Subgroups — #32097414

Order 18176.

608 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598863	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17986120	C₂	1	1	yes	maximal, characteristic
3	#22589515	C₂	1	2	yes	maximal, complemented
4	#22589511	C₂	1	2	yes	maximal, complemented
5	#22589238	C₂	1	1	yes	maximal, characteristic
6	#22589236	C₂	1	1	yes	maximal, characteristic
7	#22588465	C₂	1	1	yes	maximal, characteristic, complemented
8	#22588393	C₂	1	2	yes	maximal, complemented
9	#22588385	C₂	1	2	yes	maximal, complemented
10	#22588225	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₁×#1833	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556150	C₂²	1	1	yes	characteristic
13	C₂×#14556136	C₂²	1	2	yes	—
14	#17986552	C₂²	1	2	yes	—
15	#17986407	—	2	2	no	—
16	#17986406	—	2	2	no	—
17	#17986120	—	2	2	no	—
18	#17986120	—	2	2	no	—
19	C₂×#14556063	C₂²	1	1	yes	characteristic
20	#17986037	C₂²	1	1	yes	characteristic
21	#17986035	C₂²	1	1	yes	characteristic
22	#17986035	C₂²	1	2	yes	—
23	#17986017	C₂²	1	2	yes	—
24	#17986015	C₂²	1	2	yes	—
25	C₂×#14556054	C₂²	1	2	yes	—
26	#17985972	C₂²	1	2	yes	—
27	#17985972	C₂²	1	2	yes	—
28	#17985954	C₂²	1	1	yes	characteristic
29	C₂×#14556038	C₂²	1	1	yes	characteristic
30	#17985771	C₂²	1	2	yes	—
31	#17985760	C₂²	1	1	yes	characteristic
32	#17985758	C₂²	1	1	yes	characteristic
33	#17985756	C₂²	1	2	yes	—
34	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
35	#427×C₇₁	C₂²	1	1	yes	characteristic
36	#405×C₇₁	C₂²	1	1	yes	characteristic
37	#344×C₇₁	C₂²	1	2	yes	—
38	H(3,ℤ/4)×C₇₁	C₂²	1	2	yes	—
39	C₂×#11765865	C₂³	1	2	yes	—
40	#14556150	—	2	1	no	—
41	#14556150	—	2	1	no	—
42	#14556150	—	2	1	no	—
43	#14556150	—	2	1	no	—
44	C₂×Dic₂₈₄	Q₈	1	1	yes	characteristic, complemented
45	C₂×#11765854	Q₈	1	2	yes	complemented
46	C₂×D₅₆₈	Q₈	1	1	yes	characteristic, complemented
47	#14556136	—	2	2	no	—
48	#14556136	—	2	2	no	—
49	#14556136	—	2	2	no	—
50	#14556136	—	2	2	no	—
51	C₂×#11765849	C₂³	1	1	yes	characteristic
52	C₂×#11765848	—	2	2	no	—
53	C₂²×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
54	C₂×C₈×D₇₁	—	2	2	no	—
55	#14556127	—	2	2	no	—
56	#14556119	—	2	2	no	—
57	#14556105	—	4	2	no	—
58	C₄×Dic₁₄₂	—	4	2	no	—
59	#14556087	C₂³	1	1	yes	characteristic
60	#14556087	C₂³	1	2	yes	—
61	C₂×C₄×Dic₇₁	C₂³	1	1	yes	characteristic
62	#14556063	—	2	2	no	—
63	#14556062	—	2	1	no	—
64	#14556061	—	2	2	no	—
65	#14556061	—	2	1	no	—
66	#14556060	—	2	1	no	—
67	#14556059	—	2	2	no	—
68	#14556059	—	2	1	no	—
69	#14556056	—	4	2	no	—
70	#14556054	—	2	4	no	—
71	#14556054	—	4	2	no	—
72	#14556042	—	4	2	no	—
73	#14556042	—	4	2	no	—
74	#14556038	—	2	1	no	—
75	#14556038	D₄	1	2	yes	—
76	#14556038	D₄	1	2	yes	—
77	C₈×Dic₇₁	—	2	1	no	—
78	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
79	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
80	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
81	#117×C₇₁	C₂³	1	1	yes	characteristic
82	#117×C₇₁	C₂³	1	2	yes	—
83	#108×C₇₁	—	2	1	no	—
84	#108×C₇₁	—	2	1	no	—
85	#107×C₇₁	—	2	1	no	—
86	#106×C₇₁	—	2	1	no	—
87	#97×C₇₁	—	2	1	no	—
88	C₄×C₈×C₇₁	—	2	1	no	—
89	#11765865	—	2	2	no	—
90	#11765865	—	2	2	no	—
91	#11765865	—	2	2	no	—
92	#11765865	—	2	2	no	—
93	C₂×Dic₁₄₂	—	4	2	no	—
94	C₂×Dic₁₄₂	C₂×Q₈	1	2	yes	—
95	C₂×D₂₈₄	C₂×Q₈	1	2	yes	—
96	C₂×C₂²⋊D₇₁	—	2	2	no	—
97	C₂²×Dic₇₁	—	2	1	no	—
98	C₂×C₄×D₇₁	—	2	2	no	—
99	Dic₂₈₄	—	2	1	no	—
100	Dic₂₈₄	—	2	1	no	—
101	#11765854	—	2	2	no	—
102	#11765854	—	2	2	no	—
103	D₅₆₈	—	2	1	no	—
104	D₅₆₈	—	2	1	no	—
105	#11765849	—	2	1	no	—
106	#11765849	—	2	1	no	—
107	#11765848	—	4	2	no	—
108	#11765848	—	4	2	no	—
109	C₂×C₇₁⋊C₈	—	2	1	no	—
110	C₂×C₇₁⋊C₈	C₂×D₄	1	2	yes	—
111	C₂×C₇₁⋊C₈	#41	1	1	yes	characteristic
112	C₂×C₇₁⋊C₈	#41	1	1	yes	characteristic
113	C₂×C₇₁⋊C₈	#41	1	2	yes	—
114	C₈×D₇₁	—	4	2	no	—
115	C₈×D₇₁	—	4	2	no	—
116	C₄⋊Dic₇₁	—	2	2	no	—
117	C₄⋊Dic₇₁	—	2	1	no	—
118	C₄⋊Dic₇₁	—	4	2	no	—
119	Dic₇₁⋊C₄	—	4	2	no	—
120	Dic₇₁⋊C₄	—	4	2	no	—
121	C₂²⋊Dic₇₁	—	2	1	no	—
122	C₂²⋊Dic₇₁	—	2	2	no	—
123	C₄×Dic₇₁	—	2	1	no	—
124	C₄×Dic₇₁	—	2	1	no	—
125	C₄×Dic₇₁	—	2	2	no	—
126	C₄×Dic₇₁	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
128	M₁₆×C₇₁	—	2	2	no	—
129	M₁₆×C₇₁	—	2	1	no	—
130	M₁₆×C₇₁	—	2	2	no	—
131	M₁₆×C₇₁	—	2	1	no	—
132	C₂×C₈×C₇₁	—	2	1	no	—
133	C₂×C₈×C₇₁	—	2	2	no	—
134	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
137	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
138	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
139	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
140	C₄⋊C₄×C₇₁	—	2	2	no	—
141	C₄⋊C₄×C₇₁	—	2	1	no	—
142	C₂²⋊C₄×C₇₁	—	2	1	no	—
143	C₂²⋊C₄×C₇₁	—	2	2	no	—
144	C₄²×C₇₁	—	2	1	no	—
145	C₄²×C₇₁	—	2	2	no	—
146	C₂³×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂²×D₇₁	—	2	2	no	—
148	C₂×C₄×C₇₁	—	4	2	no	—
149	C₂×C₄×C₇₁	—	4	1	no	—
150	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
152	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²⋊D₇₁	—	4	2	no	—
157	C₂²⋊D₇₁	—	4	2	no	—
158	C₂×Dic₇₁	—	2	2	no	—
159	C₂×Dic₇₁	—	2	2	no	—
160	C₂×Dic₇₁	—	4	2	no	—
161	C₂×Dic₇₁	—	4	1	no	—
162	C₂×Dic₇₁	—	8	1	no	—
163	D₂₈₄	—	2	2	no	—
164	D₂₈₄	—	2	2	no	—
165	C₄×D₇₁	—	4	2	no	—
166	C₄×D₇₁	—	4	2	no	—
167	Dic₁₄₂	—	2	2	no	—
168	Dic₁₄₂	—	2	2	no	—
169	Dic₁₄₂	—	8	4	no	—
170	C₈×C₇₁	—	4	2	no	—
171	C₈×C₇₁	—	4	1	no	—
172	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
173	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
174	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
175	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
176	C₇₁⋊C₈	—	4	1	no	—
177	C₇₁⋊C₈	#124	1	2	yes	—
178	C₇₁⋊C₈	#124	1	2	yes	—
179	C₂²×C₇₁	—	4	1	no	—
180	C₂²×C₇₁	#535	1	1	yes	characteristic
181	C₂²×C₇₁	#554	1	1	yes	characteristic
182	C₂²×C₇₁	#555	1	1	yes	characteristic
183	C₂×D₇₁	—	4	2	no	—
184	C₂×D₇₁	—	8	2	no	—
185	C₄×C₇₁	—	8	2	no	—
186	C₄×C₇₁	—	8	1	no	—
187	C₄×C₇₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
188	C₄×C₇₁	C₂×#124	1	1	yes	characteristic
189	C₄×C₇₁	#554	1	1	yes	characteristic
190	C₄×C₇₁	#555	1	1	yes	characteristic
191	Dic₇₁	—	4	2	no	—
192	Dic₇₁	—	4	2	no	—
193	Dic₇₁	—	8	2	no	—
194	Dic₇₁	—	8	1	no	—
195	#29155	—	71	1	no	maximal, Sylow
196	C₂×C₇₁	—	2	1	no	—
197	C₂×C₇₁	—	2	1	no	—
198	C₂×C₇₁	—	2	1	no	—
199	C₂×C₇₁	#2563	1	1	yes	characteristic
200	D₇₁	—	8	2	no	—
201	#3162	—	71	1	no	—
202	C₂×#444	—	71	1	no	—
203	#2031	—	71	2	no	—
204	#2031	—	71	2	no	—
205	#1976	—	71	1	no	—
206	#1976	—	71	1	no	—
207	#1833	—	71	1	no	—
208	#1833	D₇₁	1	1	yes	characteristic, complemented
209	#1814	—	71	2	no	—
210	#1812	—	71	2	no	—
211	#1773	—	71	1	no	—
212	C₇₁	#29155	1	1	yes	characteristic, complemented, Sylow
213	C₂×#135	—	71	1	no	—
214	C₂×#131	—	71	2	no	—
215	#533	—	71	2	no	—
216	#501	—	142	2	no	—
217	#501	—	142	2	no	—
218	#444	—	142	2	no	—
219	#444	—	142	2	no	—
220	C₂×#108	—	71	1	no	—
221	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
222	#427	—	71	1	no	—
223	#427	—	71	1	no	—
224	#427	—	71	2	no	—
225	#427	C₂×D₇₁	1	1	yes	characteristic
226	#421	—	71	2	no	—
227	#421	—	71	2	no	—
228	C₂×#104	—	71	2	no	—
229	#410	—	71	2	no	—
230	#410	—	71	2	no	—
231	#405	—	71	1	no	—
232	#405	C₂×D₇₁	1	1	yes	characteristic
233	C₂×#97	—	71	1	no	—
234	#344	—	71	2	no	—
235	#344	C₂×D₇₁	1	2	yes	—
236	#339	—	71	1	no	—
237	#339	—	71	1	no	—
238	H(3,ℤ/4)	—	71	2	no	—
239	H(3,ℤ/4)	C₂×D₇₁	1	2	yes	—
240	C₂×#41	—	71	2	no	—
241	#135	—	142	1	no	—
242	#135	—	142	1	no	—
243	#135	—	142	1	no	—
244	#135	—	142	1	no	—
245	C₂×Q₁₆	—	71	1	no	—
246	C₂×SD₁₆	—	71	2	no	—
247	C₂×D₈	—	71	1	no	—
248	#131	—	142	2	no	—
249	#131	—	142	2	no	—
250	#131	—	142	2	no	—
251	#131	—	142	2	no	—
252	C₂×M₁₆	—	71	1	no	—
253	C₂×M₁₆	—	142	2	no	—
254	C₂×M₁₆	C₂²×D₇₁	1	2	yes	—
255	C₂×M₁₆	C₂²×D₇₁	1	1	yes	characteristic
256	C₂²×C₈	—	71	1	no	—
257	C₂²×C₈	—	142	2	no	—
258	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
259	C₄⋊Q₈	—	142	2	no	—
260	#125	—	142	2	no	—
261	C₂²⋊Q₈	—	284	2	no	—
262	C₄×Q₈	—	284	2	no	—
263	#117	—	71	2	no	—
264	#117	—	71	1	no	—
265	#117	C₂²×D₇₁	1	2	yes	—
266	#117	C₂²×D₇₁	1	1	yes	characteristic
267	C₂×C₄²	—	71	1	no	—
268	#108	—	2	1	no	—
269	#108	—	2	1	no	—
270	#108	—	142	2	no	—
271	#107	—	2	1	no	—
272	#107	—	142	1	no	—
273	#107	—	142	2	no	—
274	#107	—	142	1	no	—
275	#106	—	2	1	no	—
276	#106	—	142	1	no	—
277	#106	—	142	2	no	—
278	#106	—	142	1	no	—
279	#104	—	284	2	no	—
280	#104	—	284	2	no	—
281	#104	—	142	4	no	—
282	#99	—	284	2	no	—
283	#99	—	284	2	no	—
284	#97	—	2	1	no	—
285	#97	—	71	2	no	—
286	#97	—	71	2	no	—
287	#97	—	142	1	no	—
288	C₄×C₈	—	2	1	no	—
289	C₄×C₈	—	142	1	no	—
290	#41	—	142	2	no	—
291	#41	—	142	2	no	—
292	#41	—	142	2	no	—
293	#41	—	142	2	no	—
294	C₂×Q₈	—	284	2	no	—
295	C₂×Q₈	—	71	2	no	—
296	C₂×D₄	—	71	2	no	—
297	C₂×D₄	—	142	2	no	—
298	C₂²×C₄	—	142	2	no	—
299	C₂²×C₄	—	142	1	no	—
300	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
301	Q₁₆	—	142	1	no	—
302	Q₁₆	—	142	1	no	—
303	SD₁₆	—	142	2	no	—
304	SD₁₆	—	142	2	no	—
305	D₈	—	142	1	no	—
306	D₈	—	142	1	no	—
307	M₁₆	—	2	2	no	—
308	M₁₆	—	2	1	no	—
309	M₁₆	—	2	2	no	—
310	M₁₆	—	2	1	no	—
311	M₁₆	—	284	2	no	—
312	M₁₆	—	284	2	no	—
313	M₁₆	—	142	1	no	—
314	M₁₆	—	142	1	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	284	2	no	—
318	C₂×C₈	—	284	2	no	—
319	C₂×C₈	—	71	2	no	—
320	C₂×C₈	—	71	1	no	—
321	C₂×C₈	—	71	1	no	—
322	C₂×C₈	—	71	2	no	—
323	C₂×C₈	—	142	1	no	—
324	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
325	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
326	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
327	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
328	C₂×C₈	#11765865	1	1	yes	characteristic
329	C₂×C₈	#11765865	1	1	yes	characteristic
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	1	no	—
332	C₄⋊C₄	—	284	2	no	—
333	C₄⋊C₄	—	284	2	no	—
334	C₄⋊C₄	—	284	2	no	—
335	C₄⋊C₄	—	142	2	no	—
336	C₄⋊C₄	—	142	1	no	—
337	C₂²⋊C₄	—	2	1	no	—
338	C₂²⋊C₄	—	2	2	no	—
339	C₂²⋊C₄	—	142	1	no	—
340	C₂²⋊C₄	—	142	2	no	—
341	C₄²	—	2	1	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	71	2	no	—
344	C₄²	—	142	1	no	—
345	C₄²	—	142	2	no	—
346	C₄²	—	142	1	no	—
347	C₂³	—	142	2	no	—
348	C₂³	#14556174	1	1	yes	characteristic
349	Q₈	—	568	4	no	—
350	Q₈	—	142	2	no	—
351	Q₈	—	142	2	no	—
352	D₄	—	284	2	no	—
353	D₄	—	284	2	no	—
354	D₄	—	142	2	no	—
355	D₄	—	142	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	284	2	no	—
359	C₂×C₄	—	284	2	no	—
360	C₂×C₄	—	284	2	no	—
361	C₂×C₄	—	284	1	no	—
362	C₂×C₄	—	568	1	no	—
363	C₂×C₄	—	142	2	no	—
364	C₂×C₄	—	142	2	no	—
365	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
366	C₂×C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
367	C₂×C₄	#41×D₇₁	1	1	yes	characteristic
368	C₂×C₄	#41×D₇₁	1	1	yes	characteristic
369	C₂×C₄	C₂×#11765865	1	1	yes	characteristic
370	C₂×C₄	#14556174	1	1	yes	characteristic
371	C₈	—	4	1	no	—
372	C₈	—	4	2	no	—
373	C₈	—	284	1	no	—
374	C₈	—	71	2	no	—
375	C₈	—	71	2	no	—
376	C₈	#14556106	1	1	yes	characteristic
377	C₈	#14556106	1	1	yes	characteristic
378	C₈	#14556106	1	1	yes	characteristic
379	C₈	#14556106	1	1	yes	characteristic
380	C₂²	—	4	1	no	—
381	C₂²	—	284	2	no	—
382	C₂²	—	568	2	no	—
383	C₂²	#17986567	1	1	yes	characteristic
384	C₂²	#17986705	1	1	yes	characteristic
385	C₂²	#17986712	1	1	yes	characteristic
386	C₄	—	8	2	no	—
387	C₄	—	8	1	no	—
388	C₄	—	284	2	no	—
389	C₄	—	284	2	no	—
390	C₄	—	568	2	no	—
391	C₄	—	568	1	no	—
392	C₄	C₂×#14556106	1	1	yes	characteristic
393	C₄	#124×D₇₁	1	1	yes	characteristic
394	C₄	#17986705	1	1	yes	characteristic
395	C₄	#17986712	1	1	yes	characteristic
396	C₂	—	2	1	no	—
397	C₂	—	2	1	no	—
398	C₂	—	2	1	no	—
399	C₂	—	568	2	no	—
400	C₂	#22593512	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17986712
commutator, cosocle	C₂²×C₄×C₇₁	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₇₁×#1833	C₂
socle	C₂×C₇₁	#2563
prime radical	C₂×#14556150	C₂²
hypercenter	#1833	D₇₁
hypocenter	C₇₁	#29155
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1833

Complements

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

N	H ≅ G/N	type
1. #22598863	C₂	semidirect
3. #22589515	C₂	semidirect
4. #22589511	C₂	semidirect
7. #22588465	C₂	semidirect
8. #22588393	C₂	semidirect
9. #22588385	C₂	semidirect
10. #22588225	C₂	semidirect
11. C₇₁×#1833	C₂	semidirect
44. C₂×Dic₂₈₄	Q₈	semidirect
45. C₂×#11765854	Q₈	semidirect
46. C₂×D₅₆₈	Q₈	semidirect
208. #1833	D₇₁	semidirect
212. C₇₁	#29155	semidirect

Maximal subgroups

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

subgroup	index
#22598863	2
C₂×#17986120	2
#22589515	2
#22589511	2
#22589238	2
#22589236	2
#22588465	2
#22588393	2
#22588385	2
#22588225	2
C₇₁×#1833	2
#29155	71