Subgroups — #34018892

Order 18688.

614 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626720	C₂	1	2	yes	maximal
2	#22625570	C₂	1	4	yes	maximal, complemented
3	#22625519	C₂	1	4	yes	maximal, complemented
4	#22625489	C₂	1	1	yes	maximal, characteristic
5	#22625281	C₂	1	2	yes	maximal
6	C₈×#11924036	C₂	1	1	yes	maximal, characteristic
7	#22620518	C₂	1	1	yes	maximal, characteristic
8	#17989638	—	2	2	no	—
9	#17989637	C₂²	1	4	yes	—
10	#17989636	C₂²	1	4	yes	—
11	Q₈×#10596833	—	2	2	no	—
12	#17989594	—	2	2	no	—
13	#17989591	C₂²	1	2	yes	—
14	#17989552	—	2	2	no	—
15	C₄×#11924036	C₂²	1	1	yes	characteristic
16	C₄×#11924036	C₂²	1	1	yes	characteristic
17	#17989450	C₂²	1	4	yes	—
18	#17989437	C₂²	1	8	yes	—
19	C₈×#10596833	C₂²	1	2	yes	—
20	C₈×#10596833	C₂²	1	2	yes	—
21	#17989142	C₂²	1	2	yes	—
22	#17988967	C₄	1	1	yes	characteristic
23	#17988939	C₄	1	2	yes	—
24	#17988829	C₄	1	4	yes	—
25	#17988681	C₂²	1	4	yes	—
26	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
27	#500×C₇₃	C₄	1	1	yes	characteristic
28	C₂×#11924038	—	2	2	no	—
29	C₂×#11924036	—	2	1	no	—
30	C₂×#11924036	D₄	1	1	yes	characteristic
31	C₂×#11924036	D₄	1	1	yes	characteristic
32	#14556658	—	2	4	no	—
33	C₄⋊#10596833	—	2	4	no	—
34	C₄⋊#10596833	—	2	2	no	—
35	C₄⋊#10596833	C₂³	1	4	yes	—
36	#14556646	—	2	4	no	—
37	C₄×#10596833	—	2	4	no	—
38	C₄×#10596833	—	2	1	no	—
39	C₄×#10596833	D₄	1	1	yes	characteristic
40	C₄×#10596833	D₄	1	1	yes	characteristic
41	C₄×#10596833	C₂³	1	2	yes	—
42	C₄×#10596833	C₂³	1	1	yes	characteristic
43	C₄×#10596833	C₂³	1	1	yes	characteristic
44	C₂×Dic₂₉₂	C₂×C₄	1	4	yes	—
45	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
46	#14556591	C₂×C₄	1	2	yes	—
47	#14556582	—	2	2	no	—
48	#14556571	C₂×C₄	1	4	yes	—
49	#14556567	—	2	2	no	—
50	Q₈×Dic₇₃	—	2	2	no	—
51	#14556556	C₂³	1	4	yes	—
52	#14556549	—	2	2	no	—
53	C₄²×D₇₃	C₂³	1	1	yes	characteristic
54	#14556514	C₂×C₄	1	8	yes	—
55	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₇₃	C₂³	1	2	yes	—
57	C₄⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
58	#107×C₇₃	C₂×C₄	1	4	yes	—
59	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
60	#11924038	—	4	4	no	—
61	#11924036	—	2	4	no	—
62	#11924036	—	2	2	no	—
63	#11924036	Q₁₆	1	4	yes	complemented
64	C₂×#10596833	—	2	2	no	—
65	C₂×#10596833	—	2	4	no	—
66	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
67	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
68	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
69	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
70	C₂×Dic₁₄₆	C₂²×C₄	1	4	yes	—
71	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
74	Dic₂₉₂	—	2	8	no	—
75	C₂×#10596831	C₂²×C₄	1	2	yes	—
76	C₈×D₇₃	—	2	4	no	—
77	C₄⋊Dic₇₃	—	2	2	no	—
78	C₄⋊Dic₇₃	—	2	2	no	—
79	C₄⋊Dic₇₃	—	2	4	no	—
80	#11924007	—	2	4	no	—
81	#11924005	—	2	4	no	—
82	C₄×Dic₇₃	—	2	4	no	—
83	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
84	C₄×Dic₇₃	#41	1	1	yes	characteristic
85	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
86	C₂×Q₈×C₇₃	—	2	2	no	—
87	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
88	C₄⋊C₄×C₇₃	—	2	2	no	—
89	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
90	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
92	Q₈×C₇₃	—	4	4	no	—
93	C₂×C₄×C₇₃	—	2	4	no	—
94	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×Dic₇₃	—	2	4	no	—
98	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
99	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
100	C₂×Dic₇₃	C₂×#41	1	1	yes	characteristic
101	C₄×D₇₃	—	2	2	no	—
102	C₄×D₇₃	—	2	2	no	—
103	C₄×D₇₃	C₂×Q₁₆	1	2	yes	—
104	C₄×D₇₃	#135	1	2	yes	—
105	Dic₁₄₆	—	2	8	no	—
106	#10596833	—	2	4	no	—
107	#10596833	—	2	1	no	—
108	#10596833	—	2	1	no	—
109	#10596833	—	4	4	no	—
110	#10596833	C₂×Q₁₆	1	2	yes	—
111	#10596833	C₂×Q₁₆	1	2	yes	—
112	C₈×C₇₃	#131	1	4	yes	—
113	#10596831	—	2	2	no	—
114	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
115	C₂×D₇₃	—	2	1	no	—
116	C₂×D₇₃	—	2	1	no	—
117	C₂×D₇₃	C₄×Q₁₆	1	2	yes	—
118	C₄×C₇₃	—	4	4	no	—
119	C₄×C₇₃	C₄×Q₁₆	1	2	yes	—
120	C₄×C₇₃	#443	1	2	yes	—
121	C₄×C₇₃	C₂×#131	1	2	yes	—
122	Dic₇₃	—	2	1	no	—
123	Dic₇₃	—	2	1	no	—
124	Dic₇₃	—	4	4	no	—
125	Dic₇₃	C₂²×Q₁₆	1	1	yes	characteristic
126	Dic₇₃	C₂×#135	1	1	yes	characteristic
127	#17958	—	73	1	no	maximal, Sylow
128	C₂×C₇₃	#2931	1	1	yes	characteristic
129	C₂×C₇₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
130	C₂×C₇₃	C₂×#443	1	1	yes	characteristic
131	D₇₃	—	2	2	no	—
132	#3170	—	73	1	no	—
133	#2960	—	73	2	no	—
134	#1591	—	73	4	no	—
135	C₈×Q₁₆	—	73	4	no	—
136	#1567	—	73	1	no	—
137	#1493	—	73	2	no	—
138	C₂×C₈²	—	73	1	no	—
139	C₇₃	#17958	1	1	yes	characteristic, complemented, Sylow
140	#533	—	73	2	no	—
141	#500	#64408	1	1	yes	characteristic
142	#494	—	73	1	no	—
143	#487	—	73	2	no	—
144	#467	—	73	4	no	—
145	#446	—	73	4	no	—
146	#446	—	146	2	no	—
147	C₈×Q₈	—	73	4	no	—
148	C₈×Q₈	—	146	2	no	—
149	C₄×Q₁₆	—	73	4	no	—
150	#424	—	146	2	no	—
151	#423	—	73	2	no	—
152	#405	—	146	2	no	—
153	C₂×C₄×C₈	—	73	1	no	—
154	C₂×C₄×C₈	—	73	1	no	—
155	C₂×C₄×C₈	—	73	1	no	—
156	#335	—	73	4	no	—
157	Q₈⋊C₈	—	73	8	no	—
158	C₈²	—	73	2	no	—
159	C₈²	—	73	2	no	—
160	C₂×Q₁₆	—	73	4	no	—
161	C₂×M₁₆	—	146	2	no	—
162	C₂²×C₈	—	73	1	no	—
163	C₂²×C₈	—	73	1	no	—
164	C₂²×C₈	—	73	2	no	—
165	C₂²×C₈	—	146	1	no	—
166	C₄⋊Q₈	—	73	2	no	—
167	C₄⋊Q₈	C₂×#64408	1	2	yes	—
168	#125	—	146	2	no	—
169	C₂²⋊Q₈	—	73	4	no	—
170	C₂²⋊Q₈	—	146	2	no	—
171	C₄×Q₈	—	73	4	no	—
172	C₄×Q₈	—	146	2	no	—
173	#117	—	146	2	no	—
174	C₂×C₄²	—	73	1	no	—
175	#107	C₂×#64408	1	4	yes	—
176	C₄⋊C₈	—	73	4	no	—
177	C₄⋊C₈	—	146	2	no	—
178	C₄⋊C₈	—	146	4	no	—
179	C₄⋊C₈	—	146	4	no	—
180	#103	—	73	8	no	—
181	#97	—	146	4	no	—
182	C₄×C₈	—	73	1	no	—
183	C₄×C₈	—	73	1	no	—
184	C₄×C₈	—	73	1	no	—
185	C₄×C₈	—	73	1	no	—
186	C₄×C₈	—	73	2	no	—
187	C₄×C₈	—	73	2	no	—
188	C₄×C₈	—	73	1	no	—
189	C₄×C₈	—	146	1	no	—
190	C₄×C₈	—	146	4	no	—
191	C₄×C₈	C₂×#64408	1	1	yes	characteristic
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	73	4	no	—
194	C₂²×C₄	—	73	1	no	—
195	C₂²×C₄	—	73	1	no	—
196	C₂²×C₄	—	73	1	no	—
197	Q₁₆	—	146	8	no	—
198	M₁₆	—	292	4	no	—
199	C₂×C₈	—	73	1	no	—
200	C₂×C₈	—	73	1	no	—
201	C₂×C₈	—	73	4	no	—
202	C₂×C₈	—	73	1	no	—
203	C₂×C₈	—	73	1	no	—
204	C₂×C₈	—	73	2	no	—
205	C₂×C₈	—	146	4	no	—
206	C₂×C₈	—	146	2	no	—
207	C₂×C₈	—	146	2	no	—
208	C₂×C₈	—	146	4	no	—
209	C₂×C₈	—	146	4	no	—
210	C₂×C₈	C₂²×#64408	1	2	yes	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	146	4	no	—
213	C₄⋊C₄	—	146	4	no	—
214	C₄⋊C₄	—	146	2	no	—
215	C₄⋊C₄	—	146	2	no	—
216	C₄⋊C₄	C₂²×#64408	1	4	yes	—
217	C₂²⋊C₄	—	146	4	no	—
218	C₄²	—	73	1	no	—
219	C₄²	—	73	1	no	—
220	C₄²	—	73	1	no	—
221	C₄²	—	146	4	no	—
222	C₄²	C₂²×#64408	1	1	yes	characteristic
223	C₂³	—	73	1	no	—
224	Q₈	—	4	4	no	—
225	Q₈	—	146	8	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	73	1	no	—
228	C₂×C₄	—	73	2	no	—
229	C₂×C₄	—	73	2	no	—
230	C₂×C₄	—	73	1	no	—
231	C₂×C₄	—	73	1	no	—
232	C₂×C₄	—	146	4	no	—
233	C₂×C₄	—	146	2	no	—
234	C₂×C₄	—	146	2	no	—
235	C₂×C₄	D₄×#64408	1	1	yes	characteristic
236	C₂×C₄	D₄×#64408	1	1	yes	characteristic
237	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
238	C₈	—	292	4	no	—
239	C₈	—	73	2	no	—
240	C₈	—	73	2	no	—
241	C₈	—	146	1	no	—
242	C₈	—	146	1	no	—
243	C₈	—	146	4	no	—
244	C₈	—	146	2	no	—
245	C₈	#14556679	1	4	yes	—
246	C₂²	—	73	2	no	—
247	C₂²	—	146	1	no	—
248	C₂²	—	146	1	no	—
249	C₂²	C₂×D₄×#64408	1	1	yes	characteristic
250	C₄	—	4	4	no	—
251	C₄	—	292	4	no	—
252	C₄	—	73	1	no	—
253	C₄	—	73	1	no	—
254	C₄	—	146	1	no	—
255	C₄	—	146	1	no	—
256	C₄	Q₁₆×#64408	1	2	yes	—
257	C₄	#17989629	1	2	yes	—
258	C₄	C₂×#14556679	1	2	yes	—
259	C₂	—	146	2	no	—
260	C₂	#22626656	1	1	yes	characteristic
261	C₂	C₂×Q₁₆×#64408	1	1	yes	characteristic
262	C₂	C₂×#17989629	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	#500×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂²×D₇₃	C₄×D₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	#500	#64408
hypocenter	C₇₃	#17958
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
2. #22625570	C₂	semidirect
3. #22625519	C₂	semidirect
63. #11924036	Q₁₆	semidirect
139. C₇₃	#17958	semidirect

Maximal subgroups

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

subgroup	index
#22626720	2
#22625570	2
#22625519	2
#22625489	2
#22625281	2
C₈×#11924036	2
#22620518	2
#17958	73