Subgroups — #34034879

Order 18688.

526 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627156	C₂	1	1	yes	maximal, characteristic, complemented
2	#22627155	C₂	1	1	yes	maximal, characteristic, complemented
3	#22627147	C₂	1	1	yes	maximal, characteristic, complemented
4	#22627143	C₂	1	2	yes	maximal, complemented
5	D₄×#11924040	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×#11924040	C₂	1	2	yes	maximal, complemented
7	#22627136	C₂	1	2	yes	maximal, complemented
8	#22627133	C₂	1	1	yes	maximal, characteristic, complemented
9	#22627092	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×#11924040	C₂	1	1	yes	maximal, characteristic, complemented
11	#22626744	C₂	1	1	yes	maximal, characteristic
12	C₂×#14556687	C₂²	1	2	yes	—
13	C₂²×#11924040	C₂²	1	1	yes	characteristic
14	C₂²×#11924040	C₂²	1	1	yes	characteristic
15	C₂²×#11924040	C₂²	1	2	yes	—
16	#17989690	C₂²	1	2	yes	—
17	C₄⋊#11924040	C₂²	1	1	yes	characteristic
18	C₄⋊#11924040	C₂²	1	1	yes	characteristic
19	C₄⋊#11924040	C₂²	1	2	yes	—
20	C₂²⋊#11924040	C₂²	1	2	yes	—
21	C₂²⋊#11924040	C₂²	1	2	yes	—
22	C₂²⋊#11924040	C₂²	1	2	yes	—
23	#17989681	C₂²	1	2	yes	—
24	#17989681	C₂²	1	2	yes	—
25	C₄×#11924040	C₂²	1	1	yes	characteristic
26	C₄×#11924040	C₂²	1	1	yes	characteristic
27	C₄×#11924040	C₂²	1	1	yes	characteristic
28	C₄×#11924040	C₂²	1	1	yes	characteristic
29	C₄×#11924040	C₂²	1	2	yes	—
30	#17989637	C₄	1	1	yes	characteristic
31	#17989636	C₂²	1	1	yes	characteristic
32	#17989622	C₄	1	1	yes	characteristic
33	#17989618	C₄	1	2	yes	—
34	D₄×#10596833	C₂²	1	2	yes	—
35	D₄×#10596833	C₂²	1	1	yes	characteristic
36	#17989611	C₄	1	2	yes	—
37	#17989608	C₄	1	1	yes	characteristic
38	#17989551	C₂²	1	2	yes	—
39	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
40	#17989118	C₄	1	1	yes	characteristic
41	#14556687	—	2	4	no	—
42	C₂×#11924040	—	2	2	no	—
43	C₂×#11924040	—	2	4	no	—
44	C₂×#11924040	—	2	2	no	—
45	C₂×#11924040	C₂³	1	1	yes	characteristic
46	C₂×#11924040	C₂³	1	1	yes	characteristic
47	C₂×#11924040	C₂³	1	2	yes	—
48	C₂×#11924040	C₂³	1	2	yes	—
49	C₂×#11924040	C₂³	1	1	yes	characteristic
50	C₂×#11924040	C₂³	1	1	yes	characteristic
51	C₂×#11924037	C₂×C₄	1	2	yes	—
52	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
53	C₂²×#10596833	C₂³	1	1	yes	characteristic
54	C₂²×#10596833	C₂³	1	2	yes	—
55	#14556658	C₂×C₄	1	2	yes	—
56	C₄⋊#10596833	C₂×C₄	1	1	yes	characteristic
57	C₄⋊#10596833	C₂×C₄	1	1	yes	characteristic
58	C₄⋊#10596833	C₂×C₄	1	2	yes	—
59	C₂²⋊#10596833	C₂×C₄	1	2	yes	—
60	C₂²⋊#10596833	C₂×C₄	1	2	yes	—
61	C₂²⋊#10596833	C₂×C₄	1	2	yes	—
62	#14556646	C₂×C₄	1	2	yes	—
63	#14556646	C₂×C₄	1	2	yes	—
64	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
65	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
66	C₄×#10596833	C₂³	1	1	yes	characteristic
67	C₄×#10596833	C₂³	1	2	yes	—
68	C₄×#10596833	C₂³	1	1	yes	characteristic
69	#14556584	C₈	1	2	yes	—
70	#14556582	C₈	1	1	yes	characteristic
71	#14556578	C₈	1	1	yes	characteristic
72	#14556572	C₈	1	2	yes	—
73	#14556570	C₈	1	1	yes	characteristic
74	#14556556	C₂×C₄	1	1	yes	characteristic
75	D₄×Dic₇₃	C₂×C₄	1	2	yes	—
76	D₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
77	#14556547	C₂×C₄	1	2	yes	—
78	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
79	#121×C₇₃	C₈	1	1	yes	characteristic
80	#11924040	—	2	1	no	—
81	#11924040	—	2	1	no	—
82	#11924040	—	2	2	no	—
83	#11924040	—	2	2	no	—
84	#11924040	#41	1	2	yes	—
85	#11924040	#41	1	2	yes	—
86	#11924037	—	2	4	no	—
87	C₂×#10596833	—	2	2	no	—
88	C₂×#10596833	—	2	1	no	—
89	C₂×#10596833	—	2	1	no	—
90	C₂×#10596833	—	2	2	no	—
91	C₂×#10596833	—	2	2	no	—
92	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
93	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
94	C₂×#10596833	C₂²×C₄	1	2	yes	—
95	C₂×#10596833	C₂²×C₄	1	2	yes	—
96	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
97	C₂×#10596833	C₂⁴	1	1	yes	characteristic
98	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
99	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
100	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
102	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
103	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
104	#11924007	C₂×C₈	1	2	yes	—
105	#11924007	C₂×C₈	1	2	yes	—
106	#11924007	C₂×C₈	1	2	yes	—
107	C₂²⋊Dic₇₃	C₂×C₈	1	2	yes	—
108	C₂²⋊Dic₇₃	C₂×C₈	1	2	yes	—
109	C₂²⋊Dic₇₃	C₂×C₈	1	2	yes	—
110	C₂²⋊Dic₇₃	C₂×C₈	1	2	yes	—
111	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
112	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
113	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
115	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
116	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
117	C₂×D₄×C₇₃	C₂×C₈	1	1	yes	characteristic
118	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
119	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
120	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
121	C₂³×C₇₃	C₂²×C₈	1	2	yes	—
122	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
123	D₄×C₇₃	—	2	4	no	—
124	D₄×C₇₃	—	2	2	no	—
125	C₂×C₄×C₇₃	—	2	2	no	—
126	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×Dic₇₃	—	2	2	no	—
130	C₂×Dic₇₃	—	2	1	no	—
131	C₂×Dic₇₃	—	2	2	no	—
132	C₂×Dic₇₃	—	2	1	no	—
133	C₂×Dic₇₃	—	2	2	no	—
134	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
135	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
136	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
137	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
138	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
139	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
140	Dic₁₄₆	—	2	2	no	—
141	#10596833	—	2	1	no	—
142	#10596833	—	2	2	no	—
143	#10596833	—	2	1	no	—
144	#10596833	—	2	2	no	—
145	#10596833	#131	1	2	yes	—
146	#10596833	C₂×#41	1	1	yes	characteristic
147	#10596833	C₂×#41	1	1	yes	characteristic
148	C₂²×C₇₃	—	2	1	no	—
149	C₂²×C₇₃	—	2	1	no	—
150	C₂²×C₇₃	—	2	2	no	—
151	C₂²×C₇₃	—	2	2	no	—
152	C₂²×C₇₃	—	2	2	no	—
153	C₂²×C₇₃	#504	1	2	yes	—
154	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
155	C₄×C₇₃	—	2	1	no	—
156	C₄×C₇₃	—	2	2	no	—
157	C₄×C₇₃	#504	1	2	yes	—
158	Dic₇₃	—	2	2	no	—
159	Dic₇₃	—	2	1	no	—
160	Dic₇₃	—	2	1	no	—
161	Dic₇₃	—	2	1	no	—
162	Dic₇₃	—	2	2	no	—
163	Dic₇₃	C₄×#41	1	1	yes	characteristic
164	Dic₇₃	C₂×#131	1	1	yes	characteristic
165	#33480	—	73	1	no	maximal, Sylow
166	C₂×C₇₃	—	2	2	no	—
167	C₂×C₇₃	—	4	2	no	—
168	C₂×C₇₃	C₈×#41	1	1	yes	characteristic
169	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
170	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
171	#2966	—	73	1	no	—
172	#2183	—	73	1	no	—
173	Q₈×C₁₆	—	73	1	no	—
174	#2169	—	73	1	no	—
175	#2168	—	73	2	no	—
176	D₄×C₁₆	—	73	1	no	—
177	D₄×C₁₆	—	73	2	no	—
178	#2164	—	73	2	no	—
179	#2163	—	73	1	no	—
180	#2108	—	73	2	no	—
181	C₂×C₄×C₁₆	—	73	1	no	—
182	C₇₃	#33480	1	1	yes	characteristic, complemented, Sylow
183	#529	—	73	1	no	—
184	C₂×M₃₂	—	73	2	no	—
185	C₂²×C₁₆	—	73	1	no	—
186	C₂²×C₁₆	—	73	1	no	—
187	C₂²×C₁₆	—	73	2	no	—
188	#446	—	73	1	no	—
189	C₈×Q₈	—	73	1	no	—
190	#436	—	73	1	no	—
191	#435	—	73	2	no	—
192	C₈×D₄	—	73	2	no	—
193	C₈×D₄	—	73	1	no	—
194	#433	—	73	2	no	—
195	#432	—	73	1	no	—
196	#405	—	73	2	no	—
197	C₂×C₄×C₈	—	73	1	no	—
198	C₄⋊C₁₆	—	73	1	no	—
199	C₄⋊C₁₆	—	73	1	no	—
200	C₄⋊C₁₆	—	73	2	no	—
201	C₄⋊C₁₆	—	73	2	no	—
202	C₂²⋊C₁₆	—	73	2	no	—
203	C₂²⋊C₁₆	—	73	2	no	—
204	C₂²⋊C₁₆	—	73	2	no	—
205	#346	—	73	2	no	—
206	#346	—	73	2	no	—
207	C₄×C₁₆	—	73	1	no	—
208	C₄×C₁₆	—	73	1	no	—
209	C₄×C₁₆	—	73	1	no	—
210	C₄×C₁₆	—	73	1	no	—
211	C₄×C₁₆	—	73	2	no	—
212	C₂×M₁₆	—	73	2	no	—
213	C₂²×C₈	—	73	1	no	—
214	C₂²×C₈	—	73	1	no	—
215	C₂²×C₈	—	73	2	no	—
216	#126	—	73	2	no	—
217	#125	—	73	1	no	—
218	#124	—	73	1	no	—
219	#123	—	73	2	no	—
220	C₂²⋊Q₈	—	73	1	no	—
221	#121	#10596842	1	1	yes	characteristic
222	C₄×Q₈	—	73	1	no	—
223	C₄×D₄	—	73	2	no	—
224	C₄×D₄	—	73	1	no	—
225	#117	—	73	2	no	—
226	C₂×C₄²	—	73	1	no	—
227	M₃₂	—	146	4	no	—
228	C₂×C₁₆	—	73	1	no	—
229	C₂×C₁₆	—	73	1	no	—
230	C₂×C₁₆	—	73	1	no	—
231	C₂×C₁₆	—	73	1	no	—
232	C₂×C₁₆	—	73	2	no	—
233	C₂×C₁₆	—	73	2	no	—
234	C₂×C₁₆	—	146	2	no	—
235	C₂×C₁₆	—	146	4	no	—
236	C₂×C₁₆	—	146	2	no	—
237	C₄⋊C₈	—	73	1	no	—
238	C₄⋊C₈	—	73	1	no	—
239	C₄⋊C₈	—	73	2	no	—
240	C₄⋊C₈	—	73	2	no	—
241	C₂²⋊C₈	—	73	2	no	—
242	C₂²⋊C₈	—	73	2	no	—
243	C₂²⋊C₈	—	73	2	no	—
244	#97	—	73	2	no	—
245	#97	—	73	2	no	—
246	C₄×C₈	—	73	1	no	—
247	C₄×C₈	—	73	1	no	—
248	C₄×C₈	—	73	2	no	—
249	C₄×C₈	—	73	1	no	—
250	C₄×C₈	—	73	1	no	—
251	C₂×Q₈	—	73	1	no	—
252	C₂×D₄	C₂×#10596842	1	2	yes	—
253	C₂×D₄	C₂×#10596842	1	1	yes	characteristic
254	C₂²×C₄	—	73	1	no	—
255	C₂²×C₄	—	73	1	no	—
256	C₂²×C₄	—	73	2	no	—
257	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
258	M₁₆	—	146	4	no	—
259	C₂×C₈	—	73	1	no	—
260	C₂×C₈	—	73	1	no	—
261	C₂×C₈	—	73	1	no	—
262	C₂×C₈	—	73	1	no	—
263	C₂×C₈	—	73	2	no	—
264	C₂×C₈	—	73	2	no	—
265	C₂×C₈	—	146	1	no	—
266	C₂×C₈	—	146	1	no	—
267	C₂×C₈	—	146	2	no	—
268	C₂×C₈	—	146	2	no	—
269	C₂×C₈	—	146	2	no	—
270	C₄⋊C₄	—	73	2	no	—
271	C₄⋊C₄	—	73	2	no	—
272	C₄⋊C₄	—	73	2	no	—
273	C₄⋊C₄	—	73	1	no	—
274	C₄⋊C₄	—	73	2	no	—
275	C₄⋊C₄	C₂×#10596842	1	1	yes	characteristic
276	C₂²⋊C₄	—	73	2	no	—
277	C₂²⋊C₄	—	73	2	no	—
278	C₂²⋊C₄	—	73	2	no	—
279	C₂²⋊C₄	—	73	2	no	—
280	C₂²⋊C₄	C₂×#10596842	1	2	yes	—
281	C₄²	—	73	1	no	—
282	C₄²	—	73	1	no	—
283	C₄²	—	73	1	no	—
284	C₄²	—	73	1	no	—
285	C₄²	—	73	2	no	—
286	C₁₆	—	73	2	no	—
287	C₁₆	—	73	2	no	—
288	C₁₆	—	146	1	no	—
289	C₁₆	—	146	1	no	—
290	C₁₆	—	146	2	no	—
291	C₁₆	—	146	2	no	—
292	C₂³	C₂²×#10596842	1	2	yes	—
293	C₂³	C₂²×#10596842	1	1	yes	characteristic
294	Q₈	—	146	2	no	—
295	D₄	—	2	4	no	—
296	D₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	73	1	no	—
299	C₂×C₄	—	73	1	no	—
300	C₂×C₄	—	73	1	no	—
301	C₂×C₄	—	73	1	no	—
302	C₂×C₄	—	73	2	no	—
303	C₂×C₄	—	73	2	no	—
304	C₂×C₄	—	146	1	no	—
305	C₂×C₄	—	146	2	no	—
306	C₂×C₄	—	146	1	no	—
307	C₂×C₄	—	146	2	no	—
308	C₂×C₄	—	146	2	no	—
309	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
310	C₂×C₄	C₂²×#10596842	1	2	yes	—
311	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
312	C₈	—	73	1	no	—
313	C₈	—	73	1	no	—
314	C₈	—	73	2	no	—
315	C₈	—	146	1	no	—
316	C₈	—	146	2	no	—
317	C₈	—	146	1	no	—
318	C₈	—	146	2	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	#17989708	1	2	yes	—
325	C₂²	C₂³×#10596842	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	2	no	—
328	C₄	—	73	1	no	—
329	C₄	—	73	1	no	—
330	C₄	—	146	1	no	—
331	C₄	—	146	2	no	—
332	C₄	—	146	1	no	—
333	C₄	—	146	1	no	—
334	C₄	—	146	2	no	—
335	C₄	#17989708	1	2	yes	—
336	C₂	—	2	2	no	—
337	C₂	—	4	2	no	—
338	C₂	#41×#10596842	1	1	yes	characteristic
339	C₂	C₂×#17989708	1	1	yes	characteristic
340	C₂	C₂×#17989708	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10596842
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting, prime radical	#121×C₇₃	C₈
cosocle	C₂×#10596833	C₂⁴
hypercenter	#121	#10596842
hypocenter	C₇₃	#33480
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. #22627156	C₂	semidirect
2. #22627155	C₂	semidirect
3. #22627147	C₂	semidirect
4. #22627143	C₂	semidirect
5. D₄×#11924040	C₂	semidirect
6. D₄×#11924040	C₂	semidirect
7. #22627136	C₂	semidirect
8. #22627133	C₂	semidirect
9. #22627092	C₂	semidirect
10. C₂×C₄×#11924040	C₂	semidirect
182. C₇₃	#33480	semidirect

Maximal subgroups

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

subgroup	index
#22627156	2
#22627155	2
#22627147	2
#22627143	2
D₄×#11924040	2
#22627136	2
#22627133	2
#22627092	2
C₂×C₄×#11924040	2
#22626744	2
#33480	73