Subgroups — #34008126

Order 18688.

506 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625864	C₂	1	1	yes	maximal, characteristic
2	#22625646	C₂	1	1	yes	maximal, characteristic
3	#22624767	C₂	1	4	yes	maximal
4	C₄⋊Q₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556658	—	2	1	no	—
6	C₂×#14556658	—	2	1	no	—
7	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
8	C₂×C₄×#10596833	C₂²	1	2	yes	—
9	#17989453	—	2	1	no	—
10	#17989453	—	2	1	no	—
11	C₂×Q₈×Dic₇₃	—	2	1	no	—
12	#17988494	—	2	1	no	—
13	#17988486	—	2	1	no	—
14	#17988486	C₄	1	2	yes	—
15	#17988457	C₄	1	1	yes	characteristic
16	C₄⋊C₄×Dic₇₃	—	2	1	no	—
17	C₄⋊C₄×Dic₇₃	C₂²	1	2	yes	—
18	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₇₃	C₄	1	1	yes	characteristic
20	C₂²×#10596833	—	2	2	no	—
21	C₂²×#10596833	—	2	2	no	—
22	#14556658	—	4	2	no	—
23	#14556658	—	4	2	no	—
24	C₄⋊#10596833	—	2	4	no	—
25	C₄×#10596833	—	2	2	no	—
26	C₄×#10596833	Q₈	1	4	yes	complemented
27	Q₈×Dic₇₃	—	4	2	no	—
28	Q₈×Dic₇₃	—	4	2	no	—
29	C₂×C₄⋊Dic₇₃	—	2	2	no	—
30	C₂×C₄⋊Dic₇₃	—	2	1	no	—
31	C₂×C₄⋊Dic₇₃	—	2	1	no	—
32	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₇₃	—	2	2	no	—
34	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
37	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
38	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
39	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
40	#14556498	—	2	2	no	—
41	#14556498	—	2	2	no	—
42	C₂²×Q₈×C₇₃	—	2	1	no	—
43	C₄⋊Q₈×C₇₃	—	2	4	no	—
44	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂×#10596833	—	2	4	no	—
48	C₂×#10596833	—	4	2	no	—
49	C₂×#10596833	—	4	2	no	—
50	C₂×#10596833	—	4	2	no	—
51	C₂×#10596833	—	4	2	no	—
52	C₂×#10596833	—	4	2	no	—
53	C₂²×Dic₇₃	—	2	2	no	—
54	C₂²×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
58	C₄⋊Dic₇₃	—	2	4	no	—
59	C₄⋊Dic₇₃	—	4	2	no	—
60	C₄⋊Dic₇₃	—	4	2	no	—
61	C₄⋊Dic₇₃	—	4	4	no	—
62	C₄×Dic₇₃	—	2	2	no	—
63	C₄×Dic₇₃	—	4	1	no	—
64	C₄×Dic₇₃	—	4	1	no	—
65	C₄×Dic₇₃	—	4	2	no	—
66	C₄×Dic₇₃	—	4	2	no	—
67	C₄×Dic₇₃	D₈	1	2	yes	—
68	C₄×Dic₇₃	SD₁₆	1	2	yes	—
69	C₄×Dic₇₃	SD₁₆	1	2	yes	—
70	C₄×Dic₇₃	Q₁₆	1	2	yes	—
71	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
72	C₄×Dic₇₃	#41	1	2	yes	—
73	C₂×Q₈×C₇₃	—	2	4	no	—
74	C₂×Q₈×C₇₃	—	4	2	no	—
75	C₂×Q₈×C₇₃	—	4	2	no	—
76	C₂²×C₄×C₇₃	—	2	2	no	—
77	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₇₃	—	2	4	no	—
81	C₄⋊C₄×C₇₃	—	2	2	no	—
82	C₄⋊C₄×C₇₃	M₁₆	1	4	yes	—
83	C₄²×C₇₃	—	2	2	no	—
84	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	Q₈×C₇₃	—	4	8	no	—
86	C₂×C₄×C₇₃	—	2	1	no	—
87	C₂×C₄×C₇₃	—	2	1	no	—
88	C₂×C₄×C₇₃	—	2	1	no	—
89	C₂×C₄×C₇₃	—	2	1	no	—
90	C₂×C₄×C₇₃	—	2	4	no	—
91	C₂×C₄×C₇₃	—	4	2	no	—
92	C₂×C₄×C₇₃	—	4	2	no	—
93	C₂×C₄×C₇₃	#102	1	2	yes	—
94	C₂×C₄×C₇₃	#103	1	2	yes	—
95	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
96	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₇₃	#131	1	2	yes	—
98	C₂×Dic₇₃	—	2	1	no	—
99	C₂×Dic₇₃	—	2	1	no	—
100	C₂×Dic₇₃	—	2	1	no	—
101	C₂×Dic₇₃	—	2	1	no	—
102	C₂×Dic₇₃	—	2	1	no	—
103	C₂×Dic₇₃	—	2	1	no	—
104	C₂×Dic₇₃	—	4	2	no	—
105	C₂×Dic₇₃	—	4	2	no	—
106	C₂×Dic₇₃	—	4	2	no	—
107	C₂×Dic₇₃	#102	1	2	yes	—
108	C₂×Dic₇₃	#103	1	2	yes	—
109	C₂×Dic₇₃	C₂²⋊Q₈	1	2	yes	—
110	C₂×Dic₇₃	#124	1	1	yes	characteristic
111	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
112	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
113	C₂×Dic₇₃	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×Dic₇₃	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×Dic₇₃	C₂×Q₁₆	1	1	yes	characteristic
116	#10596833	—	4	4	no	—
117	#10596833	—	8	2	no	—
118	C₂²×C₇₃	#391	1	1	yes	characteristic
119	C₂²×C₇₃	#407	1	1	yes	characteristic
120	C₂²×C₇₃	#408	1	1	yes	characteristic
121	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
122	C₂²×C₇₃	C₂×#103	1	1	yes	characteristic
123	C₂²×C₇₃	#421	1	1	yes	characteristic
124	C₂²×C₇₃	#421	1	1	yes	characteristic
125	C₄×C₇₃	—	2	2	no	—
126	C₄×C₇₃	—	2	2	no	—
127	C₄×C₇₃	—	4	4	no	—
128	C₄×C₇₃	#446	1	4	yes	—
129	Dic₇₃	—	2	2	no	—
130	Dic₇₃	—	2	2	no	—
131	Dic₇₃	—	2	2	no	—
132	Dic₇₃	—	8	2	no	—
133	Dic₇₃	#486	1	1	yes	characteristic
134	Dic₇₃	#487	1	1	yes	characteristic
135	Dic₇₃	#498	1	1	yes	characteristic
136	Dic₇₃	#500	1	1	yes	characteristic
137	#9874	—	73	1	no	maximal, Sylow
138	C₂×C₇₃	#1480	1	1	yes	characteristic
139	C₂×C₇₃	#1481	1	1	yes	characteristic
140	C₂×C₇₃	#1486	1	1	yes	characteristic
141	C₂×C₇₃	#1487	1	1	yes	characteristic
142	C₂×C₇₃	#1985	1	1	yes	characteristic
143	C₂×C₇₃	#1990	1	1	yes	characteristic
144	C₂×C₇₃	#1995	1	1	yes	characteristic
145	C₄×C₄⋊Q₈	—	73	1	no	—
146	#1842	—	73	1	no	—
147	#1744	—	73	1	no	—
148	#1279	—	73	4	no	—
149	C₇₃	#9874	1	1	yes	characteristic, complemented, Sylow
150	C₂×C₄⋊Q₈	#64408	1	1	yes	characteristic
151	C₂×C₄×Q₈	—	146	1	no	—
152	C₂×C₄⋊C₈	—	73	2	no	—
153	C₂×C₄⋊C₈	—	146	1	no	—
154	C₂×C₄⋊C₈	—	146	1	no	—
155	C₂×C₄×C₈	—	73	2	no	—
156	#391	—	146	1	no	—
157	#389	—	73	2	no	—
158	#389	—	146	1	no	—
159	#384	—	73	1	no	—
160	C₄×C₄⋊C₄	—	73	2	no	—
161	C₄×C₄⋊C₄	—	146	1	no	—
162	C₄³	—	73	1	no	—
163	#336	—	146	1	no	—
164	#336	—	146	1	no	—
165	C₂²×Q₈	—	2	1	no	—
166	C₂²×C₈	—	146	2	no	—
167	C₂²×C₈	—	146	2	no	—
168	C₄⋊Q₈	—	2	4	no	—
169	C₄×Q₈	—	292	2	no	—
170	C₄×Q₈	—	292	2	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	—	73	2	no	—
173	C₂×C₄⋊C₄	—	146	2	no	—
174	C₂×C₄⋊C₄	—	146	1	no	—
175	C₂×C₄⋊C₄	—	146	1	no	—
176	C₂×C₄⋊C₄	C₂×#64408	1	2	yes	—
177	C₂×C₄²	—	73	1	no	—
178	C₂×C₄²	—	73	1	no	—
179	C₂×C₄²	—	73	1	no	—
180	C₂×C₄²	—	73	1	no	—
181	C₂×C₄²	—	73	1	no	—
182	C₂×C₄²	—	73	1	no	—
183	C₂×C₄²	—	146	2	no	—
184	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
185	C₄⋊C₈	—	292	2	no	—
186	C₄⋊C₈	—	292	2	no	—
187	C₄⋊C₈	—	146	4	no	—
188	C₄×C₈	—	73	4	no	—
189	C₄×C₈	—	146	2	no	—
190	#95	—	146	2	no	—
191	#95	—	146	2	no	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×Q₈	—	4	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	73	1	no	—
197	C₂²×C₄	—	73	1	no	—
198	C₂²×C₄	—	73	1	no	—
199	C₂²×C₄	—	73	1	no	—
200	C₂²×C₄	—	146	2	no	—
201	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
202	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
203	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
204	C₂×C₈	—	292	2	no	—
205	C₂×C₈	—	292	2	no	—
206	C₂×C₈	—	292	2	no	—
207	C₂×C₈	—	292	2	no	—
208	C₂×C₈	—	292	2	no	—
209	C₂×C₈	—	146	4	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	292	2	no	—
213	C₄⋊C₄	—	292	4	no	—
214	C₄⋊C₄	—	292	2	no	—
215	C₄⋊C₄	—	146	4	no	—
216	C₄⋊C₄	#11924037	1	4	yes	—
217	C₄²	—	2	2	no	—
218	C₄²	—	292	1	no	—
219	C₄²	—	292	1	no	—
220	C₄²	—	292	2	no	—
221	C₄²	—	292	2	no	—
222	C₄²	—	73	2	no	—
223	C₄²	—	73	2	no	—
224	C₄²	—	73	2	no	—
225	C₄²	—	73	2	no	—
226	C₄²	—	73	2	no	—
227	C₄²	—	73	2	no	—
228	C₄²	—	146	2	no	—
229	C₂³	C₂×C₂²⋊#64408	1	1	yes	characteristic
230	Q₈	—	4	8	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	292	2	no	—
239	C₂×C₄	—	292	2	no	—
240	C₂×C₄	—	292	2	no	—
241	C₂×C₄	—	73	1	no	—
242	C₂×C₄	—	73	1	no	—
243	C₂×C₄	—	73	1	no	—
244	C₂×C₄	—	73	1	no	—
245	C₂×C₄	—	73	1	no	—
246	C₂×C₄	—	73	1	no	—
247	C₂×C₄	—	73	2	no	—
248	C₂×C₄	—	73	2	no	—
249	C₂×C₄	—	73	2	no	—
250	C₂×C₄	—	146	1	no	—
251	C₂×C₄	—	146	1	no	—
252	C₂×C₄	—	146	1	no	—
253	C₂×C₄	—	146	1	no	—
254	C₂×C₄	—	146	1	no	—
255	C₂×C₄	—	146	1	no	—
256	C₂×C₄	#14556653	1	2	yes	—
257	C₂×C₄	#14556654	1	2	yes	—
258	C₂×C₄	Q₈×#64408	1	2	yes	—
259	C₂×C₄	C₂×#11924037	1	2	yes	—
260	C₂×C₄	#14556679	1	2	yes	—
261	C₈	—	292	4	no	—
262	C₈	—	584	2	no	—
263	C₂²	(C₂×#64408)⋊Q₈	1	1	yes	characteristic
264	C₂²	#17989557	1	1	yes	characteristic
265	C₂²	#17989562	1	1	yes	characteristic
266	C₂²	C₂×#14556653	1	1	yes	characteristic
267	C₂²	C₂×#14556654	1	1	yes	characteristic
268	C₂²	#17989585	1	1	yes	characteristic
269	C₂²	#17989586	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	4	no	—
273	C₄	—	584	2	no	—
274	C₄	—	73	1	no	—
275	C₄	—	73	1	no	—
276	C₄	—	73	1	no	—
277	C₄	—	73	1	no	—
278	C₄	—	146	2	no	—
279	C₄	—	146	2	no	—
280	C₄	—	146	2	no	—
281	C₄	#17989638	1	4	yes	—
282	C₂	#22625238	1	1	yes	characteristic
283	C₂	#22625241	1	1	yes	characteristic
284	C₂	#22625256	1	1	yes	characteristic
285	C₂	#22625259	1	1	yes	characteristic
286	C₂	#22626105	1	1	yes	characteristic
287	C₂	#22626112	1	1	yes	characteristic
288	C₂	#22626119	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#64408
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄⋊Q₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×C₄⋊Q₈	#64408
hypocenter	C₇₃	#9874
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
26. C₄×#10596833	Q₈	semidirect
149. C₇₃	#9874	semidirect

Maximal subgroups

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

subgroup	index
#22625864	2
#22625646	2
#22624767	2
C₄⋊Q₈×Dic₇₃	2
#9874	73