Subgroups — #34007696

Order 18688.

568 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#10596833	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10596833	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#10596833	C₂	1	4	yes	maximal
4	C₄⋊Q₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10596833	—	2	1	no	—
6	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
7	C₂×C₄×#10596833	—	2	1	no	—
8	C₂×C₄×#10596833	C₂²	1	2	yes	—
9	#17989453	—	2	1	no	—
10	#17989453	—	2	1	no	—
11	Q₈⋊#10596833	—	2	8	no	—
12	C₂×Q₈×Dic₇₃	C₂²	1	2	yes	—
13	#17988494	C₄	1	2	yes	—
14	#17988486	—	2	1	no	—
15	#17988486	—	2	1	no	—
16	#17988457	C₄	1	1	yes	characteristic
17	C₄⋊C₄×Dic₇₃	—	2	1	no	—
18	C₄⋊C₄×Dic₇₃	—	2	1	no	—
19	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Q₈×C₇₃	C₄	1	1	yes	characteristic
21	C₂²×#10596833	—	2	2	no	—
22	C₂²×#10596833	—	2	2	no	—
23	C₄⋊#10596833	—	2	4	no	—
24	C₄⋊#10596833	—	2	2	no	—
25	C₄⋊#10596833	D₄	1	4	yes	—
26	C₄×#10596833	—	2	4	no	—
27	C₄×#10596833	—	4	2	no	—
28	Q₈×Dic₇₃	—	2	4	no	—
29	Q₈×Dic₇₃	—	2	4	no	—
30	C₂×C₄⋊Dic₇₃	—	2	2	no	—
31	C₂×C₄⋊Dic₇₃	—	2	1	no	—
32	C₂×C₄⋊Dic₇₃	—	2	1	no	—
33	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₇₃	—	2	2	no	—
35	C₂×C₄×Dic₇₃	C₂×C₄	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	D₄	1	1	yes	characteristic
40	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
41	#14556498	—	2	2	no	—
42	#14556498	—	2	2	no	—
43	C₂²×Q₈×C₇₃	C₂×C₄	1	2	yes	—
44	C₄⋊Q₈×C₇₃	—	2	2	no	—
45	C₄⋊Q₈×C₇₃	C₈	1	4	yes	complemented
46	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
47	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
48	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂×#10596833	—	2	4	no	—
50	C₂×#10596833	—	2	4	no	—
51	C₂×#10596833	—	4	2	no	—
52	C₂×#10596833	—	4	2	no	—
53	C₂×#10596833	—	4	2	no	—
54	C₂×#10596833	—	4	2	no	—
55	C₂²×Dic₇₃	—	2	2	no	—
56	C₂²×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
60	C₄⋊Dic₇₃	—	2	4	no	—
61	C₄⋊Dic₇₃	—	4	2	no	—
62	C₄⋊Dic₇₃	—	4	2	no	—
63	C₄⋊Dic₇₃	—	4	2	no	—
64	C₄⋊Dic₇₃	—	4	2	no	—
65	C₄×Dic₇₃	—	2	2	no	—
66	C₄×Dic₇₃	—	2	2	no	—
67	C₄×Dic₇₃	—	2	2	no	—
68	C₄×Dic₇₃	—	4	1	no	—
69	C₄×Dic₇₃	—	4	4	no	—
70	C₄×Dic₇₃	SD₁₆	1	2	yes	—
71	C₄×Dic₇₃	Q₁₆	1	2	yes	—
72	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
73	C₄×Dic₇₃	#41	1	2	yes	—
74	C₂×Q₈×C₇₃	—	2	2	no	—
75	C₂×Q₈×C₇₃	—	2	4	no	—
76	C₂×Q₈×C₇₃	—	2	4	no	—
77	C₂×Q₈×C₇₃	C₂×C₈	1	4	yes	—
78	C₂²×C₄×C₇₃	—	2	2	no	—
79	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₇₃	—	2	4	no	—
83	C₄⋊C₄×C₇₃	—	2	4	no	—
84	C₄²×C₇₃	C₂×C₈	1	2	yes	—
85	C₄²×C₇₃	M₁₆	1	2	yes	—
86	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	Q₈×C₇₃	—	2	8	no	—
88	Q₈×C₇₃	—	4	4	no	—
89	C₂×C₄×C₇₃	—	2	1	no	—
90	C₂×C₄×C₇₃	—	2	1	no	—
91	C₂×C₄×C₇₃	—	2	4	no	—
92	C₂×C₄×C₇₃	—	4	2	no	—
93	C₂×C₄×C₇₃	—	4	2	no	—
94	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₇₃	#103	1	2	yes	—
97	C₂×C₄×C₇₃	#104	1	2	yes	—
98	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₇₃	#131	1	2	yes	—
102	C₂×Dic₇₃	—	2	1	no	—
103	C₂×Dic₇₃	—	2	1	no	—
104	C₂×Dic₇₃	—	2	1	no	—
105	C₂×Dic₇₃	—	2	1	no	—
106	C₂×Dic₇₃	—	2	1	no	—
107	C₂×Dic₇₃	—	2	1	no	—
108	C₂×Dic₇₃	—	2	1	no	—
109	C₂×Dic₇₃	—	4	2	no	—
110	C₂×Dic₇₃	—	4	2	no	—
111	C₂×Dic₇₃	—	4	2	no	—
112	C₂×Dic₇₃	#103	1	2	yes	—
113	C₂×Dic₇₃	#121	1	2	yes	—
114	C₂×Dic₇₃	#124	1	1	yes	characteristic
115	C₂×Dic₇₃	#127	1	1	yes	characteristic
116	C₂×Dic₇₃	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×Dic₇₃	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×Dic₇₃	#137	1	2	yes	—
119	#10596833	—	4	4	no	—
120	#10596833	—	4	4	no	—
121	C₂²×C₇₃	#390	1	1	yes	characteristic
122	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₂²×C₇₃	#408	1	1	yes	characteristic
124	C₂²×C₇₃	C₂×#103	1	1	yes	characteristic
125	C₂²×C₇₃	#419	1	1	yes	characteristic
126	C₂²×C₇₃	C₂×#104	1	1	yes	characteristic
127	C₂²×C₇₃	#421	1	1	yes	characteristic
128	C₄×C₇₃	—	2	2	no	—
129	C₄×C₇₃	—	4	4	no	—
130	C₄×C₇₃	Q₈⋊C₈	1	4	yes	complemented
131	C₄×C₇₃	C₈×D₄	1	2	yes	—
132	C₄×C₇₃	#436	1	2	yes	—
133	Dic₇₃	—	2	1	no	—
134	Dic₇₃	—	2	1	no	—
135	Dic₇₃	—	2	2	no	—
136	Dic₇₃	—	4	1	no	—
137	Dic₇₃	—	8	2	no	—
138	Dic₇₃	#461	1	2	yes	—
139	Dic₇₃	#462	1	2	yes	—
140	#9673	—	73	1	no	maximal, Sylow
141	C₂×C₇₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
142	C₂×C₇₃	#1479	1	1	yes	characteristic
143	C₂×C₇₃	#1481	1	1	yes	characteristic
144	C₂×C₇₃	#1485	1	1	yes	characteristic
145	C₂×C₇₃	#1954	1	1	yes	characteristic
146	C₂×C₇₃	#1980	1	1	yes	characteristic
147	C₂×C₇₃	#1981	1	1	yes	characteristic
148	C₄×C₄⋊Q₈	—	73	1	no	—
149	C₄×C₄⋊C₈	—	73	1	no	—
150	C₄×C₄⋊C₈	—	73	1	no	—
151	C₂×Q₈⋊C₈	—	73	4	no	—
152	C₇₃	#9673	1	1	yes	characteristic, complemented, Sylow
153	C₂×C₄⋊Q₈	#64408	1	1	yes	characteristic
154	C₂×C₄×Q₈	—	73	2	no	—
155	C₂×C₄⋊C₈	—	73	2	no	—
156	C₂×C₄⋊C₈	—	146	1	no	—
157	C₂×C₄×C₈	—	73	2	no	—
158	C₂×C₄×C₈	—	146	1	no	—
159	#391	—	73	2	no	—
160	#389	—	146	1	no	—
161	#389	—	146	1	no	—
162	#384	—	73	1	no	—
163	C₄×C₄⋊C₄	—	146	1	no	—
164	C₄×C₄⋊C₄	—	146	1	no	—
165	C₄³	—	73	1	no	—
166	#336	—	146	1	no	—
167	#336	—	146	1	no	—
168	Q₈⋊C₈	—	146	8	no	—
169	C₂²×Q₈	C₂×#64408	1	2	yes	—
170	C₂²×C₈	—	146	2	no	—
171	C₂²×C₈	—	146	2	no	—
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	#10596833	1	4	yes	complemented
174	C₄×Q₈	—	146	4	no	—
175	C₄×Q₈	—	146	4	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	73	2	no	—
179	C₂×C₄⋊C₄	—	146	2	no	—
180	C₂×C₄⋊C₄	—	146	1	no	—
181	C₂×C₄⋊C₄	—	146	1	no	—
182	C₂×C₄²	—	73	1	no	—
183	C₂×C₄²	—	73	1	no	—
184	C₂×C₄²	—	73	1	no	—
185	C₂×C₄²	—	73	1	no	—
186	C₂×C₄²	—	73	1	no	—
187	C₂×C₄²	—	73	1	no	—
188	C₂×C₄²	—	146	2	no	—
189	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
190	C₄⋊C₈	—	73	4	no	—
191	C₄⋊C₈	—	146	2	no	—
192	C₄⋊C₈	—	146	4	no	—
193	C₄×C₈	—	292	2	no	—
194	C₄×C₈	—	146	4	no	—
195	#95	—	146	2	no	—
196	#95	—	146	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	C₂×#10596833	1	4	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	73	1	no	—
203	C₂²×C₄	—	73	1	no	—
204	C₂²×C₄	—	73	1	no	—
205	C₂²×C₄	—	73	1	no	—
206	C₂²×C₄	—	146	2	no	—
207	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
209	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
210	C₂×C₈	—	292	2	no	—
211	C₂×C₈	—	292	2	no	—
212	C₂×C₈	—	292	2	no	—
213	C₂×C₈	—	292	2	no	—
214	C₂×C₈	—	146	4	no	—
215	C₂×C₈	—	146	4	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	292	2	no	—
219	C₄⋊C₄	—	292	2	no	—
220	C₄⋊C₄	—	292	2	no	—
221	C₄⋊C₄	—	292	2	no	—
222	C₄⋊C₄	—	146	4	no	—
223	C₄²	—	292	1	no	—
224	C₄²	—	292	4	no	—
225	C₄²	—	73	2	no	—
226	C₄²	—	73	2	no	—
227	C₄²	—	73	2	no	—
228	C₄²	—	73	2	no	—
229	C₄²	—	146	2	no	—
230	C₄²	—	146	2	no	—
231	C₄²	—	146	2	no	—
232	C₄²	C₂×#10596833	1	2	yes	—
233	C₄²	#11924037	1	2	yes	—
234	C₂³	C₂×C₂²⋊#64408	1	1	yes	characteristic
235	Q₈	—	2	8	no	—
236	Q₈	—	4	4	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	292	2	no	—
243	C₂×C₄	—	292	2	no	—
244	C₂×C₄	—	292	2	no	—
245	C₂×C₄	—	73	2	no	—
246	C₂×C₄	—	73	1	no	—
247	C₂×C₄	—	73	1	no	—
248	C₂×C₄	—	73	2	no	—
249	C₂×C₄	—	73	1	no	—
250	C₂×C₄	—	73	1	no	—
251	C₂×C₄	—	73	2	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	—	146	1	no	—
257	C₂×C₄	—	146	1	no	—
258	C₂×C₄	—	146	1	no	—
259	C₂×C₄	C₂²⋊#10596833	1	2	yes	—
260	C₂×C₄	C₂²⋊#10596833	1	2	yes	—
261	C₂×C₄	#14556654	1	2	yes	—
262	C₂×C₄	#14556656	1	2	yes	—
263	C₂×C₄	D₄×#64408	1	2	yes	—
264	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
265	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
266	C₂×C₄	#14556678	1	2	yes	—
267	C₈	—	292	4	no	—
268	C₈	—	292	4	no	—
269	C₂²	#17989539	1	1	yes	characteristic
270	C₂²	C₂×C₂²⋊#10596833	1	1	yes	characteristic
271	C₂²	#17989561	1	1	yes	characteristic
272	C₂²	C₂×#14556654	1	1	yes	characteristic
273	C₂²	#17989580	1	1	yes	characteristic
274	C₂²	C₂×#14556656	1	1	yes	characteristic
275	C₂²	#17989586	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	4	4	no	—
278	C₄	—	292	1	no	—
279	C₄	—	584	2	no	—
280	C₄	—	73	2	no	—
281	C₄	—	73	2	no	—
282	C₄	—	146	1	no	—
283	C₄	—	146	1	no	—
284	C₄	—	146	2	no	—
285	C₄	Q₈⋊#10596833	1	4	yes	complemented
286	C₄	D₄×#10596833	1	2	yes	—
287	C₄	#17989622	1	2	yes	—
288	C₂	C₂×Q₈⋊#10596833	1	1	yes	characteristic
289	C₂	#22625233	1	1	yes	characteristic
290	C₂	#22625240	1	1	yes	characteristic
291	C₂	#22625254	1	1	yes	characteristic
292	C₂	#22626052	1	1	yes	characteristic
293	C₂	#22626094	1	1	yes	characteristic
294	C₂	#22626097	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
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	#9673
LCS	G ▷ C₂×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
45. C₄⋊Q₈×C₇₃	C₈	semidirect
130. C₄×C₇₃	Q₈⋊C₈	semidirect
152. C₇₃	#9673	semidirect
173. C₄⋊Q₈	#10596833	semidirect
285. C₄	Q₈⋊#10596833	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#10596833	2
C₂×Q₈⋊#10596833	2
C₄⋊Q₈×Dic₇₃	2
#9673	73