Subgroups — #34012426

Order 18688.

474 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989643	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989474	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×#64408	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₂²⋊#64408	C₂	1	1	yes	maximal, characteristic
5	#22624968	C₂	1	1	yes	maximal, characteristic, complemented
6	#22624968	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556666	C₄	1	2	yes	complemented
9	#17989643	—	2	2	no	—
10	#17989643	—	2	2	no	—
11	C₂×#14556665	C₄	1	2	yes	complemented
12	C₂×#14556664	C₂²	1	1	yes	characteristic
13	C₂×#14556664	C₂²	1	1	yes	characteristic
14	#17989555	C₄	1	1	yes	characteristic
15	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
16	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
17	C₄×C₂²⋊#64408	C₄	1	1	yes	characteristic
18	#17989474	—	2	2	no	—
19	C₁₆×#64408	—	2	2	no	—
20	#17989452	C₄	1	1	yes	characteristic
21	#17989452	C₄	1	1	yes	characteristic
22	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
23	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
24	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
25	C₂²⋊#11924003	C₄	1	1	yes	characteristic, complemented
26	#17988366	C₄	1	1	yes	characteristic, complemented
27	#17988366	C₄	1	1	yes	characteristic, complemented
28	C₂²⋊C₁₆×C₇₃	C₄	1	1	yes	characteristic, complemented
29	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
30	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
31	C₂×C₂²⋊#64408	C₈	1	2	yes	—
32	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
34	#14556666	—	2	4	no	—
35	#14556665	—	2	4	no	—
36	#14556664	—	2	1	no	—
37	#14556664	—	2	1	no	—
38	#14556664	D₄	1	2	yes	—
39	#14556664	D₄	1	2	yes	—
40	C₂×#11924031	C₂×C₄	1	2	yes	—
41	C₂×#11924031	C₂×C₄	1	2	yes	—
42	#14556648	C₈	1	2	yes	—
43	C₂²⋊#10596833	C₈	1	2	yes	—
44	C₈×#64408	—	2	1	no	—
45	C₈×#64408	—	2	1	no	—
46	C₈×#64408	—	2	1	no	—
47	C₈×#64408	—	2	1	no	—
48	(C₂×#64408)⋊C₄	C₈	1	2	yes	—
49	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₇₃	—	2	1	no	—
52	C₂×C₈×D₇₃	—	2	1	no	—
53	C₂×C₈×D₇₃	—	2	1	no	—
54	C₂×C₈×D₇₃	—	2	1	no	—
55	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
59	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
60	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
61	C₁₆×D₇₃	—	2	1	no	—
62	C₁₆×D₇₃	—	2	1	no	—
63	C₁₆×D₇₃	—	2	1	no	—
64	C₁₆×D₇₃	—	2	1	no	—
65	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×#64408	C₂×C₈	1	2	yes	—
69	C₂²×#64408	C₂×C₈	1	2	yes	—
70	#11924036	—	2	1	no	—
71	#11924036	—	2	1	no	—
72	#11924036	—	2	1	no	—
73	#11924036	—	2	1	no	—
74	C₂×#10596833	C₂×C₈	1	2	yes	—
75	C₂×#10596833	C₂×C₈	1	2	yes	—
76	C₂²⋊#64408	—	2	4	no	—
77	C₄×#64408	—	2	1	no	—
78	C₄×#64408	—	2	1	no	—
79	C₄×#64408	—	2	1	no	—
80	C₄×#64408	—	2	1	no	—
81	#11924031	—	2	2	no	—
82	#11924031	C₂²⋊C₄	1	4	yes	complemented
83	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
84	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₇₃	—	2	1	no	—
86	C₂×C₄×D₇₃	—	2	1	no	—
87	C₂×C₄×D₇₃	—	2	1	no	—
88	C₂×C₄×D₇₃	—	2	1	no	—
89	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
90	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
91	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
92	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×#10596831	—	2	1	no	—
94	C₂×#10596831	—	2	1	no	—
95	C₂×#10596831	C₄²	1	1	yes	characteristic
96	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
97	C₈×D₇₃	—	2	1	no	—
98	C₈×D₇₃	—	2	1	no	—
99	C₈×D₇₃	—	2	1	no	—
100	C₈×D₇₃	—	2	1	no	—
101	C₈×D₇₃	—	2	1	no	—
102	C₈×D₇₃	—	2	1	no	—
103	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
104	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
105	C₈×D₇₃	#41	1	1	yes	characteristic
106	C₈×D₇₃	#41	1	1	yes	characteristic
107	#11924003	—	2	1	no	—
108	#11924003	—	2	1	no	—
109	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
110	C₂×C₈×C₇₃	—	2	1	no	—
111	C₂×C₈×C₇₃	—	2	1	no	—
112	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
113	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₁₆×C₇₃	—	2	1	no	—
115	C₁₆×C₇₃	—	2	1	no	—
116	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
117	C₂²×D₇₃	—	2	1	no	—
118	C₂²×D₇₃	—	2	1	no	—
119	C₂²×D₇₃	—	2	2	no	—
120	C₂²×D₇₃	—	2	2	no	—
121	C₂²×D₇₃	C₄×C₈	1	1	yes	characteristic
122	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×#64408	—	2	2	no	—
124	C₂×#64408	—	2	2	no	—
125	C₂×#64408	—	2	2	no	—
126	C₂×#64408	—	2	2	no	—
127	C₂×#64408	—	2	2	no	—
128	C₂×#64408	M₃₂	1	4	yes	complemented
129	C₂×C₄×C₇₃	—	2	1	no	—
130	C₂×C₄×C₇₃	—	2	1	no	—
131	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
133	C₂×Dic₇₃	—	2	1	no	—
134	C₂×Dic₇₃	—	2	1	no	—
135	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
136	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
137	C₄×D₇₃	—	2	1	no	—
138	C₄×D₇₃	—	2	1	no	—
139	C₄×D₇₃	—	2	1	no	—
140	C₄×D₇₃	—	2	1	no	—
141	C₄×D₇₃	—	2	1	no	—
142	C₄×D₇₃	—	2	1	no	—
143	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
144	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
145	C₄×D₇₃	#131	1	1	yes	characteristic
146	C₄×D₇₃	#131	1	1	yes	characteristic
147	#10596833	—	2	2	no	—
148	#10596833	—	2	2	no	—
149	C₈×C₇₃	—	2	1	no	—
150	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
151	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
152	#10596831	—	2	1	no	—
153	#10596831	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	#10596831	#117	1	1	yes	characteristic
155	C₂²×C₇₃	—	2	1	no	—
156	C₂²×C₇₃	—	2	1	no	—
157	C₂²×C₇₃	—	2	1	no	—
158	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
159	C₂×D₇₃	—	2	1	no	—
160	C₂×D₇₃	—	2	1	no	—
161	C₂×D₇₃	—	2	1	no	—
162	C₂×D₇₃	—	4	2	no	—
163	C₂×D₇₃	—	4	2	no	—
164	C₂×D₇₃	C₈×D₄	1	1	yes	characteristic
165	C₂×D₇₃	C₈×D₄	1	1	yes	characteristic
166	C₂×D₇₃	C₂×M₃₂	1	2	yes	—
167	C₂×D₇₃	#504	1	2	yes	—
168	#64408	—	2	4	no	—
169	#64408	—	4	2	no	—
170	C₄×C₇₃	—	2	1	no	—
171	C₄×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
172	C₄×C₇₃	#405	1	1	yes	characteristic
173	Dic₇₃	—	2	1	no	—
174	Dic₇₃	C₈×D₄	1	1	yes	characteristic
175	Dic₇₃	C₈×D₄	1	1	yes	characteristic
176	#12612	—	73	1	no	maximal, Sylow
177	C₂×C₇₃	—	4	1	no	—
178	C₂×C₇₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
179	C₂×C₇₃	C₄×M₃₂	1	1	yes	characteristic
180	C₂×C₇₃	#2108	1	1	yes	characteristic
181	D₇₃	—	4	1	no	—
182	D₇₃	#2168	1	2	yes	—
183	D₇₃	#2169	1	2	yes	—
184	C₂²×M₃₂	—	73	1	no	—
185	C₂×C₂²⋊C₁₆	—	73	1	no	—
186	C₂×#346	—	73	1	no	—
187	C₂×C₄×C₁₆	—	73	1	no	—
188	C₈×C₂²⋊C₄	—	73	1	no	—
189	#1374	—	73	1	no	—
190	#1374	—	73	1	no	—
191	C₇₃	#12612	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₈	—	73	1	no	—
193	C₂×M₃₂	—	73	2	no	—
194	C₂×M₃₂	—	73	2	no	—
195	C₂×M₃₂	—	146	2	no	—
196	C₂×M₃₂	—	146	2	no	—
197	C₂²×C₁₆	—	73	1	no	—
198	C₂²×C₁₆	—	73	1	no	—
199	C₂²×C₁₆	—	73	1	no	—
200	C₂²×C₁₆	—	73	1	no	—
201	C₂×C₂²⋊C₈	—	73	1	no	—
202	C₂×C₄×C₈	—	73	1	no	—
203	C₂×C₄×C₈	—	73	1	no	—
204	C₄×C₂²⋊C₄	—	73	1	no	—
205	C₂²⋊C₁₆	—	73	1	no	—
206	C₂²⋊C₁₆	—	73	1	no	—
207	C₂²⋊C₁₆	—	73	1	no	—
208	C₂²⋊C₁₆	#64408	1	1	yes	characteristic, complemented
209	#346	—	146	2	no	—
210	C₄×C₁₆	—	146	2	no	—
211	#336	—	73	1	no	—
212	#336	—	73	1	no	—
213	C₂³×C₄	—	73	1	no	—
214	C₂²×C₈	—	73	1	no	—
215	C₂²×C₈	—	73	1	no	—
216	C₂²×C₈	—	73	1	no	—
217	C₂²×C₈	—	73	1	no	—
218	C₂²×C₈	—	73	1	no	—
219	C₂²×C₈	—	73	1	no	—
220	C₂²×C₈	—	73	1	no	—
221	C₂²×C₈	—	146	1	no	—
222	C₂²×C₈	—	146	1	no	—
223	C₂²×C₈	—	146	1	no	—
224	C₂²×C₈	—	146	1	no	—
225	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
226	C₂×C₂²⋊C₄	—	73	2	no	—
227	C₂×C₄²	—	73	1	no	—
228	C₂×C₄²	—	73	1	no	—
229	M₃₂	—	146	4	no	—
230	M₃₂	—	146	4	no	—
231	C₂×C₁₆	—	73	2	no	—
232	C₂×C₁₆	—	73	2	no	—
233	C₂×C₁₆	—	73	2	no	—
234	C₂×C₁₆	—	73	2	no	—
235	C₂×C₁₆	—	73	1	no	—
236	C₂×C₁₆	—	73	1	no	—
237	C₂×C₁₆	—	146	1	no	—
238	C₂×C₁₆	—	146	1	no	—
239	C₂×C₁₆	—	146	1	no	—
240	C₂×C₁₆	—	146	1	no	—
241	C₂×C₁₆	—	146	1	no	—
242	C₂×C₁₆	—	146	1	no	—
243	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
244	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
245	C₂²⋊C₈	—	73	2	no	—
246	C₂²⋊C₈	—	73	2	no	—
247	C₄×C₈	—	146	1	no	—
248	C₄×C₈	—	146	1	no	—
249	C₄×C₈	—	146	1	no	—
250	C₄×C₈	—	146	1	no	—
251	#95	—	73	2	no	—
252	C₂⁴	—	73	1	no	—
253	C₂²×C₄	—	73	1	no	—
254	C₂²×C₄	—	73	2	no	—
255	C₂²×C₄	—	73	1	no	—
256	C₂²×C₄	—	73	1	no	—
257	C₂²×C₄	—	73	1	no	—
258	C₂²×C₄	—	73	1	no	—
259	C₂²×C₄	—	73	2	no	—
260	C₂²×C₄	—	146	1	no	—
261	C₂²×C₄	—	146	1	no	—
262	C₂²×C₄	—	146	1	no	—
263	C₂²×C₄	—	146	1	no	—
264	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	73	1	no	—
268	C₂×C₈	—	73	1	no	—
269	C₂×C₈	—	73	1	no	—
270	C₂×C₈	—	73	1	no	—
271	C₂×C₈	—	73	1	no	—
272	C₂×C₈	—	73	2	no	—
273	C₂×C₈	—	73	2	no	—
274	C₂×C₈	—	73	1	no	—
275	C₂×C₈	—	146	1	no	—
276	C₂×C₈	—	146	1	no	—
277	C₂×C₈	—	146	1	no	—
278	C₂×C₈	—	146	1	no	—
279	C₂×C₈	—	146	1	no	—
280	C₂×C₈	—	146	1	no	—
281	C₂×C₈	—	146	1	no	—
282	C₂×C₈	—	146	1	no	—
283	C₂×C₈	—	146	1	no	—
284	C₂×C₈	—	146	1	no	—
285	C₂×C₈	—	146	1	no	—
286	C₂×C₈	—	146	1	no	—
287	C₂×C₈	C₄×#64408	1	1	yes	characteristic
288	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
289	C₂²⋊C₄	—	146	4	no	—
290	C₄²	—	146	1	no	—
291	C₄²	—	146	1	no	—
292	C₄²	—	146	1	no	—
293	C₄²	—	146	1	no	—
294	C₁₆	—	2	1	no	—
295	C₁₆	—	2	1	no	—
296	C₁₆	—	73	4	no	—
297	C₁₆	—	146	2	no	—
298	C₁₆	—	146	1	no	—
299	C₁₆	—	146	1	no	—
300	C₂³	—	73	1	no	—
301	C₂³	—	73	1	no	—
302	C₂³	—	146	2	no	—
303	C₂³	—	146	1	no	—
304	C₂³	—	146	1	no	—
305	C₂³	—	146	2	no	—
306	C₂³	C₈×#64408	1	1	yes	characteristic
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	73	1	no	—
310	C₂×C₄	—	73	1	no	—
311	C₂×C₄	—	73	1	no	—
312	C₂×C₄	—	73	1	no	—
313	C₂×C₄	—	73	1	no	—
314	C₂×C₄	—	73	4	no	—
315	C₂×C₄	—	73	1	no	—
316	C₂×C₄	—	146	1	no	—
317	C₂×C₄	—	146	1	no	—
318	C₂×C₄	—	146	1	no	—
319	C₂×C₄	—	146	1	no	—
320	C₂×C₄	—	146	2	no	—
321	C₂×C₄	—	146	2	no	—
322	C₂×C₄	—	146	2	no	—
323	C₂×C₄	—	146	1	no	—
324	C₂×C₄	—	146	1	no	—
325	C₂×C₄	—	146	2	no	—
326	C₂×C₄	—	146	2	no	—
327	C₂×C₄	—	146	1	no	—
328	C₂×C₄	—	146	1	no	—
329	C₂×C₄	C₈×#64408	1	1	yes	characteristic
330	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	73	1	no	—
333	C₈	—	73	1	no	—
334	C₈	—	146	2	no	—
335	C₈	—	146	2	no	—
336	C₈	—	146	1	no	—
337	C₈	D₄×#64408	1	1	yes	characteristic
338	C₈	D₄×#64408	1	1	yes	characteristic
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	292	2	no	—
343	C₂²	—	292	2	no	—
344	C₂²	—	73	2	no	—
345	C₂²	—	73	2	no	—
346	C₂²	—	73	1	no	—
347	C₂²	—	73	1	no	—
348	C₂²	—	146	1	no	—
349	C₂²	—	146	1	no	—
350	C₂²	—	146	1	no	—
351	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	292	2	no	—
354	C₄	—	73	1	no	—
355	C₄	—	73	1	no	—
356	C₄	—	146	4	no	—
357	C₄	—	146	1	no	—
358	C₄	C₂²⋊C₄×#64408	1	1	yes	characteristic
359	C₄	#17989553	1	1	yes	characteristic
360	C₂	—	4	1	no	—
361	C₂	—	292	1	no	—
362	C₂	—	73	2	no	—
363	C₂	—	73	2	no	—
364	C₂	#22625666	1	1	yes	characteristic
365	C₂	M₃₂×#64408	1	1	yes	characteristic
366	C₂	#22626139	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₂²⋊C₁₆×C₇₃	C₄
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	C₂²⋊C₁₆	#64408
hypocenter	C₇₃	#12612
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#17989474	C₂	semidirect
3. C₂×C₁₆×#64408	C₂	semidirect
5. #22624968	C₂	semidirect
6. #22624968	C₂	semidirect
8. C₂×#14556666	C₄	semidirect
11. C₂×#14556665	C₄	semidirect
25. C₂²⋊#11924003	C₄	semidirect
26. #17988366	C₄	semidirect
27. #17988366	C₄	semidirect
28. C₂²⋊C₁₆×C₇₃	C₄	semidirect
82. #11924031	C₂²⋊C₄	semidirect
128. C₂×#64408	M₃₂	semidirect
191. C₇₃	#12612	semidirect
208. C₂²⋊C₁₆	#64408	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989643	2
C₂×#17989474	2
C₂×C₁₆×#64408	2
C₈×C₂²⋊#64408	2
#22624968	2
C₂²⋊C₁₆×D₇₃	2
#12612	73