Subgroups — #34033274

Order 18688.

514 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627049	C₂	1	2	yes	maximal
2	#22627040	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17989671	C₂	1	1	yes	maximal, characteristic, complemented
4	#22626938	C₂	1	2	yes	maximal, complemented
5	#131×#64408	C₂	1	1	yes	maximal, characteristic
6	#17989700	C₂²	1	2	yes	—
7	C₂×C₄×#10596842	C₂²	1	2	yes	—
8	Q₈⋊#10596842	C₄	1	4	yes	—
9	D₄⋊#10596842	C₄	1	4	yes	—
10	#17989671	—	2	1	no	—
11	#17989671	—	2	1	no	—
12	#17989671	—	2	1	no	—
13	C₈×#10596842	—	2	1	no	—
14	D₇₃⋊#131	C₄	1	1	yes	characteristic
15	#41×#64408	C₂²	1	1	yes	characteristic
16	#17989553	—	2	1	no	—
17	#17989553	C₄	1	1	yes	characteristic
18	#17989550	—	2	1	no	—
19	#17989550	C₄	1	1	yes	characteristic
20	M₁₆×#64408	—	2	1	no	—
21	M₁₆×#64408	C₂²	1	1	yes	characteristic
22	C₂×C₈×#64408	—	2	1	no	—
23	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
24	#131×D₇₃	C₄	1	1	yes	characteristic
25	C₂²×#10596842	—	2	2	no	—
26	C₄⋊#10596842	C₂×C₄	1	4	yes	—
27	C₂²⋊#10596842	—	2	2	no	—
28	C₄×#10596842	—	2	2	no	—
29	C₄×#10596842	—	2	2	no	—
30	C₄×#10596842	C₂×C₄	1	4	yes	—
31	#14556679	C₈	1	2	yes	complemented
32	#14556678	—	2	2	no	—
33	#14556678	C₈	1	2	yes	complemented
34	D₇₃⋊M₁₆	—	2	1	no	—
35	D₇₃⋊M₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂×#11924036	—	2	1	no	—
37	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
38	Q₈×#64408	C₂×C₄	1	2	yes	—
39	D₄×#64408	—	2	2	no	—
40	D₄×#64408	C₂×C₄	1	2	yes	—
41	#14556670	—	2	1	no	—
42	#14556670	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×#64408	—	2	1	no	—
44	C₂×C₄×#64408	C₂³	1	1	yes	characteristic
45	#14556645	—	2	1	no	—
46	#14556645	—	2	1	no	—
47	#14556645	—	2	1	no	—
48	C₈×#64408	—	2	1	no	—
49	C₈×#64408	—	2	1	no	—
50	C₈×#64408	—	2	1	no	—
51	C₈×#64408	D₄	1	1	yes	characteristic
52	C₈×#64408	D₄	1	1	yes	characteristic
53	C₈×#64408	D₄	1	1	yes	characteristic
54	C₈×#64408	D₄	1	1	yes	characteristic
55	#41×D₇₃	C₂×C₄	1	1	yes	characteristic
56	#14556599	—	2	1	no	—
57	#14556599	C₈	1	1	yes	characteristic, complemented
58	#14556598	C₈	1	1	yes	characteristic, complemented
59	#14556597	—	2	1	no	—
60	#14556597	C₈	1	1	yes	characteristic, complemented
61	M₁₆×D₇₃	—	2	1	no	—
62	M₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₈×D₇₃	—	2	1	no	—
64	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
65	#131×C₇₃	C₈	1	1	yes	characteristic, complemented
66	C₂×#10596842	—	2	4	no	—
67	C₂×#10596842	—	2	4	no	—
68	C₂×#10596842	—	4	4	no	—
69	C₂²×#64408	—	2	2	no	—
70	C₂²×#64408	#41	1	2	yes	—
71	#11924038	—	2	2	no	—
72	#11924038	C₂×C₈	1	2	yes	—
73	#11924037	—	2	1	no	—
74	#11924037	—	2	2	no	—
75	#11924036	—	2	1	no	—
76	#11924036	—	2	2	no	—
77	#11924036	C₂²⋊C₄	1	1	yes	characteristic
78	#11924036	C₂²⋊C₄	1	1	yes	characteristic
79	#11924036	C₂×C₈	1	2	yes	—
80	C₂×#10596833	—	2	2	no	—
81	C₂×#10596833	M₁₆	1	2	yes	—
82	C₄⋊#64408	—	2	2	no	—
83	C₄⋊#64408	C₄²	1	2	yes	—
84	C₂²⋊#64408	—	2	1	no	—
85	C₂²⋊#64408	—	2	2	no	—
86	C₄×#64408	—	2	2	no	—
87	C₄×#64408	—	2	1	no	—
88	C₄×#64408	C₂²×C₄	1	2	yes	—
89	C₄×#64408	C₂×D₄	1	1	yes	characteristic
90	C₄×#64408	C₂×D₄	1	1	yes	characteristic
91	#11924029	C₂×C₈	1	1	yes	characteristic
92	#11924028	—	2	1	no	—
93	#11924028	C₂×C₈	1	1	yes	characteristic
94	#11924027	—	2	1	no	—
95	#11924027	C₂×C₈	1	1	yes	characteristic
96	Q₈×D₇₃	C₄²	1	1	yes	characteristic
97	D₄×D₇₃	—	2	1	no	—
98	D₄×D₇₃	C₄²	1	1	yes	characteristic
99	C₂×C₄×D₇₃	—	2	1	no	—
100	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
101	#11924012	—	2	1	no	—
102	#11924012	C₂×C₈	1	1	yes	characteristic
103	#11924011	—	2	1	no	—
104	#11924011	—	2	1	no	—
105	#11924011	—	2	1	no	—
106	C₂×#10596831	—	2	1	no	—
107	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
108	C₈×D₇₃	—	2	1	no	—
109	C₈×D₇₃	—	2	1	no	—
110	C₈×D₇₃	—	2	1	no	—
111	C₈×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
112	C₈×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
113	C₈×D₇₃	M₁₆	1	1	yes	characteristic
114	C₈×D₇₃	M₁₆	1	1	yes	characteristic
115	#41×C₇₃	C₂×C₈	1	1	yes	characteristic
116	M₁₆×C₇₃	—	2	1	no	—
117	M₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
118	C₂×C₈×C₇₃	—	2	1	no	—
119	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
120	C₂²×D₇₃	—	2	1	no	—
121	C₂²×D₇₃	#117	1	1	yes	characteristic
122	C₂×#64408	—	2	4	no	—
123	C₂×#64408	—	2	2	no	—
124	C₂×#64408	—	4	4	no	—
125	C₂×#64408	C₄×D₄	1	2	yes	—
126	C₂×#64408	C₄×D₄	1	2	yes	—
127	#10596842	—	4	4	no	—
128	#10596842	—	4	4	no	—
129	Q₈×C₇₃	C₄×C₈	1	1	yes	characteristic
130	D₄×C₇₃	—	2	1	no	—
131	D₄×C₇₃	C₄×C₈	1	1	yes	characteristic
132	C₂×C₄×C₇₃	—	2	1	no	—
133	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
134	C₂²⋊D₇₃	—	2	1	no	—
135	C₂²⋊D₇₃	—	2	1	no	—
136	C₂²⋊D₇₃	—	2	1	no	—
137	C₂×Dic₇₃	—	2	1	no	—
138	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
139	D₂₉₂	—	2	1	no	—
140	D₂₉₂	C₄×C₈	1	1	yes	characteristic
141	C₄×D₇₃	—	2	1	no	—
142	C₄×D₇₃	—	2	1	no	—
143	C₄×D₇₃	—	2	1	no	—
144	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
145	C₄×D₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
147	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
148	Dic₁₄₆	—	2	1	no	—
149	Dic₁₄₆	C₄×C₈	1	1	yes	characteristic
150	#10596833	—	2	2	no	—
151	#10596833	C₂²⋊C₈	1	2	yes	—
152	#10596833	C₂²⋊C₈	1	2	yes	—
153	C₈×C₇₃	—	2	1	no	—
154	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
155	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
156	#10596831	—	2	1	no	—
157	#10596831	C₂²⋊C₈	1	1	yes	characteristic
158	#10596831	C₂²⋊C₈	1	1	yes	characteristic
159	C₂²×C₇₃	—	2	1	no	—
160	C₂²×C₇₃	#431	1	1	yes	characteristic
161	C₂×D₇₃	—	2	2	no	—
162	C₂×D₇₃	—	2	1	no	—
163	C₂×D₇₃	—	4	2	no	—
164	C₂×D₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×D₇₃	#431	1	1	yes	characteristic
166	#64408	—	2	2	no	—
167	#64408	—	4	2	no	—
168	#64408	#444	1	4	yes	—
169	C₄×C₇₃	—	2	1	no	—
170	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
171	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
172	Dic₇₃	—	2	1	no	—
173	Dic₇₃	C₄×M₁₆	1	1	yes	characteristic
174	Dic₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
175	#8185	—	73	1	no	maximal, Sylow
176	C₂×C₇₃	—	2	1	no	—
177	C₂×C₇₃	—	4	1	no	—
178	C₂×C₇₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
179	D₇₃	—	2	1	no	—
180	D₇₃	—	4	1	no	—
181	D₇₃	#1765	1	2	yes	—
182	C₄×#131	—	73	1	no	—
183	#1476	—	73	2	no	—
184	#1450	—	73	1	no	—
185	C₂×#322	—	73	1	no	—
186	#1278	—	73	2	no	—
187	C₇₃	#8185	1	1	yes	characteristic, complemented, Sylow
188	C₂×#131	—	73	1	no	—
189	C₂×#131	—	73	1	no	—
190	C₄×#41	—	73	1	no	—
191	#431	—	73	2	no	—
192	#405	—	73	1	no	—
193	#405	—	73	1	no	—
194	#405	—	146	1	no	—
195	#405	—	146	1	no	—
196	C₄×M₁₆	—	73	1	no	—
197	C₄×M₁₆	—	146	1	no	—
198	C₂×C₄×C₈	—	73	1	no	—
199	C₂×C₄×C₈	—	73	2	no	—
200	C₂×C₄×C₈	—	146	1	no	—
201	Q₈⋊C₈	—	73	4	no	—
202	D₄⋊C₈	—	73	4	no	—
203	#322	—	146	1	no	—
204	#322	—	146	1	no	—
205	#322	—	146	1	no	—
206	C₈²	—	146	1	no	—
207	C₂×#41	—	73	1	no	—
208	#131	—	73	2	no	—
209	#131	—	73	2	no	—
210	#131	—	73	1	no	—
211	#131	—	73	1	no	—
212	#131	—	73	1	no	—
213	#131	—	146	1	no	—
214	#131	—	146	1	no	—
215	#131	—	146	2	no	—
216	#131	#10596842	1	1	yes	characteristic, complemented
217	C₂×M₁₆	—	73	1	no	—
218	C₂×M₁₆	—	73	1	no	—
219	C₂×M₁₆	—	146	1	no	—
220	C₂×M₁₆	—	146	1	no	—
221	C₂²×C₈	—	73	1	no	—
222	C₂²×C₈	—	73	1	no	—
223	C₂²×C₈	—	146	1	no	—
224	C₂²×C₈	—	146	2	no	—
225	C₂²×C₈	—	146	1	no	—
226	C₄×Q₈	—	73	2	no	—
227	C₄×D₄	—	73	2	no	—
228	C₄×D₄	—	146	2	no	—
229	#117	—	73	1	no	—
230	#117	—	146	1	no	—
231	C₂×C₄²	—	73	1	no	—
232	C₂×C₄²	—	146	1	no	—
233	C₄⋊C₈	—	73	4	no	—
234	C₂²⋊C₈	—	146	2	no	—
235	#97	—	146	1	no	—
236	#97	—	146	1	no	—
237	#97	—	146	1	no	—
238	C₄×C₈	—	73	1	no	—
239	C₄×C₈	—	73	1	no	—
240	C₄×C₈	—	73	1	no	—
241	C₄×C₈	—	73	1	no	—
242	C₄×C₈	—	73	4	no	—
243	C₄×C₈	—	146	2	no	—
244	C₄×C₈	—	146	2	no	—
245	C₄×C₈	—	146	1	no	—
246	C₄×C₈	—	146	1	no	—
247	C₄×C₈	—	146	1	no	—
248	#41	—	73	1	no	—
249	#41	—	73	1	no	—
250	#41	—	73	1	no	—
251	#41	—	146	1	no	—
252	#41	—	146	1	no	—
253	#41	C₂×#10596842	1	1	yes	characteristic
254	C₂×Q₈	—	73	1	no	—
255	C₂×D₄	—	73	1	no	—
256	C₂×D₄	—	146	1	no	—
257	C₂²×C₄	—	73	1	no	—
258	C₂²×C₄	—	73	2	no	—
259	C₂²×C₄	—	146	1	no	—
260	C₂²×C₄	—	146	2	no	—
261	M₁₆	—	2	1	no	—
262	M₁₆	—	73	2	no	—
263	M₁₆	—	73	1	no	—
264	M₁₆	—	146	1	no	—
265	M₁₆	—	146	1	no	—
266	M₁₆	—	146	1	no	—
267	M₁₆	—	146	2	no	—
268	M₁₆	—	146	2	no	—
269	M₁₆	—	146	1	no	—
270	M₁₆	—	146	1	no	—
271	M₁₆	C₂×#10596842	1	1	yes	characteristic
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	292	4	no	—
274	C₂×C₈	—	73	2	no	—
275	C₂×C₈	—	73	1	no	—
276	C₂×C₈	—	73	1	no	—
277	C₂×C₈	—	73	2	no	—
278	C₂×C₈	—	73	1	no	—
279	C₂×C₈	—	73	1	no	—
280	C₂×C₈	—	73	1	no	—
281	C₂×C₈	—	73	1	no	—
282	C₂×C₈	—	73	1	no	—
283	C₂×C₈	—	146	1	no	—
284	C₂×C₈	—	146	1	no	—
285	C₂×C₈	—	146	4	no	—
286	C₂×C₈	—	146	4	no	—
287	C₂×C₈	—	146	1	no	—
288	C₂×C₈	—	146	2	no	—
289	C₂×C₈	—	146	2	no	—
290	C₂×C₈	—	146	1	no	—
291	C₂×C₈	—	146	1	no	—
292	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
293	C₄⋊C₄	—	73	2	no	—
294	C₄⋊C₄	—	146	2	no	—
295	C₂²⋊C₄	—	146	2	no	—
296	C₂²⋊C₄	—	146	1	no	—
297	C₄²	—	73	2	no	—
298	C₄²	—	73	1	no	—
299	C₄²	—	73	1	no	—
300	C₄²	—	146	1	no	—
301	C₄²	—	146	2	no	—
302	C₂³	—	73	1	no	—
303	C₂³	—	146	1	no	—
304	Q₈	—	73	1	no	—
305	Q₈	—	146	1	no	—
306	Q₈	C₄×#10596842	1	1	yes	characteristic
307	D₄	—	2	1	no	—
308	D₄	—	73	1	no	—
309	D₄	—	146	1	no	—
310	D₄	—	146	1	no	—
311	D₄	—	146	1	no	—
312	D₄	—	146	1	no	—
313	D₄	C₄×#10596842	1	1	yes	characteristic
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	292	4	no	—
316	C₂×C₄	—	73	2	no	—
317	C₂×C₄	—	73	1	no	—
318	C₂×C₄	—	73	1	no	—
319	C₂×C₄	—	73	1	no	—
320	C₂×C₄	—	73	1	no	—
321	C₂×C₄	—	73	1	no	—
322	C₂×C₄	—	73	2	no	—
323	C₂×C₄	—	146	1	no	—
324	C₂×C₄	—	146	1	no	—
325	C₂×C₄	—	146	2	no	—
326	C₂×C₄	—	146	1	no	—
327	C₂×C₄	—	146	4	no	—
328	C₂×C₄	—	146	1	no	—
329	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	292	4	no	—
332	C₈	—	292	4	no	—
333	C₈	—	73	2	no	—
334	C₈	—	73	2	no	—
335	C₈	—	73	1	no	—
336	C₈	—	73	1	no	—
337	C₈	—	146	2	no	—
338	C₈	—	146	1	no	—
339	C₈	C₂²⋊#10596842	1	1	yes	characteristic
340	C₈	C₂²⋊#10596842	1	1	yes	characteristic
341	C₂²	—	2	1	no	—
342	C₂²	—	292	2	no	—
343	C₂²	—	73	1	no	—
344	C₂²	—	73	1	no	—
345	C₂²	—	146	2	no	—
346	C₂²	—	146	1	no	—
347	C₂²	#17989700	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	292	2	no	—
350	C₄	—	73	4	no	—
351	C₄	—	73	1	no	—
352	C₄	—	73	1	no	—
353	C₄	—	146	2	no	—
354	C₄	—	146	1	no	—
355	C₄	C₂×C₄×#10596842	1	1	yes	characteristic
356	C₄	C₂×C₂²⋊#10596842	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	4	1	no	—
359	C₂	—	292	1	no	—
360	C₂	—	73	2	no	—
361	C₂	—	146	1	no	—
362	C₂	C₄×C₂²⋊#10596842	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊#10596842
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#10596842
Fitting	#131×C₇₃	C₈
socle	C₂×C₇₃	C₄×C₂²⋊C₈
prime radical	#41×D₇₃	C₂×C₄
cosocle	C₂×C₄×#64408	C₂³
hypercenter	#131	#10596842
hypocenter	C₇₃	#8185
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
2. #22627040	C₂	semidirect
3. C₂×#17989671	C₂	semidirect
4. #22626938	C₂	semidirect
31. #14556679	C₈	semidirect
33. #14556678	C₈	semidirect
57. #14556599	C₈	semidirect
58. #14556598	C₈	semidirect
60. #14556597	C₈	semidirect
65. #131×C₇₃	C₈	semidirect
187. C₇₃	#8185	semidirect
216. #131	#10596842	semidirect

Maximal subgroups

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

subgroup	index
#22627049	2
#22627040	2
C₂×#17989671	2
#22626938	2
#131×#64408	2
#8185	73