Subgroups — #34034967

Order 18688.

494 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	#22627158	C₂	1	4	yes	maximal, complemented
2	#22627157	C₂	1	2	yes	maximal, complemented
3	#22627155	C₂	1	2	yes	maximal, complemented
4	#22627135	C₂	1	2	yes	maximal
5	#22627134	C₂	1	1	yes	maximal, characteristic
6	#22627131	C₂	1	1	yes	maximal, characteristic
7	#22627130	C₂	1	2	yes	maximal
8	Q₈×#11924036	C₂	1	1	yes	maximal, characteristic
9	#17989690	C₂²	1	4	yes	—
10	#17989690	C₂²	1	4	yes	—
11	C₄⋊#11924040	C₂²	1	2	yes	—
12	C₄⋊#11924040	C₂²	1	2	yes	—
13	#17989684	C₂²	1	1	yes	characteristic
14	#17989684	C₂²	1	2	yes	—
15	#17989684	C₂²	1	1	yes	characteristic
16	#17989681	C₂²	1	2	yes	—
17	#17989681	C₂²	1	4	yes	—
18	C₄×#11924040	C₂²	1	4	yes	—
19	C₄×#11924040	C₂²	1	2	yes	—
20	#17989636	C₄	1	2	yes	—
21	#17989636	C₄	1	1	yes	characteristic
22	#17989636	C₂²	1	1	yes	characteristic
23	#17989636	C₂²	1	2	yes	—
24	Q₈×#10596833	C₄	1	1	yes	characteristic
25	Q₈×#10596833	C₂²	1	1	yes	characteristic
26	#17989588	C₄	1	1	yes	characteristic
27	#17989588	C₄	1	2	yes	—
28	C₄×#11924036	C₂²	1	2	yes	—
29	C₄×#11924036	C₂²	1	1	yes	characteristic
30	C₄×Q₈×D₇₃	C₄	1	1	yes	characteristic
31	C₂×#11924040	C₂³	1	2	yes	—
32	C₂×#11924040	C₂³	1	4	yes	—
33	C₂×#11924040	C₂³	1	2	yes	—
34	C₂×#11924036	C₂×C₄	1	2	yes	—
35	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
36	C₂×#11924036	C₂³	1	1	yes	characteristic
37	#14556658	C₂×C₄	1	2	yes	—
38	#14556658	C₂×C₄	1	2	yes	—
39	#14556658	C₂×C₄	1	2	yes	—
40	C₄⋊#10596833	C₂×C₄	1	1	yes	characteristic
41	C₄⋊#10596833	C₂×C₄	1	2	yes	—
42	C₄⋊#10596833	C₂×C₄	1	2	yes	—
43	C₄⋊#10596833	C₂×C₄	1	1	yes	characteristic
44	C₄×#10596833	C₂×C₄	1	2	yes	—
45	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
46	C₄×#10596833	C₂×C₄	1	2	yes	—
47	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
48	C₄×#10596833	C₂³	1	1	yes	characteristic
49	C₄×#10596833	C₂³	1	2	yes	—
50	C₄×#10596833	C₂³	1	2	yes	—
51	C₄×#10596833	C₂³	1	1	yes	characteristic
52	C₂×Q₈×D₇₃	C₈	1	1	yes	characteristic
53	Q₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
54	C₄×Dic₁₄₆	C₈	1	2	yes	—
55	C₄×Dic₁₄₆	C₈	1	1	yes	characteristic
56	#14556556	C₂×C₄	1	2	yes	—
57	#14556556	C₂×C₄	1	1	yes	characteristic
58	C₄⋊C₄×D₇₃	C₈	1	2	yes	—
59	C₄⋊C₄×D₇₃	C₈	1	1	yes	characteristic
60	C₄²×D₇₃	C₂×C₄	1	2	yes	—
61	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
62	C₄×Q₈×C₇₃	C₈	1	1	yes	characteristic
63	#11924040	—	2	2	no	—
64	#11924040	—	2	4	no	—
65	#11924040	—	2	2	no	—
66	#11924036	—	2	2	no	—
67	#11924036	—	4	2	no	—
68	#11924036	—	4	1	no	—
69	C₂×#10596833	C₂²×C₄	1	2	yes	—
70	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
71	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
72	C₂×#10596833	C₂²×C₄	1	2	yes	—
73	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
74	C₂×#10596833	C₂⁴	1	1	yes	characteristic
75	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
76	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
77	Q₈×D₇₃	—	2	4	no	—
78	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
79	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
82	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
84	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
85	#11924007	C₂×C₈	1	2	yes	—
86	#11924007	C₂×C₈	1	2	yes	—
87	#11924007	C₂×C₈	1	2	yes	—
88	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
89	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
90	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
92	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
93	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
95	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
96	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
97	C₄²×C₇₃	C₂×C₈	1	2	yes	—
98	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
99	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
100	Q₈×C₇₃	—	2	2	no	—
101	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
103	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
108	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
110	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
111	C₄×D₇₃	—	2	4	no	—
112	C₄×D₇₃	—	2	2	no	—
113	C₄×D₇₃	—	2	2	no	—
114	C₄×D₇₃	—	2	1	no	—
115	C₄×D₇₃	—	2	1	no	—
116	C₄×D₇₃	—	2	4	no	—
117	Dic₁₄₆	—	2	4	no	—
118	Dic₁₄₆	M₃₂	1	4	yes	—
119	#10596833	—	2	1	no	—
120	#10596833	—	2	2	no	—
121	#10596833	—	2	1	no	—
122	#10596833	—	2	1	no	—
123	#10596833	—	2	2	no	—
124	#10596833	2₋¹⁺⁴	1	2	yes	—
125	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
126	C₂×D₇₃	—	2	1	no	—
127	C₂×D₇₃	—	2	1	no	—
128	C₂×D₇₃	#504	1	2	yes	—
129	C₄×C₇₃	—	2	2	no	—
130	C₄×C₇₃	—	2	2	no	—
131	C₄×C₇₃	—	2	1	no	—
132	C₄×C₇₃	—	2	1	no	—
133	C₄×C₇₃	C₂×M₃₂	1	2	yes	—
134	Dic₇₃	—	2	1	no	—
135	Dic₇₃	—	2	1	no	—
136	Dic₇₃	—	2	2	no	—
137	Dic₇₃	C₂×M₃₂	1	4	yes	—
138	Dic₇₃	#519	1	1	yes	characteristic
139	Dic₇₃	#568	1	1	yes	characteristic
140	#33498	—	73	1	no	maximal, Sylow
141	C₂×C₇₃	#2982	1	1	yes	characteristic
142	C₂×C₇₃	C₂²×M₃₂	1	1	yes	characteristic
143	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
144	D₇₃	—	2	2	no	—
145	C₂×C₈×Q₈	—	73	1	no	—
146	#2183	—	73	2	no	—
147	#2183	—	73	4	no	—
148	Q₈×C₁₆	—	73	2	no	—
149	#2163	—	73	1	no	—
150	#2163	—	73	2	no	—
151	#2162	—	73	2	no	—
152	#2162	—	73	1	no	—
153	C₇₃	#33498	1	1	yes	characteristic, complemented, Sylow
154	C₂×C₄×Q₈	—	73	1	no	—
155	C₈×Q₈	—	73	1	no	—
156	C₈×Q₈	—	73	1	no	—
157	C₈×Q₈	—	73	2	no	—
158	C₈×Q₈	—	73	1	no	—
159	C₈×Q₈	—	73	1	no	—
160	C₈×Q₈	—	73	2	no	—
161	C₂×C₄⋊C₈	—	73	1	no	—
162	C₂×C₄⋊C₈	—	73	2	no	—
163	C₂×C₄×C₈	—	73	2	no	—
164	C₂×C₄×C₈	—	73	1	no	—
165	C₄⋊C₁₆	—	73	2	no	—
166	C₄⋊C₁₆	—	73	2	no	—
167	C₄⋊C₁₆	—	73	4	no	—
168	C₄⋊C₁₆	—	73	4	no	—
169	C₂²⋊C₁₆	—	73	1	no	—
170	C₂²⋊C₁₆	—	73	2	no	—
171	C₂²⋊C₁₆	—	73	1	no	—
172	#346	—	73	2	no	—
173	#346	—	73	4	no	—
174	C₄×C₁₆	—	73	4	no	—
175	C₄×C₁₆	—	73	2	no	—
176	C₂²×Q₈	—	73	1	no	—
177	C₂²×C₈	—	73	2	no	—
178	C₂²×C₈	—	73	1	no	—
179	C₂²×C₈	—	73	1	no	—
180	C₄×Q₈	—	73	1	no	—
181	C₄×Q₈	—	73	1	no	—
182	C₄×Q₈	—	73	2	no	—
183	C₄×Q₈	—	73	1	no	—
184	C₄×Q₈	—	73	2	no	—
185	C₄×Q₈	#10596842	1	1	yes	characteristic
186	C₂×C₄⋊C₄	—	73	2	no	—
187	C₂×C₄⋊C₄	—	73	1	no	—
188	C₂×C₄²	—	73	2	no	—
189	C₂×C₄²	—	73	1	no	—
190	C₂×C₁₆	—	73	2	no	—
191	C₂×C₁₆	—	73	2	no	—
192	C₂×C₁₆	—	73	4	no	—
193	C₄⋊C₈	—	73	1	no	—
194	C₄⋊C₈	—	73	2	no	—
195	C₄⋊C₈	—	73	2	no	—
196	C₄⋊C₈	—	73	1	no	—
197	C₄⋊C₈	—	73	2	no	—
198	C₄⋊C₈	—	73	2	no	—
199	C₄⋊C₈	—	73	2	no	—
200	C₄×C₈	—	73	1	no	—
201	C₄×C₈	—	73	2	no	—
202	C₄×C₈	—	73	1	no	—
203	C₄×C₈	—	73	2	no	—
204	C₄×C₈	—	73	2	no	—
205	C₄×C₈	—	73	1	no	—
206	C₄×C₈	—	73	2	no	—
207	C₄×C₈	—	73	1	no	—
208	C₂×Q₈	—	73	1	no	—
209	C₂×Q₈	—	73	2	no	—
210	C₂×Q₈	—	146	4	no	—
211	C₂×Q₈	C₂×#10596842	1	1	yes	characteristic
212	C₂²×C₄	—	73	2	no	—
213	C₂²×C₄	—	73	1	no	—
214	C₂²×C₄	—	73	2	no	—
215	C₂²×C₄	—	73	1	no	—
216	C₂²×C₄	—	73	1	no	—
217	C₂×C₈	—	292	2	no	—
218	C₂×C₈	—	292	1	no	—
219	C₂×C₈	—	73	1	no	—
220	C₂×C₈	—	73	1	no	—
221	C₂×C₈	—	73	2	no	—
222	C₂×C₈	—	73	1	no	—
223	C₂×C₈	—	73	1	no	—
224	C₂×C₈	—	73	2	no	—
225	C₂×C₈	—	146	2	no	—
226	C₄⋊C₄	—	73	2	no	—
227	C₄⋊C₄	—	73	2	no	—
228	C₄⋊C₄	—	73	2	no	—
229	C₄⋊C₄	—	73	2	no	—
230	C₄⋊C₄	—	73	1	no	—
231	C₄⋊C₄	C₂×#10596842	1	2	yes	—
232	C₄⋊C₄	C₂×#10596842	1	1	yes	characteristic
233	C₄²	—	73	2	no	—
234	C₄²	—	73	1	no	—
235	C₄²	—	73	2	no	—
236	C₄²	—	73	2	no	—
237	C₄²	—	73	1	no	—
238	C₄²	—	73	1	no	—
239	C₄²	C₂×#10596842	1	2	yes	—
240	C₄²	C₂×#10596842	1	1	yes	characteristic
241	C₁₆	—	146	2	no	—
242	C₁₆	—	146	2	no	—
243	C₁₆	—	146	4	no	—
244	C₂³	—	73	1	no	—
245	Q₈	—	2	2	no	—
246	Q₈	—	73	4	no	—
247	Q₈	—	146	4	no	—
248	C₂×C₄	—	73	1	no	—
249	C₂×C₄	—	73	2	no	—
250	C₂×C₄	—	73	2	no	—
251	C₂×C₄	—	73	1	no	—
252	C₂×C₄	—	73	1	no	—
253	C₂×C₄	—	146	1	no	—
254	C₂×C₄	—	146	1	no	—
255	C₂×C₄	—	146	4	no	—
256	C₂×C₄	—	146	4	no	—
257	C₂×C₄	—	146	2	no	—
258	C₂×C₄	—	146	2	no	—
259	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
260	C₂×C₄	C₂²×#10596842	1	2	yes	—
261	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
262	C₂×C₄	C₂²×#10596842	1	2	yes	—
263	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
264	C₈	—	73	2	no	—
265	C₈	—	146	2	no	—
266	C₈	—	146	1	no	—
267	C₈	—	146	1	no	—
268	C₈	—	146	2	no	—
269	C₈	—	146	1	no	—
270	C₂²	—	73	2	no	—
271	C₂²	—	146	1	no	—
272	C₂²	—	146	1	no	—
273	C₂²	C₂³×#10596842	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	1	no	—
278	C₄	—	73	1	no	—
279	C₄	—	73	1	no	—
280	C₄	—	73	4	no	—
281	C₄	—	146	1	no	—
282	C₄	—	146	2	no	—
283	C₄	—	146	1	no	—
284	C₄	#17989706	1	2	yes	—
285	C₂	—	146	2	no	—
286	C₂	#22627174	1	1	yes	characteristic
287	C₂	C₂×#17989706	1	1	yes	characteristic
288	C₂	#22627182	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10596842
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₄×Q₈×C₇₃	C₈
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#10596833	C₂⁴
hypercenter	C₄×Q₈	#10596842
hypocenter	C₇₃	#33498
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #22627158	C₂	semidirect
2. #22627157	C₂	semidirect
3. #22627155	C₂	semidirect
153. C₇₃	#33498	semidirect

Maximal subgroups

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

subgroup	index
#22627158	2
#22627157	2
#22627155	2
#22627135	2
#22627134	2
#22627131	2
#22627130	2
Q₈×#11924036	2
#33498	73