Subgroups — #34034973

Order 18688.

452 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627158	C₂	1	2	yes	maximal, complemented
2	#22627158	C₂	1	2	yes	maximal, complemented
3	#22627156	C₂	1	2	yes	maximal, complemented
4	#22627155	C₂	1	2	yes	maximal, complemented
5	#22627137	C₂	1	2	yes	maximal
6	#22627137	C₂	1	2	yes	maximal
7	#22627135	C₂	1	1	yes	maximal, characteristic
8	#22627131	C₂	1	1	yes	maximal, characteristic
9	#22626721	C₂	1	1	yes	maximal, characteristic
10	#17989690	C₂²	1	2	yes	—
11	#17989690	C₂²	1	2	yes	—
12	#17989690	C₂²	1	2	yes	—
13	C₄⋊#11924040	C₂²	1	2	yes	—
14	C₄⋊#11924040	C₂²	1	2	yes	—
15	C₄⋊#11924040	C₂²	1	2	yes	—
16	#17989684	C₂²	1	2	yes	—
17	#17989684	C₂²	1	2	yes	—
18	#17989681	C₂²	1	1	yes	characteristic
19	#17989681	C₂²	1	1	yes	characteristic
20	#17989681	C₂²	1	2	yes	—
21	#17989681	C₂²	1	2	yes	—
22	C₄×#11924040	C₂²	1	1	yes	characteristic
23	C₄×#11924040	C₂²	1	1	yes	characteristic
24	C₄×#11924040	C₂²	1	2	yes	—
25	C₄×#11924040	C₂²	1	2	yes	—
26	#17989639	C₄	1	2	yes	—
27	#17989639	C₄	1	2	yes	—
28	#17989636	C₂²	1	2	yes	—
29	#17989636	C₂²	1	2	yes	—
30	#17989594	C₄	1	1	yes	characteristic
31	#17989594	C₄	1	1	yes	characteristic
32	#17989592	C₄	1	1	yes	characteristic
33	#17989552	C₂²	1	1	yes	characteristic
34	#17989552	C₂²	1	1	yes	characteristic
35	C₄×#11924036	C₂²	1	1	yes	characteristic
36	#17989143	C₄	1	1	yes	characteristic
37	C₂×#11924040	C₂³	1	2	yes	—
38	C₂×#11924040	C₂³	1	2	yes	—
39	C₂×#11924040	C₂³	1	2	yes	—
40	C₂×#11924040	C₂³	1	2	yes	—
41	C₂×#11924038	C₂×C₄	1	1	yes	characteristic
42	C₂×#11924038	C₂×C₄	1	1	yes	characteristic
43	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
44	C₂×#11924036	C₂³	1	1	yes	characteristic
45	#14556658	C₂×C₄	1	2	yes	—
46	#14556658	C₂×C₄	1	2	yes	—
47	#14556658	C₂×C₄	1	2	yes	—
48	#14556658	C₂×C₄	1	2	yes	—
49	C₄⋊#10596833	C₂×C₄	1	2	yes	—
50	C₄⋊#10596833	C₂×C₄	1	2	yes	—
51	#14556646	C₂×C₄	1	2	yes	—
52	#14556646	C₂×C₄	1	2	yes	—
53	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
54	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
55	C₄×#10596833	C₂³	1	1	yes	characteristic
56	C₄×#10596833	C₂³	1	2	yes	—
57	C₄×#10596833	C₂³	1	2	yes	—
58	C₄×#10596833	C₂³	1	1	yes	characteristic
59	#14556590	C₈	1	1	yes	characteristic
60	#14556590	C₈	1	1	yes	characteristic
61	#14556583	C₈	1	1	yes	characteristic
62	#14556569	C₈	1	2	yes	—
63	#14556569	C₈	1	2	yes	—
64	#14556556	C₂×C₄	1	2	yes	—
65	#14556556	C₂×C₄	1	2	yes	—
66	#14556549	C₂×C₄	1	1	yes	characteristic
67	#14556549	C₂×C₄	1	1	yes	characteristic
68	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
69	#125×C₇₃	C₈	1	1	yes	characteristic
70	#11924040	—	2	2	no	—
71	#11924040	—	2	2	no	—
72	#11924040	—	2	2	no	—
73	#11924040	—	2	2	no	—
74	#11924038	—	4	1	no	—
75	#11924038	—	4	1	no	—
76	#11924036	—	2	2	no	—
77	#11924036	—	4	1	no	—
78	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
79	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
80	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
81	C₂×#10596833	C₂²×C₄	1	2	yes	—
82	C₂×#10596833	C₂²×C₄	1	2	yes	—
83	C₂×#10596833	C₂⁴	1	1	yes	characteristic
84	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
85	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
86	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
89	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
90	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
91	#11924007	C₂×C₈	1	2	yes	—
92	#11924007	C₂×C₈	1	2	yes	—
93	#11924007	C₂×C₈	1	2	yes	—
94	#11924007	C₂×C₈	1	2	yes	—
95	#11924005	C₂×C₈	1	2	yes	—
96	#11924005	C₂×C₈	1	2	yes	—
97	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
98	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
101	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
102	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
103	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
104	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
105	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
106	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
107	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
113	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
115	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
116	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
118	C₄×D₇₃	—	2	1	no	—
119	C₄×D₇₃	—	2	1	no	—
120	C₄×D₇₃	—	2	2	no	—
121	C₄×D₇₃	—	4	1	no	—
122	Dic₁₄₆	—	2	4	no	—
123	Dic₁₄₆	—	4	2	no	—
124	#10596833	—	2	1	no	—
125	#10596833	—	2	1	no	—
126	#10596833	—	2	1	no	—
127	#10596833	—	2	2	no	—
128	#10596833	—	2	2	no	—
129	#10596833	2₋¹⁺⁴	1	2	yes	—
130	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
131	C₂×D₇₃	—	2	1	no	—
132	C₂×D₇₃	—	2	1	no	—
133	C₂×D₇₃	—	2	1	no	—
134	C₄×C₇₃	—	2	2	no	—
135	C₄×C₇₃	—	2	1	no	—
136	C₄×C₇₃	—	2	2	no	—
137	C₄×C₇₃	—	2	1	no	—
138	C₄×C₇₃	—	2	1	no	—
139	Dic₇₃	—	2	2	no	—
140	Dic₇₃	—	2	1	no	—
141	Dic₇₃	—	2	2	no	—
142	Dic₇₃	#504	1	2	yes	—
143	Dic₇₃	#519	1	1	yes	characteristic
144	Dic₇₃	#568	1	1	yes	characteristic
145	#33499	—	73	1	no	maximal, Sylow
146	C₂×C₇₃	#2982	1	1	yes	characteristic
147	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
148	C₂×C₇₃	#3407	1	1	yes	characteristic
149	D₇₃	—	4	1	no	—
150	#2960	—	73	1	no	—
151	#2183	—	73	2	no	—
152	#2183	—	73	2	no	—
153	#2183	—	73	2	no	—
154	Q₈×C₁₆	—	73	2	no	—
155	#2164	—	73	2	no	—
156	#2164	—	73	2	no	—
157	#2163	—	73	1	no	—
158	#2162	—	73	1	no	—
159	C₇₃	#33499	1	1	yes	characteristic, complemented, Sylow
160	#533	—	73	1	no	—
161	#446	—	73	2	no	—
162	#446	—	73	2	no	—
163	C₈×Q₈	—	73	2	no	—
164	C₈×Q₈	—	73	2	no	—
165	#424	—	73	1	no	—
166	#424	—	73	1	no	—
167	#423	—	73	1	no	—
168	#405	—	73	1	no	—
169	#405	—	73	1	no	—
170	C₂×C₄×C₈	—	73	1	no	—
171	C₄⋊C₁₆	—	73	2	no	—
172	C₄⋊C₁₆	—	73	2	no	—
173	C₄⋊C₁₆	—	73	2	no	—
174	C₄⋊C₁₆	—	73	2	no	—
175	C₄⋊C₁₆	—	73	2	no	—
176	C₄⋊C₁₆	—	73	2	no	—
177	C₂²⋊C₁₆	—	73	2	no	—
178	C₂²⋊C₁₆	—	73	2	no	—
179	#346	—	73	1	no	—
180	#346	—	73	1	no	—
181	#346	—	73	2	no	—
182	#346	—	73	2	no	—
183	C₄×C₁₆	—	73	1	no	—
184	C₄×C₁₆	—	73	1	no	—
185	C₄×C₁₆	—	73	2	no	—
186	C₄×C₁₆	—	73	2	no	—
187	C₂×M₁₆	—	73	1	no	—
188	C₂×M₁₆	—	73	1	no	—
189	C₂²×C₈	—	73	1	no	—
190	C₂²×C₈	—	73	1	no	—
191	C₄⋊Q₈	—	73	1	no	—
192	C₄⋊Q₈	—	73	1	no	—
193	#125	—	73	1	no	—
194	#125	#10596842	1	1	yes	characteristic
195	C₂²⋊Q₈	—	73	2	no	—
196	C₂²⋊Q₈	—	73	2	no	—
197	C₄×Q₈	—	73	2	no	—
198	C₄×Q₈	—	73	2	no	—
199	#117	—	73	1	no	—
200	#117	—	73	1	no	—
201	C₂×C₄²	—	73	1	no	—
202	C₂×C₁₆	—	73	2	no	—
203	C₂×C₁₆	—	73	2	no	—
204	C₂×C₁₆	—	73	2	no	—
205	C₂×C₁₆	—	73	2	no	—
206	C₄⋊C₈	—	73	2	no	—
207	C₄⋊C₈	—	73	2	no	—
208	C₄⋊C₈	—	73	2	no	—
209	C₄⋊C₈	—	73	2	no	—
210	C₄⋊C₈	—	73	2	no	—
211	C₄⋊C₈	—	73	2	no	—
212	#97	—	73	2	no	—
213	#97	—	73	2	no	—
214	C₄×C₈	—	73	1	no	—
215	C₄×C₈	—	73	2	no	—
216	C₄×C₈	—	73	2	no	—
217	C₄×C₈	—	73	1	no	—
218	C₄×C₈	—	73	1	no	—
219	C₄×C₈	—	73	1	no	—
220	C₂×Q₈	—	73	2	no	—
221	C₂×Q₈	—	73	2	no	—
222	C₂²×C₄	—	73	1	no	—
223	C₂²×C₄	—	73	1	no	—
224	C₂²×C₄	—	73	1	no	—
225	M₁₆	—	292	1	no	—
226	M₁₆	—	292	1	no	—
227	C₂×C₈	—	292	1	no	—
228	C₂×C₈	—	73	1	no	—
229	C₂×C₈	—	73	1	no	—
230	C₂×C₈	—	73	1	no	—
231	C₂×C₈	—	73	1	no	—
232	C₂×C₈	—	73	2	no	—
233	C₂×C₈	—	73	2	no	—
234	C₂×C₈	—	146	2	no	—
235	C₄⋊C₄	—	73	2	no	—
236	C₄⋊C₄	—	73	2	no	—
237	C₄⋊C₄	—	73	2	no	—
238	C₄⋊C₄	—	73	2	no	—
239	C₄⋊C₄	—	73	1	no	—
240	C₄⋊C₄	—	73	1	no	—
241	C₄⋊C₄	C₂×#10596842	1	1	yes	characteristic
242	C₄⋊C₄	C₂×#10596842	1	1	yes	characteristic
243	C₄⋊C₄	C₂×#10596842	1	2	yes	—
244	C₄⋊C₄	C₂×#10596842	1	2	yes	—
245	C₂²⋊C₄	—	73	2	no	—
246	C₂²⋊C₄	—	73	2	no	—
247	C₄²	—	73	1	no	—
248	C₄²	—	73	1	no	—
249	C₄²	—	73	1	no	—
250	C₄²	—	73	2	no	—
251	C₄²	—	73	2	no	—
252	C₄²	C₂×#10596842	1	1	yes	characteristic
253	C₁₆	—	146	2	no	—
254	C₁₆	—	146	2	no	—
255	C₁₆	—	146	2	no	—
256	C₁₆	—	146	2	no	—
257	C₂³	—	73	1	no	—
258	Q₈	—	292	2	no	—
259	Q₈	—	146	4	no	—
260	C₂×C₄	—	292	1	no	—
261	C₂×C₄	—	73	1	no	—
262	C₂×C₄	—	73	1	no	—
263	C₂×C₄	—	73	1	no	—
264	C₂×C₄	—	73	2	no	—
265	C₂×C₄	—	73	2	no	—
266	C₂×C₄	—	146	1	no	—
267	C₂×C₄	—	146	1	no	—
268	C₂×C₄	—	146	2	no	—
269	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
270	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
271	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
272	C₂×C₄	C₂²×#10596842	1	2	yes	—
273	C₂×C₄	C₂²×#10596842	1	2	yes	—
274	C₈	—	73	2	no	—
275	C₈	—	146	1	no	—
276	C₈	—	146	1	no	—
277	C₈	—	146	1	no	—
278	C₈	—	146	2	no	—
279	C₈	—	146	2	no	—
280	C₂²	—	146	1	no	—
281	C₂²	—	146	1	no	—
282	C₂²	—	146	1	no	—
283	C₂²	C₂³×#10596842	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	2	no	—
289	C₄	—	73	1	no	—
290	C₄	—	73	1	no	—
291	C₄	—	73	2	no	—
292	C₄	—	146	2	no	—
293	C₄	—	146	1	no	—
294	C₄	—	146	2	no	—
295	C₂	—	292	1	no	—
296	C₂	#22627174	1	1	yes	characteristic
297	C₂	#22627182	1	1	yes	characteristic
298	C₂	#22627185	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10596842
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	#125×C₇₃	C₈
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#10596833	C₂⁴
hypercenter	#125	#10596842
hypocenter	C₇₃	#33499
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #22627158	C₂	semidirect
2. #22627158	C₂	semidirect
3. #22627156	C₂	semidirect
4. #22627155	C₂	semidirect
159. C₇₃	#33499	semidirect

Maximal subgroups

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

subgroup	index
#22627158	2
#22627156	2
#22627155	2
#22627137	2
#22627135	2
#22627131	2
#22626721	2
#33499	73