Subgroups — #33055466

Order 18688.

714 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22619042	C₂	1	2	yes	maximal, complemented
2	SD₁₆⋊#10596831	C₂	1	4	yes	maximal, complemented
3	Q₁₆×#10596831	C₂	1	2	yes	maximal, complemented
4	D₈×#10596831	C₂	1	2	yes	maximal, complemented
5	C₂×#17988340	C₂	1	1	yes	maximal, characteristic, complemented
6	#22608318	C₂	1	2	yes	maximal, complemented
7	#22608214	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₇₃	C₂	1	1	yes	maximal, characteristic
9	Q₈×#10596831	C₂²	1	4	yes	—
10	D₄×#10596831	—	2	4	no	—
11	D₄×#10596831	C₂²	1	4	yes	—
12	#17988640	—	2	2	no	—
13	#17988640	C₂²	1	1	yes	characteristic
14	#17988640	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#10596831	—	2	2	no	—
16	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
17	#17988340	C₂²	1	2	yes	—
18	#17988340	C₂²	1	2	yes	—
19	Q₈⋊#10596831	C₂²	1	4	yes	—
20	D₄⋊#10596831	C₂²	1	4	yes	—
21	#17988308	C₂²	1	2	yes	—
22	C₈×#10596831	C₂²	1	2	yes	—
23	C₂×#135×C₇₃	C₄	1	1	yes	characteristic
24	C₄×#41×C₇₃	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₇₃	C₄	1	1	yes	characteristic
26	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
27	C₄×SD₁₆×C₇₃	C₄	1	2	yes	—
28	C₄×SD₁₆×C₇₃	C₂²	1	2	yes	—
29	C₄×D₈×C₇₃	C₄	1	1	yes	characteristic
30	C₄×D₈×C₇₃	C₂²	1	1	yes	characteristic
31	#427×C₇₃	C₄	1	1	yes	characteristic
32	#416×C₇₃	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
34	C₂²×#10596831	—	2	4	no	—
35	C₂²×#10596831	—	2	2	no	—
36	C₄⋊#10596831	—	2	4	no	—
37	C₄⋊#10596831	—	2	2	no	—
38	C₄⋊#10596831	—	2	1	no	—
39	C₄⋊#10596831	D₄	1	1	yes	characteristic
40	C₄⋊#10596831	D₄	1	1	yes	characteristic
41	C₄⋊#10596831	C₂³	1	4	yes	—
42	C₂²⋊#10596831	—	2	4	no	—
43	C₂²⋊#10596831	—	2	1	no	—
44	C₂²⋊#10596831	D₄	1	2	yes	—
45	C₄×#10596831	—	2	4	no	—
46	C₄×#10596831	C₂³	1	2	yes	—
47	C₄×#10596831	C₂³	1	2	yes	—
48	C₂×#41×C₇₃	C₂×C₄	1	2	yes	—
49	#135×C₇₃	C₈	1	8	yes	complemented
50	C₂×Q₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂×SD₁₆×C₇₃	C₂×C₄	1	2	yes	—
52	C₂×D₈×C₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
55	C₄×Q₈×C₇₃	C₂×C₄	1	2	yes	—
56	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
57	C₄×D₄×C₇₃	—	2	2	no	—
58	C₄×D₄×C₇₃	—	2	2	no	—
59	C₄×D₄×C₇₃	C₂×C₄	1	2	yes	—
60	C₄×D₄×C₇₃	C₂³	1	2	yes	—
61	#117×C₇₃	—	2	2	no	—
62	#117×C₇₃	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₇₃	—	2	2	no	—
64	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
65	#107×C₇₃	C₂×C₄	1	1	yes	characteristic
66	#107×C₇₃	C₂×C₄	1	1	yes	characteristic
67	#106×C₇₃	C₂×C₄	1	2	yes	—
68	#103×C₇₃	C₂×C₄	1	2	yes	—
69	#103×C₇₃	C₂×C₄	1	2	yes	—
70	#102×C₇₃	C₂×C₄	1	2	yes	—
71	#102×C₇₃	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
76	C₂×#10596831	—	2	4	no	—
77	C₂×#10596831	—	2	2	no	—
78	C₂×#10596831	—	8	4	no	—
79	C₂×#10596831	—	8	2	no	—
80	C₂×#10596831	C₂×D₄	1	2	yes	—
81	C₂×#10596831	C₂×D₄	1	2	yes	—
82	#41×C₇₃	—	2	4	no	—
83	#41×C₇₃	C₂×C₈	1	8	yes	complemented
84	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
85	C₂×D₄×C₇₃	—	2	2	no	—
86	C₂×D₄×C₇₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₇₃	—	2	2	no	—
88	C₂²×C₄×C₇₃	—	2	2	no	—
89	C₂²×C₄×C₇₃	—	2	2	no	—
90	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
93	Q₁₆×C₇₃	C₂×C₈	1	4	yes	complemented
94	SD₁₆×C₇₃	C₂×C₈	1	8	yes	complemented
95	D₈×C₇₃	C₂×C₈	1	4	yes	complemented
96	C₂×C₈×C₇₃	—	2	2	no	—
97	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	complemented
98	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
102	C₄⋊C₄×C₇₃	—	2	2	no	—
103	C₄⋊C₄×C₇₃	—	2	2	no	—
104	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
105	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
106	C₂²⋊C₄×C₇₃	—	2	2	no	—
107	C₂²⋊C₄×C₇₃	—	2	2	no	—
108	C₂²⋊C₄×C₇₃	—	2	2	no	—
109	C₄²×C₇₃	—	2	2	no	—
110	C₄²×C₇₃	—	2	2	no	—
111	C₄²×C₇₃	—	2	2	no	—
112	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₇₃	#41	1	1	yes	characteristic
115	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
116	C₂³×C₇₃	—	2	2	no	—
117	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
118	Q₈×C₇₃	—	2	2	no	—
119	Q₈×C₇₃	C₂²×C₈	1	4	yes	—
120	D₄×C₇₃	—	2	8	no	—
121	D₄×C₇₃	—	2	2	no	—
122	D₄×C₇₃	C₂²×C₈	1	4	yes	—
123	C₂×C₄×C₇₃	—	2	2	no	—
124	C₂×C₄×C₇₃	—	2	1	no	—
125	C₂×C₄×C₇₃	—	2	1	no	—
126	C₂×C₄×C₇₃	—	2	8	no	—
127	C₂×C₄×C₇₃	—	2	2	no	—
128	C₂×C₄×C₇₃	—	2	2	no	—
129	C₂×C₄×C₇₃	—	4	1	no	—
130	C₂×C₄×C₇₃	—	4	2	no	—
131	C₂×C₄×C₇₃	—	4	2	no	—
132	C₂×C₄×C₇₃	—	8	2	no	—
133	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₇₃	#131	1	2	yes	—
138	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
141	C₈×C₇₃	—	2	1	no	—
142	C₈×C₇₃	—	2	1	no	—
143	C₈×C₇₃	C₂²×C₈	1	2	yes	—
144	C₈×C₇₃	C₂²×C₈	1	2	yes	—
145	#10596831	—	2	2	no	—
146	#10596831	—	2	2	no	—
147	#10596831	—	4	2	no	—
148	#10596831	—	4	4	no	—
149	C₂²×C₇₃	—	2	1	no	—
150	C₂²×C₇₃	—	2	1	no	—
151	C₂²×C₇₃	—	2	4	no	—
152	C₂²×C₇₃	—	4	2	no	—
153	C₂²×C₇₃	—	4	2	no	—
154	C₂²×C₇₃	C₈×D₄	1	2	yes	—
155	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
156	C₄×C₇₃	—	2	4	no	—
157	C₄×C₇₃	—	2	1	no	—
158	C₄×C₇₃	—	2	1	no	—
159	C₄×C₇₃	—	2	1	no	—
160	C₄×C₇₃	—	4	2	no	—
161	C₄×C₇₃	—	4	2	no	—
162	C₄×C₇₃	C₈×D₄	1	2	yes	—
163	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
164	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
165	C₄×C₇₃	#576	1	1	yes	characteristic
166	C₄×C₇₃	#578	1	1	yes	characteristic
167	#17857	—	73	1	no	maximal, Sylow
168	C₂×C₇₃	—	2	2	no	—
169	C₂×C₇₃	—	4	4	no	—
170	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
171	C₂×C₇₃	#2951	1	1	yes	characteristic
172	C₂×C₇₃	#2957	1	1	yes	characteristic
173	#2988	—	73	2	no	—
174	C₄×#135	D₇₃	1	1	yes	characteristic
175	SD₁₆⋊C₈	—	73	4	no	—
176	C₈×Q₁₆	—	73	2	no	—
177	C₈×D₈	—	73	2	no	—
178	C₂×#335	—	73	1	no	—
179	#1477	—	73	2	no	—
180	#1450	—	73	1	no	—
181	C₇₃	#17857	1	1	yes	characteristic, complemented, Sylow
182	C₂×#135	Dic₇₃	1	1	yes	characteristic
183	C₄×#41	C₂×D₇₃	1	2	yes	—
184	C₈×Q₈	—	73	4	no	—
185	C₄×Q₁₆	Dic₇₃	1	1	yes	characteristic
186	C₄×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
187	C₄×SD₁₆	Dic₇₃	1	2	yes	—
188	C₄×SD₁₆	C₂×D₇₃	1	2	yes	—
189	C₄×D₈	Dic₇₃	1	1	yes	characteristic
190	C₄×D₈	C₂×D₇₃	1	1	yes	characteristic
191	C₈×D₄	—	73	4	no	—
192	C₈×D₄	—	146	4	no	—
193	#431	—	73	1	no	—
194	#431	—	73	1	no	—
195	#431	—	146	2	no	—
196	#427	Dic₇₃	1	1	yes	characteristic
197	C₂×C₄⋊C₈	—	73	2	no	—
198	C₂×C₄⋊C₈	—	146	2	no	—
199	#416	Dic₇₃	1	2	yes	—
200	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
201	#335	—	73	2	no	—
202	#335	—	73	2	no	—
203	Q₈⋊C₈	—	73	4	no	—
204	D₄⋊C₈	—	73	4	no	—
205	#322	—	73	2	no	—
206	C₈²	—	73	2	no	—
207	C₂×#41	C₂×Dic₇₃	1	2	yes	—
208	#135	#10596831	1	8	yes	complemented
209	C₂×Q₁₆	C₂×Dic₇₃	1	1	yes	characteristic
210	C₂×SD₁₆	C₂×Dic₇₃	1	2	yes	—
211	C₂×D₈	C₂×Dic₇₃	1	1	yes	characteristic
212	C₂²×C₈	—	146	4	no	—
213	C₂²×C₈	—	146	2	no	—
214	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
215	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
216	C₄×Q₈	C₂×Dic₇₃	1	2	yes	—
217	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
218	C₄×D₄	—	2	2	no	—
219	C₄×D₄	—	2	2	no	—
220	C₄×D₄	C₂×Dic₇₃	1	2	yes	—
221	C₄×D₄	C₂²×D₇₃	1	2	yes	—
222	#117	—	2	2	no	—
223	#117	C₂×Dic₇₃	1	2	yes	—
224	C₂×C₄²	—	2	2	no	—
225	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
226	#107	C₂×Dic₇₃	1	1	yes	characteristic
227	#107	C₂×Dic₇₃	1	1	yes	characteristic
228	#106	C₂×Dic₇₃	1	2	yes	—
229	C₄⋊C₈	—	73	1	no	—
230	C₄⋊C₈	—	73	1	no	—
231	C₄⋊C₈	—	73	4	no	—
232	C₄⋊C₈	—	146	4	no	—
233	C₄⋊C₈	—	146	2	no	—
234	C₄⋊C₈	—	146	1	no	—
235	#103	C₂×Dic₇₃	1	2	yes	—
236	#103	C₂×Dic₇₃	1	2	yes	—
237	#102	C₂×Dic₇₃	1	2	yes	—
238	#102	C₂×Dic₇₃	1	2	yes	—
239	C₂²⋊C₈	—	73	2	no	—
240	C₂²⋊C₈	—	146	4	no	—
241	C₂²⋊C₈	—	146	1	no	—
242	C₄×C₈	—	73	2	no	—
243	C₄×C₈	—	73	2	no	—
244	C₄×C₈	—	146	4	no	—
245	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
246	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
249	#41	—	2	4	no	—
250	#41	C₂×#10596831	1	8	yes	complemented
251	C₂×Q₈	C₂²×Dic₇₃	1	2	yes	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	C₂²×Dic₇₃	1	2	yes	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
259	C₂²×C₄	#11924027	1	1	yes	characteristic
260	Q₁₆	C₂×#10596831	1	4	yes	complemented
261	SD₁₆	C₂×#10596831	1	8	yes	complemented
262	D₈	C₂×#10596831	1	4	yes	complemented
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	584	2	no	—
265	C₂×C₈	—	584	4	no	—
266	C₂×C₈	—	73	2	no	—
267	C₂×C₈	—	73	2	no	—
268	C₂×C₈	—	146	4	no	—
269	C₂×C₈	—	146	2	no	—
270	C₂×C₈	C₂×#10596831	1	4	yes	complemented
271	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
272	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
273	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
274	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	C₂²×Dic₇₃	1	2	yes	—
278	C₄⋊C₄	C₂²×Dic₇₃	1	2	yes	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
286	C₄²	D₄×D₇₃	1	1	yes	characteristic
287	C₄²	#11924027	1	1	yes	characteristic
288	C₄²	C₂³×D₇₃	1	1	yes	characteristic
289	C₂³	—	2	2	no	—
290	C₂³	D₄×Dic₇₃	1	1	yes	characteristic
291	Q₈	—	2	2	no	—
292	Q₈	C₂²×#10596831	1	4	yes	—
293	D₄	—	2	8	no	—
294	D₄	—	2	2	no	—
295	D₄	C₂²×#10596831	1	4	yes	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
307	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
308	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
309	C₂×C₄	C₂²×#10596831	1	2	yes	—
310	C₂×C₄	#14556598	1	2	yes	—
311	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
313	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	292	2	no	—
317	C₈	—	292	4	no	—
318	C₈	—	146	2	no	—
319	C₈	—	146	2	no	—
320	C₈	C₂²×#10596831	1	2	yes	—
321	C₈	C₂²×#10596831	1	2	yes	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	4	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	D₄×#10596831	1	2	yes	—
328	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	4	no	—
333	C₄	—	4	2	no	—
334	C₄	—	4	2	no	—
335	C₄	D₄×#10596831	1	2	yes	—
336	C₄	C₂³×#10596831	1	1	yes	characteristic
337	C₄	C₂×#14556598	1	1	yes	characteristic
338	C₄	#17989389	1	1	yes	characteristic
339	C₄	#17989401	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	4	4	no	—
342	C₂	C₂×D₄×#10596831	1	1	yes	characteristic
343	C₂	#22618807	1	1	yes	characteristic
344	C₂	#22618848	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₄²	C₂³×D₇₃
Fitting	C₄×#135×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂×#135×C₇₃	C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	C₄×#135	D₇₃
hypocenter	C₇₃	#17857
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #22619042	C₂	semidirect
2. SD₁₆⋊#10596831	C₂	semidirect
3. Q₁₆×#10596831	C₂	semidirect
4. D₈×#10596831	C₂	semidirect
5. C₂×#17988340	C₂	semidirect
6. #22608318	C₂	semidirect
7. #22608214	C₂	semidirect
49. #135×C₇₃	C₈	semidirect
83. #41×C₇₃	C₂×C₈	semidirect
93. Q₁₆×C₇₃	C₂×C₈	semidirect
94. SD₁₆×C₇₃	C₂×C₈	semidirect
95. D₈×C₇₃	C₂×C₈	semidirect
97. C₂×C₈×C₇₃	C₂×C₈	semidirect
181. C₇₃	#17857	semidirect
208. #135	#10596831	semidirect
250. #41	C₂×#10596831	semidirect
260. Q₁₆	C₂×#10596831	semidirect
261. SD₁₆	C₂×#10596831	semidirect
262. D₈	C₂×#10596831	semidirect
270. C₂×C₈	C₂×#10596831	semidirect

Maximal subgroups

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

subgroup	index
#22619042	2
SD₁₆⋊#10596831	2
Q₁₆×#10596831	2
D₈×#10596831	2
C₂×#17988340	2
#22608318	2
#22608214	2
C₄×#135×C₇₃	2
#17857	73