Subgroups — #34032745

Order 18688.

352 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627075	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10596842	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×#64408	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#10596842	—	2	1	no	—
5	C₂×C₄⋊#10596842	—	2	1	no	—
6	C₂×C₄⋊#10596842	—	2	1	no	—
7	C₂×C₄⋊#10596842	C₄	1	2	yes	—
8	C₂×C₄×#10596842	—	2	1	no	—
9	(C₂×#10596842)⋊C₄	C₄	1	2	yes	—
10	#17989588	C₄	1	1	yes	characteristic
11	C₂×C₈×#64408	—	2	1	no	—
12	C₄²×#64408	C₂²	1	1	yes	characteristic
13	#17989452	—	2	1	no	—
14	C₄⋊C₈×D₇₃	C₄	1	1	yes	characteristic
15	C₂²×#10596842	—	2	2	no	—
16	C₂²×#10596842	—	2	2	no	—
17	C₄⋊#10596842	—	2	2	no	—
18	C₄⋊#10596842	—	2	2	no	—
19	C₄⋊#10596842	—	4	2	no	—
20	C₄⋊#10596842	—	4	1	no	—
21	C₄⋊#10596842	—	4	1	no	—
22	C₄⋊#10596842	—	4	1	no	—
23	C₄⋊#10596842	—	4	1	no	—
24	C₄×#10596842	—	4	1	no	—
25	C₄×#10596842	—	4	1	no	—
26	C₂×#11924036	—	2	1	no	—
27	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×#64408	D₄	1	1	yes	characteristic
30	C₂×C₄×#64408	D₄	1	1	yes	characteristic
31	C₂×C₄×#64408	D₄	1	1	yes	characteristic
32	C₂×C₄×#64408	Q₈	1	1	yes	characteristic
33	#14556658	C₈	1	2	yes	complemented
34	C₄⋊#10596833	—	2	1	no	—
35	C₈×#64408	—	4	1	no	—
36	C₈×#64408	—	4	1	no	—
37	C₂×C₈×D₇₃	—	2	1	no	—
38	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄⋊#10596831	C₈	1	1	yes	characteristic, complemented
40	#14556520	—	2	1	no	—
41	C₄⋊C₈×C₇₃	C₈	1	1	yes	characteristic, complemented
42	C₂×#10596842	—	2	4	no	—
43	C₂×#10596842	—	4	2	no	—
44	C₂×#10596842	—	4	2	no	—
45	C₂×#10596842	—	4	2	no	—
46	C₂×#10596842	—	4	2	no	—
47	C₂×#10596842	—	4	2	no	—
48	C₂²×#64408	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×#64408	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×#64408	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×#64408	C₄⋊C₄	1	1	yes	characteristic
52	#11924036	—	4	1	no	—
53	#11924036	—	4	1	no	—
54	C₂×#10596833	—	2	2	no	—
55	C₄×#64408	—	2	1	no	—
56	C₄×#64408	—	2	1	no	—
57	C₄×#64408	—	2	1	no	—
58	C₄×#64408	—	2	1	no	—
59	C₄×#64408	—	2	1	no	—
60	C₄×#64408	—	2	1	no	—
61	C₄×#64408	—	4	1	no	—
62	C₄×#64408	D₈	1	1	yes	characteristic
63	C₄×#64408	D₈	1	1	yes	characteristic
64	C₄×#64408	SD₁₆	1	1	yes	characteristic
65	C₄×#64408	SD₁₆	1	1	yes	characteristic
66	C₄×#64408	SD₁₆	1	1	yes	characteristic
67	C₄×#64408	SD₁₆	1	1	yes	characteristic
68	C₄×#64408	Q₁₆	1	1	yes	characteristic
69	C₄×#64408	Q₁₆	1	1	yes	characteristic
70	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
71	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
73	C₂×#10596831	—	2	1	no	—
74	C₈×D₇₃	—	4	1	no	—
75	C₈×D₇₃	—	4	1	no	—
76	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
77	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
78	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
79	C₂×C₈×C₇₃	—	2	1	no	—
80	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
81	C₂²×D₇₃	#95	1	1	yes	characteristic
82	C₂×#64408	—	2	1	no	—
83	C₂×#64408	—	2	1	no	—
84	C₂×#64408	—	2	1	no	—
85	C₂×#64408	—	2	1	no	—
86	C₂×#64408	—	2	1	no	—
87	C₂×#64408	—	2	1	no	—
88	C₂×#64408	—	2	1	no	—
89	C₂×#64408	#100	1	1	yes	characteristic
90	C₂×#64408	#101	1	1	yes	characteristic
91	C₂×#64408	#102	1	1	yes	characteristic
92	C₂×#64408	#102	1	1	yes	characteristic
93	C₂×#64408	#102	1	1	yes	characteristic
94	C₂×#64408	#103	1	1	yes	characteristic
95	C₂×#64408	#103	1	1	yes	characteristic
96	C₂×#64408	#103	1	1	yes	characteristic
97	C₂×#64408	#106	1	1	yes	characteristic
98	C₂×#64408	#107	1	1	yes	characteristic
99	#10596842	—	4	4	no	—
100	#10596842	—	8	2	no	—
101	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
103	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
104	C₂×Dic₇₃	#97	1	1	yes	characteristic
105	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
106	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
107	C₄×D₇₃	—	2	1	no	—
108	C₄×D₇₃	—	2	1	no	—
109	C₄×D₇₃	#102	1	1	yes	characteristic
110	C₄×D₇₃	#103	1	1	yes	characteristic
111	C₄×D₇₃	#104	1	1	yes	characteristic
112	C₄×D₇₃	#104	1	1	yes	characteristic
113	C₄×D₇₃	#106	1	1	yes	characteristic
114	C₄×D₇₃	#107	1	1	yes	characteristic
115	C₄×D₇₃	#108	1	1	yes	characteristic
116	C₄×D₇₃	#108	1	1	yes	characteristic
117	#10596833	—	4	2	no	—
118	C₈×C₇₃	—	4	1	no	—
119	#10596831	—	4	1	no	—
120	C₂²×C₇₃	#336	1	1	yes	characteristic
121	C₂×D₇₃	#339	1	1	yes	characteristic
122	C₂×D₇₃	#340	1	1	yes	characteristic
123	C₂×D₇₃	#340	1	1	yes	characteristic
124	C₂×D₇₃	#341	1	1	yes	characteristic
125	C₂×D₇₃	#343	1	1	yes	characteristic
126	C₂×D₇₃	#344	1	1	yes	characteristic
127	#64408	—	2	1	no	—
128	#64408	—	2	1	no	—
129	#64408	—	2	1	no	—
130	#64408	—	2	1	no	—
131	#64408	—	4	1	no	—
132	#64408	#331	1	1	yes	characteristic
133	#64408	#332	1	1	yes	characteristic
134	#64408	#332	1	1	yes	characteristic
135	#64408	#333	1	1	yes	characteristic
136	C₄×C₇₃	—	2	1	no	—
137	C₄×C₇₃	D₄⋊C₈	1	1	yes	characteristic
138	C₄×C₇₃	Q₈⋊C₈	1	1	yes	characteristic
139	C₄×C₇₃	#334	1	1	yes	characteristic
140	C₄×C₇₃	#335	1	1	yes	characteristic
141	Dic₇₃	—	2	1	no	—
142	Dic₇₃	D₄⋊C₈	1	1	yes	characteristic
143	Dic₇₃	Q₈⋊C₈	1	1	yes	characteristic
144	Dic₇₃	#334	1	1	yes	characteristic
145	Dic₇₃	#335	1	1	yes	characteristic
146	#7031	—	73	1	no	maximal, Sylow
147	C₂×C₇₃	#1277	1	1	yes	characteristic
148	C₂×C₇₃	#1279	1	1	yes	characteristic
149	C₂×C₇₃	#1283	1	1	yes	characteristic
150	D₇₃	#1296	1	1	yes	characteristic
151	D₇₃	#1299	1	1	yes	characteristic
152	D₇₃	#1301	1	1	yes	characteristic
153	D₇₃	#1303	1	1	yes	characteristic
154	#1842	—	73	1	no	—
155	C₄×C₄⋊C₈	—	73	1	no	—
156	C₄×C₄⋊C₈	—	73	1	no	—
157	C₇₃	#7031	1	1	yes	characteristic, complemented, Sylow
158	C₂×C₄⋊C₈	—	73	2	no	—
159	C₂×C₄⋊C₈	—	73	1	no	—
160	C₂×C₄⋊C₈	—	73	1	no	—
161	C₂×C₄⋊C₈	—	146	1	no	—
162	C₂×C₄⋊C₈	—	146	1	no	—
163	C₂×C₄⋊C₈	—	146	1	no	—
164	C₂×C₄×C₈	—	146	1	no	—
165	C₂×C₄×C₈	—	146	1	no	—
166	C₄³	—	73	1	no	—
167	#336	—	73	2	no	—
168	#336	—	146	1	no	—
169	C₂²×C₈	—	146	1	no	—
170	C₂²×C₈	—	146	2	no	—
171	C₂²×C₈	—	146	1	no	—
172	C₂²×C₈	—	146	2	no	—
173	C₂×C₄²	—	73	1	no	—
174	C₂×C₄²	—	73	1	no	—
175	C₂×C₄²	—	73	1	no	—
176	C₂×C₄²	—	73	1	no	—
177	C₂×C₄²	—	73	1	no	—
178	C₂×C₄²	—	73	1	no	—
179	C₂×C₄²	—	73	1	no	—
180	C₄⋊C₈	—	292	1	no	—
181	C₄⋊C₈	—	292	1	no	—
182	C₄⋊C₈	—	292	1	no	—
183	C₄⋊C₈	—	292	1	no	—
184	C₄⋊C₈	—	292	2	no	—
185	C₄⋊C₈	—	73	2	no	—
186	C₄⋊C₈	—	73	1	no	—
187	C₄⋊C₈	—	146	1	no	—
188	C₄⋊C₈	—	146	1	no	—
189	C₄⋊C₈	—	146	2	no	—
190	C₄⋊C₈	—	146	2	no	—
191	C₄⋊C₈	#10596842	1	1	yes	characteristic, complemented
192	C₄×C₈	—	292	1	no	—
193	C₄×C₈	—	292	1	no	—
194	C₄×C₈	—	292	1	no	—
195	C₄×C₈	—	292	1	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	—	73	1	no	—
201	C₂²×C₄	—	73	1	no	—
202	C₂²×C₄	—	73	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	292	1	no	—
205	C₂×C₈	—	292	1	no	—
206	C₂×C₈	—	292	1	no	—
207	C₂×C₈	—	292	1	no	—
208	C₂×C₈	—	292	2	no	—
209	C₂×C₈	—	292	2	no	—
210	C₂×C₈	—	292	2	no	—
211	C₂×C₈	—	292	2	no	—
212	C₂×C₈	—	292	2	no	—
213	C₂×C₈	—	146	2	no	—
214	C₂×C₈	—	146	4	no	—
215	C₂×C₈	—	146	1	no	—
216	C₄²	—	292	1	no	—
217	C₄²	—	73	1	no	—
218	C₄²	—	73	1	no	—
219	C₄²	—	73	1	no	—
220	C₄²	—	73	1	no	—
221	C₄²	—	73	1	no	—
222	C₄²	—	73	1	no	—
223	C₄²	—	73	1	no	—
224	C₄²	—	73	1	no	—
225	C₄²	—	73	1	no	—
226	C₄²	—	73	1	no	—
227	C₄²	—	73	1	no	—
228	C₄²	—	146	1	no	—
229	C₄²	—	146	1	no	—
230	C₄²	—	146	1	no	—
231	C₄²	—	146	1	no	—
232	C₄²	—	146	1	no	—
233	C₄²	—	146	1	no	—
234	C₄²	C₂×#10596842	1	1	yes	characteristic
235	C₂³	—	73	1	no	—
236	C₂×C₄	—	73	1	no	—
237	C₂×C₄	—	73	1	no	—
238	C₂×C₄	—	73	1	no	—
239	C₂×C₄	—	73	1	no	—
240	C₂×C₄	—	73	1	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	1	no	—
248	C₂×C₄	—	73	1	no	—
249	C₂×C₄	—	73	1	no	—
250	C₂×C₄	—	73	1	no	—
251	C₂×C₄	—	73	1	no	—
252	C₂×C₄	—	73	1	no	—
253	C₂×C₄	—	73	1	no	—
254	C₂×C₄	—	73	1	no	—
255	C₂×C₄	—	73	1	no	—
256	C₂×C₄	—	73	1	no	—
257	C₂×C₄	—	146	1	no	—
258	C₂×C₄	—	146	1	no	—
259	C₂×C₄	—	146	1	no	—
260	C₂×C₄	—	146	1	no	—
261	C₂×C₄	—	146	1	no	—
262	C₂×C₄	—	146	1	no	—
263	C₂×C₄	—	146	1	no	—
264	C₂×C₄	—	146	1	no	—
265	C₂×C₄	—	146	1	no	—
266	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
267	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
268	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
269	C₈	—	4	1	no	—
270	C₈	—	292	2	no	—
271	C₈	—	292	4	no	—
272	C₈	—	292	1	no	—
273	C₈	—	584	2	no	—
274	C₂²	—	73	1	no	—
275	C₂²	—	73	1	no	—
276	C₂²	—	73	1	no	—
277	C₂²	—	73	1	no	—
278	C₂²	—	73	1	no	—
279	C₂²	—	73	1	no	—
280	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	292	1	no	—
283	C₄	—	73	1	no	—
284	C₄	—	73	1	no	—
285	C₄	—	73	1	no	—
286	C₄	—	73	1	no	—
287	C₄	—	73	1	no	—
288	C₄	—	73	1	no	—
289	C₄	—	73	1	no	—
290	C₄	—	73	1	no	—
291	C₄	—	146	1	no	—
292	C₄	—	146	1	no	—
293	C₄	—	146	1	no	—
294	C₄	—	146	1	no	—
295	C₄	—	146	1	no	—
296	C₄	D₄⋊#10596842	1	1	yes	characteristic
297	C₄	Q₈⋊#10596842	1	1	yes	characteristic
298	C₄	#17989676	1	1	yes	characteristic
299	C₄	#17989677	1	1	yes	characteristic
300	C₂	—	73	1	no	—
301	C₂	—	73	1	no	—
302	C₂	—	73	1	no	—
303	C₂	—	73	1	no	—
304	C₂	#22626937	1	1	yes	characteristic
305	C₂	#22626939	1	1	yes	characteristic
306	C₂	#22626943	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×#10596842)⋊C₄
commutator	C₂×C₄×C₇₃	C₄×C₈
Frattini	C₄²	C₂×#10596842
Fitting	C₄⋊C₈×C₇₃	C₈
socle	C₂²×C₇₃	#336
prime radical	C₂²×D₇₃	#95
cosocle	C₄²×#64408	C₂²
hypercenter	C₄⋊C₈	#10596842
hypocenter	C₇₃	#7031
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
33. #14556658	C₈	semidirect
39. C₄⋊#10596831	C₈	semidirect
41. C₄⋊C₈×C₇₃	C₈	semidirect
157. C₇₃	#7031	semidirect
191. C₄⋊C₈	#10596842	semidirect

Maximal subgroups

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

subgroup	index
#22627075	2
C₄×C₄⋊#10596842	2
C₄⋊C₈×#64408	2
#7031	73