Subgroups — #34032727

Order 18688.

414 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	C₄×C₄⋊#10596842	C₂	1	1	yes	maximal, characteristic
2	C₄²×#10596842	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×#64408	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#10596842	C₄	1	2	yes	—
5	C₂×C₄×#10596842	—	2	1	no	—
6	C₂×C₄×#10596842	—	2	1	no	—
7	C₂×C₄×#10596842	—	2	1	no	—
8	C₂×C₄×#10596842	C₄	1	2	yes	—
9	(C₂×#10596842)⋊C₄	—	2	1	no	—
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	C₈	1	4	yes	complemented
19	C₄×#10596842	—	2	4	no	—
20	C₄×#10596842	—	2	2	no	—
21	C₄×#10596842	—	2	2	no	—
22	C₄×#10596842	—	2	2	no	—
23	C₄×#10596842	—	4	1	no	—
24	C₄×#10596842	—	4	1	no	—
25	C₄×#10596842	C₈	1	4	yes	complemented
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	—	2	1	no	—
34	C₄⋊#10596833	C₈	1	2	yes	complemented
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	—	2	4	no	—
44	C₂×#10596842	—	2	4	no	—
45	C₂×#10596842	—	2	4	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	2	no	—
56	C₄×#64408	—	2	2	no	—
57	C₄×#64408	—	2	1	no	—
58	C₄×#64408	—	2	1	no	—
59	C₄×#64408	C₂×C₈	1	2	yes	—
60	C₄×#64408	C₂×C₈	1	2	yes	—
61	C₄×#64408	M₁₆	1	2	yes	—
62	C₄×#64408	M₁₆	1	2	yes	—
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	Q₁₆	1	1	yes	characteristic
67	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
68	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
70	C₂×#10596831	—	2	1	no	—
71	C₈×D₇₃	—	4	1	no	—
72	C₈×D₇₃	—	4	1	no	—
73	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
74	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
75	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
76	C₂×C₈×C₇₃	—	2	1	no	—
77	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
78	C₂²×D₇₃	#95	1	1	yes	characteristic
79	C₂×#64408	—	2	1	no	—
80	C₂×#64408	—	2	1	no	—
81	C₂×#64408	—	2	1	no	—
82	C₂×#64408	—	2	1	no	—
83	C₂×#64408	C₂²⋊C₈	1	2	yes	—
84	C₂×#64408	C₂²⋊C₈	1	2	yes	—
85	C₂×#64408	#102	1	1	yes	characteristic
86	C₂×#64408	#103	1	1	yes	characteristic
87	C₂×#64408	#104	1	2	yes	—
88	C₂×#64408	C₄⋊C₈	1	2	yes	—
89	C₂×#64408	C₄⋊C₈	1	2	yes	—
90	C₂×#64408	#106	1	1	yes	characteristic
91	C₂×#64408	#107	1	1	yes	characteristic
92	C₂×#64408	#108	1	2	yes	—
93	#10596842	—	2	8	no	—
94	#10596842	—	4	4	no	—
95	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
96	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
97	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
98	C₂×Dic₇₃	#97	1	1	yes	characteristic
99	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
101	C₄×D₇₃	—	2	1	no	—
102	C₄×D₇₃	—	2	1	no	—
103	C₄×D₇₃	C₄×C₈	1	1	yes	characteristic
104	C₄×D₇₃	C₄×C₈	1	1	yes	characteristic
105	C₄×D₇₃	#97	1	1	yes	characteristic
106	C₄×D₇₃	#97	1	1	yes	characteristic
107	C₄×D₇₃	#102	1	1	yes	characteristic
108	C₄×D₇₃	#103	1	1	yes	characteristic
109	C₄×D₇₃	#104	1	1	yes	characteristic
110	C₄×D₇₃	#104	1	1	yes	characteristic
111	#10596833	—	4	2	no	—
112	C₈×C₇₃	—	4	1	no	—
113	#10596831	—	4	1	no	—
114	C₂²×C₇₃	#336	1	1	yes	characteristic
115	C₂×D₇₃	#336	1	1	yes	characteristic
116	C₂×D₇₃	#336	1	1	yes	characteristic
117	C₂×D₇₃	#339	1	1	yes	characteristic
118	C₂×D₇₃	#339	1	1	yes	characteristic
119	C₂×D₇₃	#340	1	1	yes	characteristic
120	C₂×D₇₃	#341	1	1	yes	characteristic
121	#64408	—	2	2	no	—
122	#64408	—	2	2	no	—
123	#64408	D₄⋊C₈	1	2	yes	—
124	#64408	Q₈⋊C₈	1	2	yes	—
125	#64408	#334	1	2	yes	—
126	#64408	#335	1	2	yes	—
127	C₄×C₇₃	—	2	1	no	—
128	C₄×C₇₃	C₈²	1	1	yes	characteristic
129	C₄×C₇₃	#322	1	1	yes	characteristic
130	C₄×C₇₃	D₄⋊C₈	1	1	yes	characteristic
131	C₄×C₇₃	Q₈⋊C₈	1	1	yes	characteristic
132	Dic₇₃	—	2	1	no	—
133	Dic₇₃	#322	1	1	yes	characteristic
134	Dic₇₃	#322	1	1	yes	characteristic
135	Dic₇₃	D₄⋊C₈	1	1	yes	characteristic
136	Dic₇₃	Q₈⋊C₈	1	1	yes	characteristic
137	#7015	—	73	1	no	maximal, Sylow
138	C₂×C₇₃	#1273	1	1	yes	characteristic
139	C₂×C₇₃	#1278	1	1	yes	characteristic
140	C₂×C₇₃	#1279	1	1	yes	characteristic
141	D₇₃	#1276	1	1	yes	characteristic
142	D₇₃	#1277	1	1	yes	characteristic
143	D₇₃	#1278	1	1	yes	characteristic
144	D₇₃	#1279	1	1	yes	characteristic
145	C₄×C₄⋊C₈	—	73	1	no	—
146	C₄×C₄⋊C₈	—	73	1	no	—
147	C₄²×C₈	—	73	1	no	—
148	C₇₃	#7015	1	1	yes	characteristic, complemented, Sylow
149	C₂×C₄⋊C₈	—	73	2	no	—
150	C₂×C₄⋊C₈	—	73	1	no	—
151	C₂×C₄⋊C₈	—	73	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	—	146	1	no	—
156	C₂×C₄×C₈	—	146	1	no	—
157	C₄³	—	73	1	no	—
158	#336	—	146	1	no	—
159	#336	—	146	1	no	—
160	C₂²×C₈	—	146	1	no	—
161	C₂²×C₈	—	146	1	no	—
162	C₂²×C₈	—	146	2	no	—
163	C₂²×C₈	—	146	2	no	—
164	C₂×C₄²	—	73	1	no	—
165	C₂×C₄²	—	73	1	no	—
166	C₂×C₄²	—	73	1	no	—
167	C₂×C₄²	—	73	1	no	—
168	C₂×C₄²	—	73	1	no	—
169	C₂×C₄²	—	73	1	no	—
170	C₂×C₄²	—	73	1	no	—
171	C₄⋊C₈	—	73	4	no	—
172	C₄⋊C₈	—	73	2	no	—
173	C₄⋊C₈	—	73	1	no	—
174	C₄⋊C₈	—	146	1	no	—
175	C₄⋊C₈	—	146	2	no	—
176	C₄⋊C₈	—	146	1	no	—
177	C₄⋊C₈	#10596842	1	1	yes	characteristic, complemented
178	C₄×C₈	—	292	1	no	—
179	C₄×C₈	—	292	1	no	—
180	C₄×C₈	—	292	1	no	—
181	C₄×C₈	—	292	1	no	—
182	C₄×C₈	—	73	4	no	—
183	C₄×C₈	—	146	2	no	—
184	C₄×C₈	—	146	4	no	—
185	C₄×C₈	—	146	2	no	—
186	C₄×C₈	—	146	2	no	—
187	C₂²×C₄	—	73	1	no	—
188	C₂²×C₄	—	73	1	no	—
189	C₂²×C₄	—	73	1	no	—
190	C₂²×C₄	—	73	1	no	—
191	C₂²×C₄	—	73	1	no	—
192	C₂²×C₄	—	73	1	no	—
193	C₂²×C₄	—	73	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	292	1	no	—
196	C₂×C₈	—	292	1	no	—
197	C₂×C₈	—	292	1	no	—
198	C₂×C₈	—	292	1	no	—
199	C₂×C₈	—	292	2	no	—
200	C₂×C₈	—	292	2	no	—
201	C₂×C₈	—	146	2	no	—
202	C₂×C₈	—	146	4	no	—
203	C₂×C₈	—	146	1	no	—
204	C₂×C₈	—	146	4	no	—
205	C₂×C₈	—	146	4	no	—
206	C₂×C₈	—	146	4	no	—
207	C₄²	—	73	2	no	—
208	C₄²	—	73	2	no	—
209	C₄²	—	73	2	no	—
210	C₄²	—	73	2	no	—
211	C₄²	—	73	1	no	—
212	C₄²	—	73	1	no	—
213	C₄²	—	73	1	no	—
214	C₄²	—	73	1	no	—
215	C₄²	—	73	1	no	—
216	C₄²	—	73	1	no	—
217	C₄²	—	73	1	no	—
218	C₄²	—	146	2	no	—
219	C₄²	—	146	1	no	—
220	C₄²	—	146	1	no	—
221	C₄²	—	146	2	no	—
222	C₄²	C₂×#10596842	1	1	yes	characteristic
223	C₂³	—	73	1	no	—
224	C₂×C₄	—	73	1	no	—
225	C₂×C₄	—	73	2	no	—
226	C₂×C₄	—	73	1	no	—
227	C₂×C₄	—	73	2	no	—
228	C₂×C₄	—	73	1	no	—
229	C₂×C₄	—	73	1	no	—
230	C₂×C₄	—	73	2	no	—
231	C₂×C₄	—	73	2	no	—
232	C₂×C₄	—	73	1	no	—
233	C₂×C₄	—	73	1	no	—
234	C₂×C₄	—	73	1	no	—
235	C₂×C₄	—	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	2	no	—
243	C₂×C₄	—	73	2	no	—
244	C₂×C₄	—	73	1	no	—
245	C₂×C₄	—	146	1	no	—
246	C₂×C₄	—	146	1	no	—
247	C₂×C₄	—	146	1	no	—
248	C₂×C₄	—	146	1	no	—
249	C₂×C₄	—	146	1	no	—
250	C₂×C₄	—	146	1	no	—
251	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
252	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
253	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
254	C₈	—	4	1	no	—
255	C₈	—	292	2	no	—
256	C₈	—	292	4	no	—
257	C₈	—	292	1	no	—
258	C₈	—	146	8	no	—
259	C₂²	—	73	1	no	—
260	C₂²	—	73	1	no	—
261	C₂²	—	73	1	no	—
262	C₂²	—	73	1	no	—
263	C₂²	—	73	1	no	—
264	C₂²	—	73	1	no	—
265	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	73	2	no	—
268	C₄	—	73	2	no	—
269	C₄	—	73	2	no	—
270	C₄	—	73	2	no	—
271	C₄	—	73	1	no	—
272	C₄	—	73	1	no	—
273	C₄	—	73	1	no	—
274	C₄	—	73	1	no	—
275	C₄	—	146	2	no	—
276	C₄	—	146	2	no	—
277	C₄	—	146	1	no	—
278	C₄	C₈×#10596842	1	1	yes	characteristic
279	C₄	#17989671	1	1	yes	characteristic
280	C₄	D₄⋊#10596842	1	1	yes	characteristic
281	C₄	Q₈⋊#10596842	1	1	yes	characteristic
282	C₂	—	73	1	no	—
283	C₂	—	73	1	no	—
284	C₂	—	73	1	no	—
285	C₂	—	73	1	no	—
286	C₂	(C₂×#10596842)⋊C₈	1	1	yes	characteristic
287	C₂	#22626938	1	1	yes	characteristic
288	C₂	#22626939	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#10596842
commutator	C₄×C₇₃	C₈²
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	#7015
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
18. C₄⋊#10596842	C₈	semidirect
25. C₄×#10596842	C₈	semidirect
34. C₄⋊#10596833	C₈	semidirect
39. C₄⋊#10596831	C₈	semidirect
41. C₄⋊C₈×C₇₃	C₈	semidirect
148. C₇₃	#7015	semidirect
177. 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
C₄×C₄⋊#10596842	2
C₄²×#10596842	2
C₄⋊C₈×#64408	2
#7015	73