Subgroups — #32985218

Order 18688.

586 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988960	C₂	1	1	yes	maximal, characteristic
2	#22610883	C₂	1	1	yes	maximal, characteristic, complemented
3	#22610858	C₂	1	1	yes	maximal, characteristic
4	C₂×#17988338	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17988334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988330	C₂	1	1	yes	maximal, characteristic
7	C₈×C₄⋊C₄×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11924017	—	2	1	no	—
9	C₂²×#11924017	—	2	1	no	—
10	C₂×#14556591	C₂²	1	1	yes	characteristic
11	C₂×#14556588	C₂²	1	1	yes	characteristic
12	#17988960	—	2	2	no	—
13	#17988960	C₄	1	4	yes	complemented
14	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
16	C₂×#14556511	—	2	2	no	—
17	#17988349	—	2	2	no	—
18	#17988338	—	2	2	no	—
19	#17988334	—	2	2	no	—
20	#17988330	—	2	2	no	—
21	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₇₃	—	2	1	no	—
23	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
25	#336×C₇₃	—	2	1	no	—
26	C₂²×Dic₁₄₆	—	2	1	no	—
27	C₂²×D₂₉₂	—	2	1	no	—
28	C₂²×D₂₉₂	—	2	1	no	—
29	C₂²×D₂₉₂	—	2	1	no	—
30	C₂×#11924017	—	2	4	no	—
31	C₂×#11924017	—	2	4	no	—
32	C₂×#11924017	—	4	2	no	—
33	C₂×#11924017	—	4	2	no	—
34	C₂×#11924017	—	4	2	no	—
35	C₂×#11924017	—	4	2	no	—
36	C₂²×#10596831	—	2	1	no	—
37	C₂²×#10596831	—	2	1	no	—
38	#14556591	—	2	1	no	—
39	#14556591	—	2	1	no	—
40	#14556591	—	2	1	no	—
41	#14556591	C₂×C₄	1	2	yes	—
42	#14556588	—	2	1	no	—
43	#14556588	—	2	1	no	—
44	#14556588	—	2	1	no	—
45	#14556588	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊Dic₇₃	—	2	1	no	—
47	C₂×C₄⋊Dic₇₃	—	2	1	no	—
48	C₄⋊#10596831	—	2	1	no	—
49	C₄⋊#10596831	—	2	1	no	—
50	C₄⋊#10596831	—	2	1	no	—
51	C₄⋊#10596831	—	2	1	no	—
52	#14556511	—	4	4	no	—
53	C₂²×C₈×C₇₃	—	2	1	no	—
54	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₇₃	D₄	1	2	yes	—
57	C₂×C₄²×C₇₃	—	2	1	no	—
58	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₇₃	—	2	1	no	—
60	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
61	C₄×C₈×C₇₃	—	4	2	no	—
62	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₇₃	D₄	1	2	yes	—
64	#95×C₇₃	—	2	1	no	—
65	C₂³×D₇₃	—	4	1	no	—
66	C₂×Dic₁₄₆	—	2	2	no	—
67	C₂×Dic₁₄₆	—	4	1	no	—
68	C₂×Dic₁₄₆	—	4	1	no	—
69	C₂×Dic₁₄₆	—	4	1	no	—
70	C₂×Dic₁₄₆	—	4	1	no	—
71	C₂×Dic₁₄₆	—	4	1	no	—
72	C₂×D₂₉₂	—	2	2	no	—
73	C₂×D₂₉₂	—	2	2	no	—
74	C₂×D₂₉₂	—	2	2	no	—
75	C₂×D₂₉₂	—	4	1	no	—
76	C₂×D₂₉₂	—	4	2	no	—
77	C₂×D₂₉₂	—	4	1	no	—
78	C₂×D₂₉₂	—	4	2	no	—
79	C₂×D₂₉₂	—	4	1	no	—
80	C₂×D₂₉₂	—	4	1	no	—
81	C₂×D₂₉₂	—	4	1	no	—
82	C₂×D₂₉₂	—	4	1	no	—
83	C₂×D₂₉₂	—	4	1	no	—
84	C₂×D₂₉₂	—	8	2	no	—
85	C₂²×Dic₇₃	—	4	1	no	—
86	#11924017	—	4	8	no	—
87	#11924017	—	4	8	no	—
88	C₂×#10596831	—	4	1	no	—
89	C₂×#10596831	—	4	1	no	—
90	C₂×#10596831	—	4	2	no	—
91	C₂×#10596831	—	4	2	no	—
92	C₄⋊Dic₇₃	—	2	2	no	—
93	C₄⋊Dic₇₃	—	2	2	no	—
94	C₄⋊Dic₇₃	—	4	1	no	—
95	C₄⋊Dic₇₃	—	4	1	no	—
96	C₂²×C₄×C₇₃	—	2	2	no	—
97	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
100	C₂×C₈×C₇₃	—	2	1	no	—
101	C₂×C₈×C₇₃	—	2	1	no	—
102	C₂×C₈×C₇₃	—	2	2	no	—
103	C₂×C₈×C₇₃	—	4	1	no	—
104	C₂×C₈×C₇₃	—	4	1	no	—
105	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
106	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
107	C₂×C₈×C₇₃	D₈	1	2	yes	—
108	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
109	C₄⋊C₄×C₇₃	—	2	2	no	—
110	C₄⋊C₄×C₇₃	SD₁₆	1	4	yes	complemented
111	C₄²×C₇₃	—	4	1	no	—
112	C₄²×C₇₃	—	4	1	no	—
113	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₇₃	#41	1	1	yes	characteristic
117	C₂³×C₇₃	#121	1	1	yes	characteristic
118	C₂²×D₇₃	—	4	2	no	—
119	C₂²×D₇₃	—	8	1	no	—
120	C₂²×D₇₃	—	8	1	no	—
121	C₂²×D₇₃	—	8	2	no	—
122	C₂²×D₇₃	—	8	2	no	—
123	C₂×C₄×C₇₃	—	2	2	no	—
124	C₂×C₄×C₇₃	—	2	4	no	—
125	C₂×C₄×C₇₃	—	4	2	no	—
126	C₂×C₄×C₇₃	—	4	2	no	—
127	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
132	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
136	C₂×C₄×C₇₃	#135	1	2	yes	—
137	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×Dic₇₃	—	4	2	no	—
140	C₂×Dic₇₃	—	8	1	no	—
141	C₂×Dic₇₃	—	8	1	no	—
142	D₂₉₂	—	4	2	no	—
143	D₂₉₂	—	4	2	no	—
144	D₂₉₂	—	4	4	no	—
145	D₂₉₂	—	8	2	no	—
146	D₂₉₂	—	8	2	no	—
147	D₂₉₂	—	8	1	no	—
148	D₂₉₂	—	8	1	no	—
149	D₂₉₂	—	8	1	no	—
150	D₂₉₂	—	8	1	no	—
151	Dic₁₄₆	—	4	2	no	—
152	Dic₁₄₆	—	4	2	no	—
153	Dic₁₄₆	—	8	1	no	—
154	Dic₁₄₆	—	8	1	no	—
155	C₈×C₇₃	—	2	2	no	—
156	C₈×C₇₃	—	4	2	no	—
157	C₈×C₇₃	#102	1	4	yes	complemented
158	#10596831	—	8	1	no	—
159	#10596831	—	8	1	no	—
160	C₂²×C₇₃	#390	1	1	yes	characteristic
161	C₂²×C₇₃	#459	1	1	yes	characteristic
162	C₂²×C₇₃	#460	1	1	yes	characteristic
163	C₂²×C₇₃	#465	1	2	yes	—
164	C₂²×C₇₃	#471	1	2	yes	—
165	C₂×D₇₃	—	8	2	no	—
166	C₂×D₇₃	—	8	2	no	—
167	C₂×D₇₃	—	8	2	no	—
168	C₂×D₇₃	—	16	2	no	—
169	C₂×D₇₃	—	16	2	no	—
170	C₄×C₇₃	—	2	1	no	—
171	C₄×C₇₃	—	2	1	no	—
172	C₄×C₇₃	—	4	4	no	—
173	C₄×C₇₃	C₂×#102	1	1	yes	characteristic
174	C₄×C₇₃	#418	1	1	yes	characteristic
175	C₄×C₇₃	C₄×SD₁₆	1	2	yes	—
176	C₄×C₇₃	#459	1	1	yes	characteristic
177	C₄×C₇₃	#460	1	1	yes	characteristic
178	C₄×C₇₃	#492	1	1	yes	characteristic
179	C₄×C₇₃	#495	1	1	yes	characteristic
180	Dic₇₃	—	8	2	no	—
181	#9732	—	73	1	no	maximal, Sylow
182	C₂×C₇₃	#1665	1	2	yes	—
183	C₂×C₇₃	#1666	1	2	yes	—
184	C₂×C₇₃	#1968	1	1	yes	characteristic
185	C₂×C₇₃	#1977	1	1	yes	characteristic
186	C₂×C₇₃	#1982	1	1	yes	characteristic
187	D₇₃	—	16	2	no	—
188	C₂×#492	—	73	1	no	—
189	#1988	—	73	1	no	—
190	#1982	—	73	1	no	—
191	C₈×C₄⋊C₄	D₇₃	1	1	yes	characteristic, complemented
192	C₂×#334	—	73	1	no	—
193	C₂×#332	—	73	1	no	—
194	C₂×#331	—	73	1	no	—
195	C₇₃	#9732	1	1	yes	characteristic, complemented, Sylow
196	C₂²×SD₁₆	—	146	1	no	—
197	C₂²×SD₁₆	—	146	1	no	—
198	C₂×C₄⋊Q₈	—	73	1	no	—
199	C₂×#127	—	73	1	no	—
200	#492	—	73	4	no	—
201	#492	—	146	2	no	—
202	C₂×C₄⋊C₈	—	73	1	no	—
203	C₂×C₄⋊C₈	—	73	1	no	—
204	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
205	C₂×#102	—	146	2	no	—
206	C₂×C₄×C₈	—	2	1	no	—
207	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
208	C₄×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
209	#340	—	146	2	no	—
210	#336	—	2	1	no	—
211	#334	—	146	2	no	—
212	#332	—	146	2	no	—
213	#331	—	146	2	no	—
214	C₂²×Q₈	—	146	1	no	—
215	C₂²×D₄	—	146	1	no	—
216	C₂²×D₄	—	146	1	no	—
217	C₂²×D₄	—	146	1	no	—
218	C₂×SD₁₆	—	292	2	no	—
219	C₂×SD₁₆	—	292	2	no	—
220	C₂×SD₁₆	—	292	2	no	—
221	C₂×SD₁₆	—	292	2	no	—
222	C₂×SD₁₆	—	146	4	no	—
223	C₂×SD₁₆	—	146	4	no	—
224	C₂²×C₈	—	2	1	no	—
225	C₂²×C₈	—	146	1	no	—
226	C₂²×C₈	—	146	1	no	—
227	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
228	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
229	C₄⋊Q₈	—	73	2	no	—
230	C₄⋊Q₈	—	146	1	no	—
231	C₄⋊Q₈	—	146	1	no	—
232	C₄⋊Q₈	—	146	1	no	—
233	#127	—	73	2	no	—
234	#127	—	146	1	no	—
235	#127	—	146	1	no	—
236	#127	—	146	1	no	—
237	C₂×C₄⋊C₄	—	146	1	no	—
238	C₂×C₄⋊C₄	—	146	1	no	—
239	C₂×C₄⋊C₄	D₂₉₂	1	2	yes	—
240	C₂×C₄²	—	2	1	no	—
241	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	146	1	no	—
244	C₄⋊C₈	—	146	1	no	—
245	C₄⋊C₈	—	146	1	no	—
246	C₄⋊C₈	—	146	1	no	—
247	C₄⋊C₈	D₂₉₂	1	2	yes	—
248	#102	—	292	4	no	—
249	C₄×C₈	—	4	2	no	—
250	C₄×C₈	C₄×D₇₃	1	2	yes	—
251	C₄×C₈	D₂₉₂	1	2	yes	—
252	#95	—	2	1	no	—
253	C₂⁴	—	292	1	no	—
254	C₂×Q₈	—	292	1	no	—
255	C₂×Q₈	—	292	1	no	—
256	C₂×Q₈	—	292	1	no	—
257	C₂×Q₈	—	292	1	no	—
258	C₂×Q₈	—	292	1	no	—
259	C₂×Q₈	—	146	2	no	—
260	C₂×D₄	—	292	1	no	—
261	C₂×D₄	—	292	1	no	—
262	C₂×D₄	—	292	1	no	—
263	C₂×D₄	—	292	1	no	—
264	C₂×D₄	—	292	1	no	—
265	C₂×D₄	—	292	2	no	—
266	C₂×D₄	—	292	2	no	—
267	C₂×D₄	—	292	1	no	—
268	C₂×D₄	—	292	1	no	—
269	C₂×D₄	—	584	2	no	—
270	C₂×D₄	—	146	2	no	—
271	C₂×D₄	—	146	2	no	—
272	C₂×D₄	—	146	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	292	1	no	—
275	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
277	C₂²×C₄	#11924028	1	1	yes	characteristic
278	SD₁₆	—	292	8	no	—
279	SD₁₆	—	292	8	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	292	1	no	—
286	C₂×C₈	—	292	1	no	—
287	C₂×C₈	—	292	2	no	—
288	C₂×C₈	—	292	2	no	—
289	C₂×C₈	#11924005	1	2	yes	—
290	C₂×C₈	#11924005	1	2	yes	—
291	C₂×C₈	#11924013	1	2	yes	—
292	C₂×C₈	#11924015	1	2	yes	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	292	1	no	—
295	C₄⋊C₄	—	292	1	no	—
296	C₄⋊C₄	—	146	2	no	—
297	C₄⋊C₄	—	146	2	no	—
298	C₄⋊C₄	#11924017	1	4	yes	complemented
299	C₄²	—	4	1	no	—
300	C₄²	—	4	1	no	—
301	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
302	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
303	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
304	C₄²	#11924028	1	1	yes	characteristic
305	C₂³	—	292	2	no	—
306	C₂³	—	584	1	no	—
307	C₂³	—	584	1	no	—
308	C₂³	—	584	2	no	—
309	C₂³	—	584	2	no	—
310	C₂³	#14556565	1	1	yes	characteristic
311	Q₈	—	292	2	no	—
312	Q₈	—	292	2	no	—
313	Q₈	—	584	1	no	—
314	Q₈	—	584	1	no	—
315	D₄	—	292	4	no	—
316	D₄	—	292	2	no	—
317	D₄	—	292	2	no	—
318	D₄	—	584	2	no	—
319	D₄	—	584	2	no	—
320	D₄	—	584	1	no	—
321	D₄	—	584	1	no	—
322	D₄	—	584	1	no	—
323	D₄	—	584	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	292	2	no	—
329	C₂×C₄	—	584	1	no	—
330	C₂×C₄	—	584	1	no	—
331	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
332	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
333	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
334	C₂×C₄	#14556565	1	1	yes	characteristic
335	C₂×C₄	#14556579	1	1	yes	characteristic
336	C₂×C₄	#14556588	1	1	yes	characteristic
337	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
338	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
339	C₂×C₄	C₂×#11924017	1	2	yes	—
340	C₂×C₄	#14556613	1	2	yes	—
341	C₂×C₄	#14556616	1	1	yes	characteristic
342	C₂×C₄	#14556616	1	1	yes	characteristic
343	C₈	—	2	2	no	—
344	C₈	—	4	2	no	—
345	C₈	—	584	1	no	—
346	C₈	—	584	1	no	—
347	C₈	#14556511	1	4	yes	complemented
348	C₂²	—	584	2	no	—
349	C₂²	—	584	2	no	—
350	C₂²	—	584	2	no	—
351	C₂²	—	1168	2	no	—
352	C₂²	—	1168	2	no	—
353	C₂²	#17988489	1	1	yes	characteristic
354	C₂²	#17988782	1	1	yes	characteristic
355	C₂²	#17988787	1	1	yes	characteristic
356	C₂²	#17988816	1	2	yes	—
357	C₂²	#17988850	1	2	yes	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	4	4	no	—
361	C₄	—	584	2	no	—
362	C₄	C₂×#14556511	1	1	yes	characteristic
363	C₄	#17988591	1	1	yes	characteristic
364	C₄	C₄×#11924017	1	2	yes	—
365	C₄	#17988782	1	1	yes	characteristic
366	C₄	#17988787	1	1	yes	characteristic
367	C₄	#17988960	1	1	yes	characteristic
368	C₄	#17988971	1	1	yes	characteristic
369	C₂	—	1168	2	no	—
370	C₂	#22609272	1	2	yes	—
371	C₂	#22609279	1	2	yes	—
372	C₂	#22610785	1	1	yes	characteristic
373	C₂	#22610834	1	1	yes	characteristic
374	C₂	#22610858	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556565
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₈×C₄⋊C₄×C₇₃	C₂
socle	C₂³×C₇₃	#121
prime radical	C₂×#14556588	C₂²
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₈×C₄⋊C₄	D₇₃
hypocenter	C₇₃	#9732
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #22610883	C₂	semidirect
4. C₂×#17988338	C₂	semidirect
5. C₂×#17988334	C₂	semidirect
7. C₈×C₄⋊C₄×C₇₃	C₂	semidirect
13. #17988960	C₄	semidirect
110. C₄⋊C₄×C₇₃	SD₁₆	semidirect
157. C₈×C₇₃	#102	semidirect
191. C₈×C₄⋊C₄	D₇₃	semidirect
195. C₇₃	#9732	semidirect
298. C₄⋊C₄	#11924017	semidirect
347. C₈	#14556511	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988960	2
#22610883	2
#22610858	2
C₂×#17988338	2
C₂×#17988334	2
C₂×#17988330	2
C₈×C₄⋊C₄×C₇₃	2
#9732	73