Subgroups — #64408⋊#350 #34034097

Order 18688.

490 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627111	C₂	1	2	yes	maximal
2	#64408⋊M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#64408⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#22627012	C₂	1	2	yes	maximal, complemented
5	#131×#64408	C₂	1	1	yes	maximal, characteristic
6	#17989706	C₂²	1	2	yes	—
7	C₂×#14556686	C₂²	1	2	yes	—
8	#17989688	C₄	1	4	yes	complemented
9	#17989687	C₄	1	4	yes	complemented
10	#17989682	—	2	1	no	—
11	#64408⋊C₁₆	—	2	1	no	—
12	#64408⋊C₁₆	—	2	1	no	—
13	#64408⋊C₁₆	—	2	1	no	—
14	D₇₃⋊#131	C₂²	1	1	yes	characteristic
15	#41×#64408	C₄	1	1	yes	characteristic
16	#17989553	—	2	1	no	—
17	#17989553	C₂²	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	C₄	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	#14556688	C₂×C₄	1	4	yes	complemented
26	#14556687	—	2	2	no	—
27	#14556686	—	2	2	no	—
28	#14556686	—	2	2	no	—
29	#14556686	C₂×C₄	1	4	yes	complemented
30	C₂×#11924040	—	2	2	no	—
31	#14556679	C₂×C₄	1	2	yes	—
32	#14556678	—	2	2	no	—
33	#14556678	C₂×C₄	1	2	yes	—
34	D₇₃⋊M₁₆	—	2	1	no	—
35	D₇₃⋊M₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂×#11924036	—	2	1	no	—
37	C₂×#11924036	C₂³	1	1	yes	characteristic
38	Q₈×#64408	C₈	1	2	yes	—
39	D₄×#64408	—	2	2	no	—
40	D₄×#64408	C₈	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	C₂×C₄	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
58	#14556598	C₈	1	1	yes	characteristic
59	#14556597	—	2	1	no	—
60	#14556597	C₈	1	1	yes	characteristic
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
66	#11924040	—	2	4	no	—
67	#11924040	—	2	4	no	—
68	C₂²×#64408	—	2	2	no	—
69	C₂²×#64408	M₁₆	1	2	yes	—
70	#11924038	—	2	2	no	—
71	#11924038	C₄²	1	2	yes	—
72	#11924037	—	2	1	no	—
73	#11924037	—	2	2	no	—
74	#11924036	—	2	1	no	—
75	#11924036	—	2	2	no	—
76	#11924036	C₂²×C₄	1	2	yes	—
77	#11924036	C₂×D₄	1	1	yes	characteristic
78	#11924036	C₂×D₄	1	1	yes	characteristic
79	C₂×#10596833	—	2	2	no	—
80	C₂×#10596833	#41	1	2	yes	—
81	C₄⋊#64408	—	2	2	no	—
82	C₄⋊#64408	C₂×C₈	1	2	yes	—
83	C₂²⋊#64408	—	2	1	no	—
84	C₂²⋊#64408	—	2	2	no	—
85	C₄×#64408	—	2	2	no	—
86	C₄×#64408	—	2	1	no	—
87	C₄×#64408	C₂²⋊C₄	1	1	yes	characteristic
88	C₄×#64408	C₂²⋊C₄	1	1	yes	characteristic
89	C₄×#64408	C₂×C₈	1	2	yes	—
90	#11924029	C₄²	1	1	yes	characteristic
91	#11924028	—	2	1	no	—
92	#11924028	C₂×C₈	1	1	yes	characteristic
93	#11924027	—	2	1	no	—
94	#11924027	C₄²	1	1	yes	characteristic
95	Q₈×D₇₃	C₂×C₈	1	1	yes	characteristic
96	D₄×D₇₃	—	2	1	no	—
97	D₄×D₇₃	C₂×C₈	1	1	yes	characteristic
98	C₂×C₄×D₇₃	—	2	1	no	—
99	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
100	#11924012	—	2	1	no	—
101	#11924012	C₂×C₈	1	1	yes	characteristic
102	#11924011	—	2	1	no	—
103	#11924011	—	2	1	no	—
104	#11924011	—	2	1	no	—
105	C₂×#10596831	—	2	1	no	—
106	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
107	C₈×D₇₃	—	2	1	no	—
108	C₈×D₇₃	—	2	1	no	—
109	C₈×D₇₃	—	2	1	no	—
110	C₈×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
111	C₈×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
112	C₈×D₇₃	M₁₆	1	1	yes	characteristic
113	C₈×D₇₃	M₁₆	1	1	yes	characteristic
114	#41×C₇₃	C₂×C₈	1	1	yes	characteristic
115	M₁₆×C₇₃	—	2	1	no	—
116	M₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
117	C₂×C₈×C₇₃	—	2	1	no	—
118	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
119	C₂²×D₇₃	—	2	1	no	—
120	C₂²×D₇₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂×#64408	—	2	4	no	—
122	C₂×#64408	—	2	2	no	—
123	C₂×#64408	—	4	4	no	—
124	C₂×#64408	C₂²⋊C₈	1	2	yes	—
125	C₂×#64408	C₂²⋊C₈	1	2	yes	—
126	Q₈×C₇₃	C₄×C₈	1	1	yes	characteristic
127	D₄×C₇₃	—	2	1	no	—
128	D₄×C₇₃	C₄×C₈	1	1	yes	characteristic
129	C₂×C₄×C₇₃	—	2	1	no	—
130	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
131	C₂²⋊D₇₃	—	2	1	no	—
132	C₂²⋊D₇₃	—	2	1	no	—
133	C₂²⋊D₇₃	—	2	1	no	—
134	C₂×Dic₇₃	—	2	1	no	—
135	C₂×Dic₇₃	#117	1	1	yes	characteristic
136	D₂₉₂	—	2	1	no	—
137	D₂₉₂	C₄×C₈	1	1	yes	characteristic
138	C₄×D₇₃	—	2	1	no	—
139	C₄×D₇₃	—	2	1	no	—
140	C₄×D₇₃	—	2	1	no	—
141	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
142	C₄×D₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
144	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
145	Dic₁₄₆	—	2	1	no	—
146	Dic₁₄₆	C₄×C₈	1	1	yes	characteristic
147	#10596833	—	2	2	no	—
148	#10596833	C₄×D₄	1	2	yes	—
149	#10596833	C₄×D₄	1	2	yes	—
150	C₈×C₇₃	—	2	1	no	—
151	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
152	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
153	#10596831	—	2	1	no	—
154	#10596831	C₂²⋊C₈	1	1	yes	characteristic
155	#10596831	C₂²⋊C₈	1	1	yes	characteristic
156	C₂²×C₇₃	—	2	1	no	—
157	C₂²×C₇₃	#431	1	1	yes	characteristic
158	C₂×D₇₃	—	2	2	no	—
159	C₂×D₇₃	—	2	1	no	—
160	C₂×D₇₃	—	4	2	no	—
161	C₂×D₇₃	C₄×M₁₆	1	1	yes	characteristic
162	C₂×D₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
163	#64408	—	2	2	no	—
164	#64408	—	4	2	no	—
165	#64408	#350	1	4	yes	complemented
166	C₄×C₇₃	—	2	1	no	—
167	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
168	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
169	Dic₇₃	—	2	1	no	—
170	Dic₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
171	Dic₇₃	#431	1	1	yes	characteristic
172	C₄×#350	—	73	1	no	maximal, Sylow
173	C₂×C₇₃	—	2	1	no	—
174	C₂×C₇₃	—	4	1	no	—
175	C₂×C₇₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
176	D₇₃	—	2	1	no	—
177	D₇₃	—	4	1	no	—
178	D₇₃	C₂×#350	1	2	yes	—
179	C₄×#131	—	73	1	no	—
180	C₂×#350	—	73	2	no	—
181	C₄×M₃₂	—	73	1	no	—
182	C₂×C₄×C₁₆	—	73	1	no	—
183	#1379	—	73	2	no	—
184	C₇₃	C₄×#350	1	1	yes	characteristic, complemented, Sylow
185	C₂×#131	—	73	1	no	—
186	C₂×#131	—	73	1	no	—
187	C₄×#41	—	73	1	no	—
188	C₂×M₃₂	—	73	2	no	—
189	C₂²×C₁₆	—	73	2	no	—
190	#405	—	73	1	no	—
191	#405	—	73	1	no	—
192	#405	—	146	1	no	—
193	#405	—	146	1	no	—
194	C₄×M₁₆	—	73	1	no	—
195	C₄×M₁₆	—	146	1	no	—
196	C₂×C₄×C₈	—	73	1	no	—
197	C₂×C₄×C₈	—	146	1	no	—
198	#350	—	73	4	no	—
199	#350	—	73	4	no	—
200	#346	—	146	1	no	—
201	C₄×C₁₆	—	146	1	no	—
202	C₄×C₁₆	—	146	1	no	—
203	C₄×C₁₆	—	146	1	no	—
204	C₂×#41	—	73	1	no	—
205	#131	—	73	2	no	—
206	#131	—	73	2	no	—
207	#131	—	73	1	no	—
208	#131	—	73	1	no	—
209	#131	—	73	1	no	—
210	#131	—	146	1	no	—
211	#131	—	146	1	no	—
212	#131	—	146	2	no	—
213	#131	#10596842	1	1	yes	characteristic
214	C₂×M₁₆	—	73	1	no	—
215	C₂×M₁₆	—	73	1	no	—
216	C₂×M₁₆	—	146	1	no	—
217	C₂×M₁₆	—	146	1	no	—
218	C₂²×C₈	—	73	1	no	—
219	C₂²×C₈	—	73	1	no	—
220	C₂²×C₈	—	146	1	no	—
221	C₂²×C₈	—	146	1	no	—
222	C₄×Q₈	—	73	2	no	—
223	C₄×D₄	—	73	2	no	—
224	C₄×D₄	—	146	2	no	—
225	#117	—	73	1	no	—
226	#117	—	146	1	no	—
227	C₂×C₄²	—	73	1	no	—
228	C₂×C₄²	—	146	1	no	—
229	M₃₂	—	73	4	no	—
230	M₃₂	—	146	2	no	—
231	C₂×C₁₆	—	73	4	no	—
232	C₂×C₁₆	—	146	2	no	—
233	C₂×C₁₆	—	146	2	no	—
234	C₂×C₁₆	—	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	—	146	1	no	—
243	C₄×C₈	—	146	1	no	—
244	C₄×C₈	—	146	1	no	—
245	#41	—	73	1	no	—
246	#41	—	73	1	no	—
247	#41	—	73	1	no	—
248	#41	—	146	1	no	—
249	#41	—	146	1	no	—
250	#41	C₂×#10596842	1	1	yes	characteristic
251	C₂×Q₈	—	73	1	no	—
252	C₂×D₄	—	73	1	no	—
253	C₂×D₄	—	146	1	no	—
254	C₂²×C₄	—	73	1	no	—
255	C₂²×C₄	—	73	2	no	—
256	C₂²×C₄	—	146	1	no	—
257	C₂²×C₄	—	146	2	no	—
258	M₁₆	—	2	1	no	—
259	M₁₆	—	73	2	no	—
260	M₁₆	—	73	1	no	—
261	M₁₆	—	146	1	no	—
262	M₁₆	—	146	1	no	—
263	M₁₆	—	146	1	no	—
264	M₁₆	—	146	2	no	—
265	M₁₆	—	146	2	no	—
266	M₁₆	—	146	1	no	—
267	M₁₆	—	146	1	no	—
268	M₁₆	C₂×#10596842	1	1	yes	characteristic
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	73	2	no	—
271	C₂×C₈	—	73	2	no	—
272	C₂×C₈	—	73	1	no	—
273	C₂×C₈	—	73	1	no	—
274	C₂×C₈	—	73	1	no	—
275	C₂×C₈	—	73	1	no	—
276	C₂×C₈	—	73	1	no	—
277	C₂×C₈	—	73	1	no	—
278	C₂×C₈	—	73	1	no	—
279	C₂×C₈	—	146	1	no	—
280	C₂×C₈	—	146	1	no	—
281	C₂×C₈	—	146	1	no	—
282	C₂×C₈	—	146	2	no	—
283	C₂×C₈	—	146	2	no	—
284	C₂×C₈	—	146	1	no	—
285	C₂×C₈	—	146	1	no	—
286	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
287	C₄⋊C₄	—	73	2	no	—
288	C₄⋊C₄	—	146	2	no	—
289	C₂²⋊C₄	—	146	2	no	—
290	C₂²⋊C₄	—	146	1	no	—
291	C₄²	—	73	1	no	—
292	C₄²	—	73	1	no	—
293	C₄²	—	73	2	no	—
294	C₄²	—	146	1	no	—
295	C₄²	—	146	2	no	—
296	C₁₆	—	146	4	no	—
297	C₁₆	—	146	4	no	—
298	C₂³	—	73	1	no	—
299	C₂³	—	146	1	no	—
300	Q₈	—	73	1	no	—
301	Q₈	—	146	1	no	—
302	Q₈	C₄×#10596842	1	1	yes	characteristic
303	D₄	—	2	1	no	—
304	D₄	—	73	1	no	—
305	D₄	—	146	1	no	—
306	D₄	—	146	1	no	—
307	D₄	—	146	1	no	—
308	D₄	—	146	1	no	—
309	D₄	C₄×#10596842	1	1	yes	characteristic
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	292	4	no	—
312	C₂×C₄	—	73	2	no	—
313	C₂×C₄	—	73	1	no	—
314	C₂×C₄	—	73	1	no	—
315	C₂×C₄	—	73	1	no	—
316	C₂×C₄	—	73	1	no	—
317	C₂×C₄	—	73	1	no	—
318	C₂×C₄	—	73	2	no	—
319	C₂×C₄	—	146	1	no	—
320	C₂×C₄	—	146	1	no	—
321	C₂×C₄	—	146	2	no	—
322	C₂×C₄	—	146	1	no	—
323	C₂×C₄	—	146	4	no	—
324	C₂×C₄	—	146	1	no	—
325	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	73	2	no	—
328	C₈	—	73	2	no	—
329	C₈	—	73	1	no	—
330	C₈	—	73	1	no	—
331	C₈	—	146	2	no	—
332	C₈	—	146	1	no	—
333	C₈	C₂²⋊#10596842	1	1	yes	characteristic
334	C₈	C₂²⋊#10596842	1	1	yes	characteristic
335	C₂²	—	2	1	no	—
336	C₂²	—	292	2	no	—
337	C₂²	—	73	1	no	—
338	C₂²	—	73	1	no	—
339	C₂²	—	146	2	no	—
340	C₂²	—	146	1	no	—
341	C₂²	#17989700	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	292	2	no	—
344	C₄	—	73	4	no	—
345	C₄	—	73	1	no	—
346	C₄	—	73	1	no	—
347	C₄	—	146	2	no	—
348	C₄	—	146	1	no	—
349	C₄	C₂×C₄×#10596842	1	1	yes	characteristic
350	C₄	C₂×C₂²⋊#10596842	1	1	yes	characteristic
351	C₂	—	2	1	no	—
352	C₂	—	4	1	no	—
353	C₂	—	292	1	no	—
354	C₂	—	73	2	no	—
355	C₂	—	146	1	no	—
356	C₂	C₄×C₂²⋊#10596842	1	1	yes	characteristic

	subgroup	G/H
center	C₈	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×#11924036	C₂³
hypercenter	#131	#10596842
hypocenter	C₇₃	C₄×#350
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
2. #64408⋊M₃₂	C₂	semidirect
3. C₂×#64408⋊C₁₆	C₂	semidirect
4. #22627012	C₂	semidirect
8. #17989688	C₄	semidirect
9. #17989687	C₄	semidirect
25. #14556688	C₂×C₄	semidirect
29. #14556686	C₂×C₄	semidirect
165. #64408	#350	semidirect
184. C₇₃	C₄×#350	semidirect

Maximal subgroups

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

subgroup	index
#22627111	2
#64408⋊M₃₂	2
C₂×#64408⋊C₁₆	2
#22627012	2
#131×#64408	2
C₄×#350	73