Subgroups — #33239667

Order 18688.

536 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	C₈×#11924027	C₂	1	1	yes	maximal, characteristic
2	#22611839	C₂	1	1	yes	maximal, characteristic, complemented
3	#22611836	C₂	1	1	yes	maximal, characteristic, complemented
4	#22611756	C₂	1	2	yes	maximal, complemented
5	#22611753	C₂	1	2	yes	maximal, complemented
6	D₄×#11924003	C₂	1	1	yes	maximal, characteristic, complemented
7	#22611731	C₂	1	2	yes	maximal, complemented
8	#22611728	C₂	1	1	yes	maximal, characteristic, complemented
9	#22611685	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#17988405	C₂	1	2	yes	maximal, complemented
11	D₄×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11924027	C₄	1	1	yes	characteristic
13	C₂×#14556530	C₂²	1	2	yes	—
14	C₂²×#11924003	C₂²	1	2	yes	—
15	C₈×Dic₁₄₆	C₂²	1	1	yes	characteristic
16	#17988711	C₄	1	1	yes	characteristic
17	D₄×#10596831	C₄	1	1	yes	characteristic
18	C₈×C₂²⋊D₇₃	C₂²	1	2	yes	—
19	#17988650	C₄	1	2	yes	—
20	#17988639	C₄	1	1	yes	characteristic
21	#17988637	C₄	1	2	yes	—
22	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
23	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
24	C₄⋊#11924003	C₂²	1	1	yes	characteristic
25	#17988405	C₂²	1	1	yes	characteristic
26	#17988405	C₂²	1	1	yes	characteristic
27	#17988405	C₂²	1	2	yes	—
28	#17988405	C₂²	1	2	yes	—
29	#17988405	C₂²	1	2	yes	—
30	C₂²⋊#11924003	C₂²	1	2	yes	—
31	#17988366	C₂²	1	1	yes	characteristic
32	#17988366	C₂²	1	1	yes	characteristic
33	#17988362	C₂²	1	2	yes	—
34	#17988362	C₂²	1	2	yes	—
35	C₄×#11924003	C₂²	1	1	yes	characteristic
36	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
37	C₈×D₄×C₇₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₇₃	C₂²	1	1	yes	characteristic
39	C₂²⋊C₁₆×C₇₃	C₂²	1	2	yes	—
40	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
41	C₂×#11924027	C₈	1	1	yes	characteristic
42	C₂²×#10596831	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
45	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
46	#14556556	C₈	1	1	yes	characteristic
47	D₄×Dic₇₃	C₈	1	1	yes	characteristic
48	C₄×C₂²⋊D₇₃	C₂×C₄	1	2	yes	—
49	#14556551	C₈	1	2	yes	—
50	#14556549	C₈	1	1	yes	characteristic
51	#14556547	C₈	1	2	yes	—
52	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
53	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
54	#14556530	—	4	2	no	—
55	C₂×#11924003	—	4	2	no	—
56	C₂×#11924003	C₂³	1	1	yes	characteristic
57	C₂×#11924003	C₂³	1	1	yes	characteristic
58	C₂×#11924003	C₂³	1	2	yes	—
59	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
60	#14556520	C₂×C₄	1	2	yes	—
61	#14556520	C₂×C₄	1	2	yes	—
62	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
63	#14556503	C₂×C₄	1	2	yes	—
64	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
67	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
68	C₈×Dic₇₃	C₂³	1	2	yes	—
69	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
71	C₄×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₁₆×C₇₃	—	4	2	no	—
73	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
75	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
77	C₂²⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
78	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
79	#11924027	—	2	4	no	—
80	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
81	C₂×C₂²⋊D₇₃	C₂×C₈	1	2	yes	—
82	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
83	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
84	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×#10596831	—	4	2	no	—
88	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
89	C₂×#10596831	C₂²×C₄	1	2	yes	—
90	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
91	C₈×D₇₃	—	2	1	no	—
92	C₈×D₇₃	—	2	1	no	—
93	C₈×D₇₃	—	4	1	no	—
94	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
95	#11924007	C₂×C₈	1	2	yes	—
96	#11924007	C₂×C₈	1	2	yes	—
97	C₂²⋊Dic₇₃	C₂×C₈	1	2	yes	—
98	#11924005	C₂×C₈	1	2	yes	—
99	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
100	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
101	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
102	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
103	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
104	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
105	#11924003	—	2	1	no	—
106	#11924003	—	2	2	no	—
107	#11924003	—	2	1	no	—
108	C₂×D₄×C₇₃	C₂×C₈	1	1	yes	characteristic
109	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₇₃	—	2	2	no	—
111	C₂×C₈×C₇₃	—	2	2	no	—
112	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
116	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
117	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
118	C₁₆×C₇₃	—	2	1	no	—
119	C₁₆×C₇₃	—	2	2	no	—
120	C₁₆×C₇₃	—	2	1	no	—
121	C₂³×C₇₃	C₂²×C₈	1	2	yes	—
122	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
123	D₄×C₇₃	M₃₂	1	4	yes	complemented
124	C₂×C₄×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×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
130	C₂²⋊D₇₃	—	2	4	no	—
131	C₂×Dic₇₃	—	2	4	no	—
132	C₂×Dic₇₃	—	2	4	no	—
133	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
135	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
136	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
137	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
138	C₄×D₇₃	—	2	1	no	—
139	C₄×D₇₃	—	2	1	no	—
140	C₄×D₇₃	—	2	2	no	—
141	C₄×D₇₃	—	2	2	no	—
142	Dic₁₄₆	—	2	2	no	—
143	C₈×C₇₃	—	2	2	no	—
144	C₈×C₇₃	—	2	1	no	—
145	C₈×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
146	C₈×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
147	#10596831	—	2	1	no	—
148	#10596831	—	2	1	no	—
149	#10596831	—	2	2	no	—
150	C₂²×C₇₃	—	2	2	no	—
151	C₂²×C₇₃	—	2	2	no	—
152	C₂²×C₇₃	C₂×M₃₂	1	4	yes	—
153	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
154	C₂×D₇₃	—	2	1	no	—
155	C₂×D₇₃	—	2	1	no	—
156	C₂×D₇₃	—	2	1	no	—
157	C₄×C₇₃	—	2	1	no	—
158	C₄×C₇₃	—	2	2	no	—
159	C₄×C₇₃	C₂×M₃₂	1	2	yes	—
160	C₄×C₇₃	#520	1	1	yes	characteristic
161	C₄×C₇₃	#568	1	1	yes	characteristic
162	Dic₇₃	—	2	2	no	—
163	Dic₇₃	—	2	1	no	—
164	Dic₇₃	—	2	2	no	—
165	Dic₇₃	—	2	1	no	—
166	Dic₇₃	#504	1	2	yes	—
167	#33504	—	73	1	no	maximal, Sylow
168	C₂×C₇₃	—	2	4	no	—
169	C₂×C₇₃	#2988	1	1	yes	characteristic
170	C₂×C₇₃	C₂²×M₃₂	1	1	yes	characteristic
171	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
172	D₇₃	—	4	1	no	—
173	C₈×#41	—	73	1	no	—
174	#2183	—	73	1	no	—
175	#2183	—	73	1	no	—
176	#2168	—	73	2	no	—
177	#2168	—	73	2	no	—
178	D₄×C₁₆	—	73	1	no	—
179	D₄×C₁₆	D₇₃	1	1	yes	characteristic, complemented
180	#2163	—	73	2	no	—
181	#2162	—	73	1	no	—
182	#2150	—	73	1	no	—
183	C₂×C₄⋊C₁₆	—	73	2	no	—
184	C₇₃	#33504	1	1	yes	characteristic, complemented, Sylow
185	C₄×#41	—	73	1	no	—
186	C₂×M₃₂	—	73	2	no	—
187	C₂²×C₁₆	—	73	2	no	—
188	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
189	C₈×Q₈	—	73	1	no	—
190	C₈×Q₈	—	73	1	no	—
191	C₈×D₄	—	73	2	no	—
192	C₈×D₄	—	73	1	no	—
193	C₈×D₄	—	73	2	no	—
194	C₈×D₄	C₂×D₇₃	1	1	yes	characteristic
195	#431	—	73	1	no	—
196	#431	—	73	2	no	—
197	C₂×C₄×C₈	—	73	2	no	—
198	C₂×C₄×C₈	—	73	1	no	—
199	C₄⋊C₁₆	—	73	1	no	—
200	C₄⋊C₁₆	—	73	2	no	—
201	C₄⋊C₁₆	—	73	2	no	—
202	C₄⋊C₁₆	—	73	2	no	—
203	C₄⋊C₁₆	—	73	1	no	—
204	C₄⋊C₁₆	—	73	1	no	—
205	C₄⋊C₁₆	C₂×D₇₃	1	1	yes	characteristic
206	C₂²⋊C₁₆	—	73	2	no	—
207	C₂²⋊C₁₆	—	73	1	no	—
208	C₂²⋊C₁₆	—	73	1	no	—
209	C₂²⋊C₁₆	C₂×D₇₃	1	2	yes	—
210	#346	—	73	2	no	—
211	#346	—	73	2	no	—
212	C₄×C₁₆	—	73	1	no	—
213	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
214	C₂×#41	—	73	1	no	—
215	C₂²×C₈	—	73	2	no	—
216	C₂²×C₈	—	73	1	no	—
217	C₂²×C₈	—	73	1	no	—
218	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
219	C₄×Q₈	—	73	1	no	—
220	C₄×Q₈	—	73	1	no	—
221	C₄×D₄	—	73	1	no	—
222	C₄×D₄	—	73	2	no	—
223	C₄×D₄	—	73	2	no	—
224	C₄×D₄	C₄×D₇₃	1	1	yes	characteristic
225	#117	—	73	1	no	—
226	#117	—	73	2	no	—
227	C₂×C₄²	—	73	2	no	—
228	C₂×C₄²	—	73	1	no	—
229	M₃₂	—	292	2	no	—
230	C₂×C₁₆	—	4	2	no	—
231	C₂×C₁₆	—	292	2	no	—
232	C₂×C₁₆	—	73	1	no	—
233	C₂×C₁₆	—	73	1	no	—
234	C₂×C₁₆	—	73	2	no	—
235	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
237	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
238	C₄⋊C₈	—	73	2	no	—
239	C₄⋊C₈	—	73	2	no	—
240	C₄⋊C₈	—	73	1	no	—
241	C₄⋊C₈	C₄×D₇₃	1	1	yes	characteristic
242	C₂²⋊C₈	—	73	2	no	—
243	C₂²⋊C₈	—	73	2	no	—
244	C₂²⋊C₈	C₄×D₇₃	1	2	yes	—
245	C₄×C₈	—	73	2	no	—
246	C₄×C₈	—	73	1	no	—
247	C₄×C₈	—	73	2	no	—
248	C₄×C₈	—	73	2	no	—
249	C₄×C₈	—	73	1	no	—
250	C₄×C₈	—	73	1	no	—
251	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
252	#41	—	146	4	no	—
253	C₂×Q₈	—	73	1	no	—
254	C₂×D₄	—	73	2	no	—
255	C₂×D₄	C₈×D₇₃	1	1	yes	characteristic
256	C₂²×C₄	—	73	2	no	—
257	C₂²×C₄	—	73	2	no	—
258	C₂²×C₄	—	73	1	no	—
259	C₂²×C₄	—	73	1	no	—
260	C₂²×C₄	—	73	1	no	—
261	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	292	1	no	—
265	C₂×C₈	—	292	2	no	—
266	C₂×C₈	—	73	1	no	—
267	C₂×C₈	—	73	1	no	—
268	C₂×C₈	—	73	2	no	—
269	C₂×C₈	—	146	1	no	—
270	C₂×C₈	—	146	1	no	—
271	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
272	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
273	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
274	C₄⋊C₄	—	73	2	no	—
275	C₄⋊C₄	—	73	2	no	—
276	C₄⋊C₄	—	73	1	no	—
277	C₄⋊C₄	C₈×D₇₃	1	1	yes	characteristic
278	C₂²⋊C₄	—	73	2	no	—
279	C₂²⋊C₄	—	73	2	no	—
280	C₂²⋊C₄	C₈×D₇₃	1	2	yes	—
281	C₄²	—	73	2	no	—
282	C₄²	—	73	1	no	—
283	C₄²	—	73	1	no	—
284	C₄²	—	73	2	no	—
285	C₄²	—	73	1	no	—
286	C₄²	—	73	2	no	—
287	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
288	C₁₆	—	2	1	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	—	2	1	no	—
291	C₁₆	—	146	1	no	—
292	C₁₆	—	146	2	no	—
293	C₁₆	—	146	1	no	—
294	C₂³	—	73	1	no	—
295	C₂³	C₂×C₈×D₇₃	1	2	yes	—
296	Q₈	—	146	2	no	—
297	D₄	—	146	4	no	—
298	D₄	#14556530	1	4	yes	complemented
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	73	1	no	—
302	C₂×C₄	—	73	1	no	—
303	C₂×C₄	—	73	2	no	—
304	C₂×C₄	—	73	1	no	—
305	C₂×C₄	—	73	2	no	—
306	C₂×C₄	—	146	2	no	—
307	C₂×C₄	—	146	4	no	—
308	C₂×C₄	—	146	4	no	—
309	C₂×C₄	—	146	2	no	—
310	C₂×C₄	—	146	1	no	—
311	C₂×C₄	—	146	1	no	—
312	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
313	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
314	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
316	C₈	—	2	2	no	—
317	C₈	—	2	1	no	—
318	C₈	—	146	1	no	—
319	C₈	—	146	1	no	—
320	C₈	—	146	2	no	—
321	C₈	#14556636	1	1	yes	characteristic
322	C₈	#14556638	1	1	yes	characteristic
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	146	1	no	—
326	C₂²	—	146	1	no	—
327	C₂²	—	146	1	no	—
328	C₂²	C₂×#14556530	1	4	yes	—
329	C₂²	C₂²×C₈×D₇₃	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	—	73	2	no	—
333	C₄	—	146	1	no	—
334	C₄	—	146	2	no	—
335	C₄	—	146	2	no	—
336	C₄	—	146	1	no	—
337	C₄	C₂×#14556530	1	2	yes	—
338	C₄	#17989066	1	1	yes	characteristic
339	C₄	#17989352	1	1	yes	characteristic
340	C₂	—	2	4	no	—
341	C₂	—	292	1	no	—
342	C₂	#22619038	1	1	yes	characteristic
343	C₂	C₂²×#14556530	1	1	yes	characteristic
344	C₂	#504×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₃
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	D₄×C₁₆×C₇₃	C₂
prime radical	C₂×#11924027	C₈
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	D₄×C₁₆	D₇₃
hypocenter	C₇₃	#33504
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22611839	C₂	semidirect
3. #22611836	C₂	semidirect
4. #22611756	C₂	semidirect
5. #22611753	C₂	semidirect
6. D₄×#11924003	C₂	semidirect
7. #22611731	C₂	semidirect
8. #22611728	C₂	semidirect
9. #22611685	C₂	semidirect
10. C₂×#17988405	C₂	semidirect
11. D₄×C₁₆×C₇₃	C₂	semidirect
123. D₄×C₇₃	M₃₂	semidirect
179. D₄×C₁₆	D₇₃	semidirect
184. C₇₃	#33504	semidirect
298. D₄	#14556530	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11924027	2
#22611839	2
#22611836	2
#22611756	2
#22611753	2
D₄×#11924003	2
#22611731	2
#22611728	2
#22611685	2
C₂×#17988405	2
D₄×C₁₆×C₇₃	2
#33504	73