂×C₄	—	73	1	no	—
216	C₂×C₄	—	73	1	no	—
217	C₂×C₄	—	73	4	no	—
218	C₂×C₄	—	73	1	no	—
219	C₂×C₄	—	73	1	no	—
220	C₂×C₄	—	73	1	no	—
221	C₂×C₄	—	73	1	no	—
222	C₂×C₄	—	73	4	no	—
223	C₂×C₄	—	73	4	no	—
224	C₂×C₄	—	73	1	no	—
225	C₂×C₄	—	146	2	no	—
226	C₂×C₄	—	146	2	no	—
227	C₂×C₄	—	146	2	no	—
228	C₂×C₄	—	146	2	no	—
229	C₂×C₄	—	146	2	no	—
230	C₂×C₄	C₈×#64408	1	1	yes	characteristic
231	C₂×C₄	C₈×#64408	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
233	C₈	—	2	1	no	—
234	C₈	—	73	4	no	—
235	C₈	—	73	1	no	—
236	C₈	—	73	1	no	—
237	C₈	—	146	2	no	—
238	C₈	—	146	1	no	—
239	C₈	D₄×#64408	1	1	yes	characteristic
240	C₈	Q₈×#64408	1	1	yes	characteristic
241	C₂²	—	73	2	no	—
242	C₂²	—	73	2	no	—
243	C₂²	—	73	1	no	—
244	C₂²	—	73	1	no	—
245	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	73	2	no	—
248	C₄	—	73	1	no	—
249	C₄	—	73	1	no	—
250	C₄	—	146	4	no	—
251	C₄	—	146	4	no	—
252	C₄	—	146	1	no	—
253	C₄	#17989474	1	2	yes	complemented
254	C₄	C₄⋊C₄×#64408	1	1	yes	characteristic
255	C₄	M₁₆×#64408	1	1	yes	characteristic
256	C₂	—	73	2	no	—
257	C₂	—	73	2	no	—
258	C₂	C₄⋊C₈×#64408	1	1	yes	characteristic
259	C₂	C₂×#17989474	1	1	yes	characteristic
260	C₂	M₃₂×#64408	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₄⋊C₁₆×C₇₃	C₄
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	C₄⋊C₁₆	#64408
hypocenter	C₇₃	#12653
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
13. #17989508	C₄	semidirect
14. C₄⋊#11924031	C₄	semidirect
18. C₄⋊#11924003	C₄	semidirect
19. #17988405	C₄	semidirect
20. C₄⋊C₁₆×C₇₃	C₄	semidirect
118. C₄×C₇₃	#346	semidirect
122. Dic₇₃	#346	semidirect
136. C₇₃	#12653	semidirect
150. C₄⋊C₁₆	#64408	semidirect
253. C₄	#17989474	semidirect

Maximal subgroups

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

subgroup	index
#22626261	2
C₂×#17989474	2
C₄×C₈×#64408	2
#22624968	2
C₄⋊C₁₆×D₇₃	2
#12653	73