Subgroups — #34012363

Order 18688.

514 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989550	C₂	1	1	yes	maximal, characteristic
2	#22626156	C₂	1	1	yes	maximal, characteristic
3	C₂×#17989474	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×#64408	C₂	1	1	yes	maximal, characteristic, complemented
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×D₇₃⋊M₁₆	C₄	1	1	yes	characteristic
9	C₂×#14556670	C₄	1	1	yes	characteristic
10	C₂×#14556664	C₂²	1	1	yes	characteristic
11	C₂×#14556664	C₂²	1	1	yes	characteristic
12	#17989550	—	2	2	no	—
13	#17989550	—	2	2	no	—
14	C₂×#14556645	C₄	1	1	yes	characteristic
15	C₂×#14556645	C₄	1	1	yes	characteristic
16	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
17	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
18	#17989482	C₄	1	2	yes	complemented
19	C₂²⋊#11924031	C₄	1	2	yes	complemented
20	#17989474	—	2	2	no	—
21	C₁₆×#64408	—	2	2	no	—
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×#11924038	C₈	1	2	yes	—
30	D₇₃⋊M₁₆	—	2	2	no	—
31	D₇₃⋊M₁₆	—	2	2	no	—
32	C₂×#11924037	C₈	1	2	yes	—
33	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
34	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
35	#14556670	—	2	2	no	—
36	#14556670	—	2	2	no	—
37	C₂×C₄⋊#64408	C₈	1	2	yes	—
38	C₂×C₂²⋊#64408	C₈	1	2	yes	—
39	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
41	#14556664	—	2	1	no	—
42	#14556664	—	2	1	no	—
43	#14556664	—	2	1	no	—
44	#14556664	—	2	1	no	—
45	C₂×#11924031	C₂×C₄	1	2	yes	—
46	C₂×#11924031	C₂×C₄	1	2	yes	—
47	#14556645	—	2	2	no	—
48	#14556645	—	2	2	no	—
49	C₈×#64408	—	2	1	no	—
50	C₈×#64408	—	2	1	no	—
51	C₈×#64408	D₄	1	2	yes	—
52	C₈×#64408	D₄	1	2	yes	—
53	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₇₃	—	2	1	no	—
56	C₂×C₈×D₇₃	—	2	1	no	—
57	C₂×C₈×D₇₃	—	2	1	no	—
58	C₂×C₈×D₇₃	—	2	1	no	—
59	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
63	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
64	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
65	C₁₆×D₇₃	—	2	1	no	—
66	C₁₆×D₇₃	—	2	1	no	—
67	C₁₆×D₇₃	—	2	1	no	—
68	C₁₆×D₇₃	—	2	1	no	—
69	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
72	C₂²×#64408	C₂×C₈	1	2	yes	—
73	C₂²×#64408	C₂×C₈	1	2	yes	—
74	#11924038	—	2	4	no	—
75	#11924037	—	2	4	no	—
76	#11924036	—	2	1	no	—
77	#11924036	—	2	1	no	—
78	#11924036	C₂²⋊C₄	1	2	yes	—
79	#11924036	M₁₆	1	2	yes	—
80	C₂×#10596833	C₂×C₈	1	2	yes	—
81	C₂×#10596833	C₂×C₈	1	2	yes	—
82	C₄⋊#64408	—	2	4	no	—
83	C₂²⋊#64408	—	2	4	no	—
84	C₄×#64408	—	2	1	no	—
85	C₄×#64408	—	2	1	no	—
86	C₄×#64408	C₂²⋊C₄	1	2	yes	—
87	C₄×#64408	M₁₆	1	2	yes	—
88	#11924031	—	2	2	no	—
89	#11924031	—	2	2	no	—
90	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
91	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
92	C₂×C₄×D₇₃	—	2	1	no	—
93	C₂×C₄×D₇₃	—	2	1	no	—
94	C₂×C₄×D₇₃	—	2	1	no	—
95	C₂×C₄×D₇₃	—	2	1	no	—
96	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
97	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
98	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
99	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×#10596831	—	2	1	no	—
101	C₂×#10596831	—	2	1	no	—
102	C₂×#10596831	C₄²	1	1	yes	characteristic
103	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
104	C₈×D₇₃	—	2	1	no	—
105	C₈×D₇₃	—	2	1	no	—
106	C₈×D₇₃	—	2	1	no	—
107	C₈×D₇₃	—	2	1	no	—
108	C₈×D₇₃	—	2	1	no	—
109	C₈×D₇₃	—	2	1	no	—
110	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
111	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
112	C₈×D₇₃	#41	1	1	yes	characteristic
113	C₈×D₇₃	#41	1	1	yes	characteristic
114	#11924003	—	2	1	no	—
115	#11924003	—	2	1	no	—
116	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
117	C₂×C₈×C₇₃	—	2	1	no	—
118	C₂×C₈×C₇₃	—	2	1	no	—
119	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
120	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
121	C₁₆×C₇₃	—	2	1	no	—
122	C₁₆×C₇₃	—	2	1	no	—
123	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
124	C₂²×D₇₃	—	2	1	no	—
125	C₂²×D₇₃	—	2	1	no	—
126	C₂²×D₇₃	—	2	2	no	—
127	C₂²×D₇₃	—	2	2	no	—
128	C₂²×D₇₃	C₄×C₈	1	1	yes	characteristic
129	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×#64408	—	2	2	no	—
131	C₂×#64408	—	2	2	no	—
132	C₂×#64408	—	2	2	no	—
133	C₂×#64408	—	2	2	no	—
134	C₂×#64408	—	2	2	no	—
135	C₂×#64408	C₂²⋊C₈	1	4	yes	—
136	C₂×C₄×C₇₃	—	2	1	no	—
137	C₂×C₄×C₇₃	—	2	1	no	—
138	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
139	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
140	C₂×Dic₇₃	—	2	1	no	—
141	C₂×Dic₇₃	—	2	1	no	—
142	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
143	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
144	C₄×D₇₃	—	2	1	no	—
145	C₄×D₇₃	—	2	1	no	—
146	C₄×D₇₃	—	2	1	no	—
147	C₄×D₇₃	—	2	1	no	—
148	C₄×D₇₃	—	2	1	no	—
149	C₄×D₇₃	—	2	1	no	—
150	C₄×D₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₄×D₇₃	#117	1	1	yes	characteristic
152	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
153	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
154	#10596833	—	2	2	no	—
155	#10596833	C₂²⋊C₈	1	4	yes	—
156	C₈×C₇₃	—	2	1	no	—
157	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
158	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
159	#10596831	—	2	1	no	—
160	#10596831	C₄×D₄	1	1	yes	characteristic
161	#10596831	C₄×D₄	1	1	yes	characteristic
162	C₂²×C₇₃	—	2	1	no	—
163	C₂²×C₇₃	—	2	1	no	—
164	C₂²×C₇₃	—	2	1	no	—
165	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
166	C₂×D₇₃	—	2	1	no	—
167	C₂×D₇₃	—	2	1	no	—
168	C₂×D₇₃	—	2	1	no	—
169	C₂×D₇₃	—	4	2	no	—
170	C₂×D₇₃	—	4	2	no	—
171	C₂×D₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
172	C₂×D₇₃	#431	1	1	yes	characteristic
173	C₂×D₇₃	#504	1	2	yes	—
174	C₂×D₇₃	#504	1	2	yes	—
175	#64408	—	2	4	no	—
176	#64408	—	4	2	no	—
177	C₄×C₇₃	—	2	1	no	—
178	C₄×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
179	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
180	Dic₇₃	—	2	1	no	—
181	Dic₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
182	Dic₇₃	#431	1	1	yes	characteristic
183	#12600	—	73	1	no	maximal, Sylow
184	C₂×C₇₃	—	4	1	no	—
185	C₂×C₇₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
186	C₂×C₇₃	#2108	1	1	yes	characteristic
187	C₂×C₇₃	#2108	1	1	yes	characteristic
188	D₇₃	—	4	1	no	—
189	D₇₃	#2113	1	2	yes	—
190	D₇₃	#2164	1	2	yes	—
191	C₂×#405	—	73	1	no	—
192	C₂×C₂²⋊C₁₆	—	73	1	no	—
193	C₂×C₂²⋊C₁₆	—	73	1	no	—
194	C₂×#346	—	73	1	no	—
195	C₂×C₄×C₁₆	—	73	1	no	—
196	#1374	—	73	1	no	—
197	#1374	—	73	1	no	—
198	C₇₃	#12600	1	1	yes	characteristic, complemented, Sylow
199	C₂²×M₁₆	—	73	1	no	—
200	C₂³×C₈	—	73	1	no	—
201	C₂×#117	—	73	1	no	—
202	C₂²×C₁₆	—	73	1	no	—
203	C₂²×C₁₆	—	73	1	no	—
204	C₂²×C₁₆	—	73	1	no	—
205	C₂²×C₁₆	—	73	1	no	—
206	#405	—	146	2	no	—
207	#405	—	146	2	no	—
208	C₂×#97	—	73	1	no	—
209	C₂×#97	—	73	1	no	—
210	C₂×C₄×C₈	—	73	1	no	—
211	C₂×C₄×C₈	—	73	1	no	—
212	C₂²⋊C₁₆	—	73	2	no	—
213	C₂²⋊C₁₆	—	73	2	no	—
214	C₂²⋊C₁₆	—	73	1	no	—
215	C₂²⋊C₁₆	—	73	1	no	—
216	C₂²⋊C₁₆	—	73	1	no	—
217	C₂²⋊C₁₆	#64408	1	1	yes	characteristic, complemented
218	#346	—	146	2	no	—
219	C₄×C₁₆	—	146	2	no	—
220	C₂³×C₄	—	73	1	no	—
221	C₂×M₁₆	—	73	2	no	—
222	C₂×M₁₆	—	73	2	no	—
223	C₂×M₁₆	—	146	2	no	—
224	C₂×M₁₆	—	146	2	no	—
225	C₂²×C₈	—	73	1	no	—
226	C₂²×C₈	—	73	1	no	—
227	C₂²×C₈	—	73	1	no	—
228	C₂²×C₈	—	73	1	no	—
229	C₂²×C₈	—	73	1	no	—
230	C₂²×C₈	—	73	1	no	—
231	C₂²×C₈	—	73	1	no	—
232	C₂²×C₈	—	146	1	no	—
233	C₂²×C₈	—	146	1	no	—
234	C₂²×C₈	—	146	1	no	—
235	C₂²×C₈	—	146	1	no	—
236	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
237	#117	—	146	2	no	—
238	#117	—	146	2	no	—
239	C₂×C₄⋊C₄	—	73	2	no	—
240	C₂×C₂²⋊C₄	—	73	2	no	—
241	C₂×C₄²	—	73	1	no	—
242	C₂×C₄²	—	73	1	no	—
243	C₂×C₁₆	—	73	2	no	—
244	C₂×C₁₆	—	73	2	no	—
245	C₂×C₁₆	—	73	1	no	—
246	C₂×C₁₆	—	73	1	no	—
247	C₂×C₁₆	—	146	1	no	—
248	C₂×C₁₆	—	146	1	no	—
249	C₂×C₁₆	—	146	1	no	—
250	C₂×C₁₆	—	146	1	no	—
251	C₂×C₁₆	—	146	1	no	—
252	C₂×C₁₆	—	146	1	no	—
253	C₂×C₁₆	—	146	1	no	—
254	C₂×C₁₆	—	146	1	no	—
255	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
256	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
257	#97	—	146	2	no	—
258	#97	—	146	2	no	—
259	C₄×C₈	—	73	2	no	—
260	C₄×C₈	—	73	2	no	—
261	C₄×C₈	—	146	1	no	—
262	C₄×C₈	—	146	1	no	—
263	C₂⁴	—	73	1	no	—
264	C₂²×C₄	—	73	1	no	—
265	C₂²×C₄	—	73	2	no	—
266	C₂²×C₄	—	73	1	no	—
267	C₂²×C₄	—	73	1	no	—
268	C₂²×C₄	—	73	1	no	—
269	C₂²×C₄	—	73	1	no	—
270	C₂²×C₄	—	73	2	no	—
271	C₂²×C₄	—	146	1	no	—
272	C₂²×C₄	—	146	1	no	—
273	C₂²×C₄	—	146	1	no	—
274	C₂²×C₄	—	146	1	no	—
275	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
276	M₁₆	—	146	4	no	—
277	M₁₆	—	146	4	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	73	2	no	—
281	C₂×C₈	—	73	2	no	—
282	C₂×C₈	—	73	2	no	—
283	C₂×C₈	—	73	1	no	—
284	C₂×C₈	—	73	1	no	—
285	C₂×C₈	—	73	1	no	—
286	C₂×C₈	—	73	1	no	—
287	C₂×C₈	—	73	1	no	—
288	C₂×C₈	—	73	2	no	—
289	C₂×C₈	—	73	1	no	—
290	C₂×C₈	—	146	1	no	—
291	C₂×C₈	—	146	1	no	—
292	C₂×C₈	—	146	1	no	—
293	C₂×C₈	—	146	1	no	—
294	C₂×C₈	—	146	1	no	—
295	C₂×C₈	—	146	1	no	—
296	C₂×C₈	—	146	1	no	—
297	C₂×C₈	—	146	1	no	—
298	C₂×C₈	—	146	1	no	—
299	C₂×C₈	—	146	1	no	—
300	C₂×C₈	C₄×#64408	1	1	yes	characteristic
301	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
302	C₄⋊C₄	—	146	4	no	—
303	C₂²⋊C₄	—	146	4	no	—
304	C₄²	—	73	2	no	—
305	C₄²	—	73	2	no	—
306	C₄²	—	146	1	no	—
307	C₄²	—	146	1	no	—
308	C₁₆	—	2	1	no	—
309	C₁₆	—	2	1	no	—
310	C₁₆	—	146	2	no	—
311	C₁₆	—	146	2	no	—
312	C₁₆	—	146	1	no	—
313	C₁₆	—	146	1	no	—
314	C₂³	—	73	1	no	—
315	C₂³	—	73	1	no	—
316	C₂³	—	146	1	no	—
317	C₂³	—	146	1	no	—
318	C₂³	—	146	2	no	—
319	C₂³	—	146	2	no	—
320	C₂³	C₈×#64408	1	1	yes	characteristic
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	73	1	no	—
324	C₂×C₄	—	73	1	no	—
325	C₂×C₄	—	73	1	no	—
326	C₂×C₄	—	73	1	no	—
327	C₂×C₄	—	73	1	no	—
328	C₂×C₄	—	73	4	no	—
329	C₂×C₄	—	73	1	no	—
330	C₂×C₄	—	146	1	no	—
331	C₂×C₄	—	146	1	no	—
332	C₂×C₄	—	146	1	no	—
333	C₂×C₄	—	146	1	no	—
334	C₂×C₄	—	146	2	no	—
335	C₂×C₄	—	146	2	no	—
336	C₂×C₄	—	146	2	no	—
337	C₂×C₄	—	146	1	no	—
338	C₂×C₄	—	146	1	no	—
339	C₂×C₄	—	146	2	no	—
340	C₂×C₄	—	146	2	no	—
341	C₂×C₄	—	146	1	no	—
342	C₂×C₄	—	146	1	no	—
343	C₂×C₄	C₈×#64408	1	1	yes	characteristic
344	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	73	4	no	—
347	C₈	—	73	1	no	—
348	C₈	—	73	1	no	—
349	C₈	—	146	2	no	—
350	C₈	—	146	1	no	—
351	C₈	D₄×#64408	1	1	yes	characteristic
352	C₈	D₄×#64408	1	1	yes	characteristic
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	292	2	no	—
357	C₂²	—	292	2	no	—
358	C₂²	—	73	2	no	—
359	C₂²	—	73	2	no	—
360	C₂²	—	73	1	no	—
361	C₂²	—	73	1	no	—
362	C₂²	—	146	1	no	—
363	C₂²	—	146	1	no	—
364	C₂²	—	146	1	no	—
365	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	292	2	no	—
368	C₄	—	73	1	no	—
369	C₄	—	73	1	no	—
370	C₄	—	146	4	no	—
371	C₄	—	146	1	no	—
372	C₄	C₂²⋊C₄×#64408	1	1	yes	characteristic
373	C₄	M₁₆×#64408	1	1	yes	characteristic
374	C₂	—	4	1	no	—
375	C₂	—	292	1	no	—
376	C₂	—	73	2	no	—
377	C₂	—	73	2	no	—
378	C₂	C₂²⋊C₈×#64408	1	1	yes	characteristic
379	C₂	#22626139	1	1	yes	characteristic
380	C₂	#22626143	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	#12600
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
3. C₂×#17989474	C₂	semidirect
4. C₂×C₁₆×#64408	C₂	semidirect
5. #22624968	C₂	semidirect
6. #22624968	C₂	semidirect
18. #17989482	C₄	semidirect
19. C₂²⋊#11924031	C₄	semidirect
25. C₂²⋊#11924003	C₄	semidirect
26. #17988366	C₄	semidirect
27. #17988366	C₄	semidirect
28. C₂²⋊C₁₆×C₇₃	C₄	semidirect
198. C₇₃	#12600	semidirect
217. 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×#17989550	2
#22626156	2
C₂×#17989474	2
C₂×C₁₆×#64408	2
#22624968	2
C₂²⋊C₁₆×D₇₃	2
#12600	73