Subgroups — #32097407

Order 18176.

612 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598854	C₂	1	1	yes	maximal, characteristic, complemented
2	#22589512	C₂	1	4	yes	maximal, complemented
3	#22589239	C₂	1	2	yes	maximal
4	#22588465	C₂	1	2	yes	maximal, complemented
5	#22588394	C₂	1	2	yes	maximal, complemented
6	#22588386	C₂	1	2	yes	maximal, complemented
7	#22588219	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#443×C₇₁	C₂	1	1	yes	maximal, characteristic
9	#17986556	C₂²	1	2	yes	—
10	#17986404	—	2	1	no	—
11	#17986401	—	2	1	no	—
12	#17986401	—	2	1	no	—
13	#17986397	—	2	1	no	—
14	C₂×#14556064	C₂²	1	1	yes	characteristic
15	#17986037	C₂²	1	2	yes	—
16	#17986037	C₂²	1	1	yes	characteristic
17	#17986037	C₂²	1	2	yes	—
18	#17986016	C₂²	1	4	yes	—
19	#17985973	C₂²	1	4	yes	—
20	#17985954	C₂²	1	2	yes	—
21	#17985771	C₂²	1	4	yes	—
22	#17985759	C₂²	1	2	yes	—
23	#17985756	C₂²	1	4	yes	—
24	C₂×#135×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×#131×C₇₁	C₂²	1	2	yes	—
26	#443×C₇₁	—	2	1	no	—
27	#443×C₇₁	—	2	1	no	—
28	#443×C₇₁	—	2	1	no	—
29	#443×C₇₁	—	2	1	no	—
30	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
31	C₂×#104×C₇₁	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
33	C₂×#11765849	C₂³	1	2	yes	—
34	#14556128	—	2	2	no	—
35	#14556120	—	2	2	no	—
36	#14556107	—	4	2	no	—
37	C₄×Dic₁₄₂	—	4	2	no	—
38	#14556087	C₂³	1	2	yes	—
39	#14556087	C₂³	1	4	yes	—
40	#14556064	—	2	1	no	—
41	#14556064	—	2	1	no	—
42	#14556062	—	2	2	no	—
43	#14556062	—	2	2	no	—
44	#14556062	—	2	1	no	—
45	#14556060	—	2	2	no	—
46	#14556060	—	2	2	no	—
47	#14556060	—	2	1	no	—
48	#14556055	—	4	4	no	—
49	#14556043	—	4	4	no	—
50	#14556038	—	2	2	no	—
51	C₈×Dic₇₁	—	2	2	no	—
52	C₂×#41×C₇₁	C₂³	1	2	yes	—
53	#135×C₇₁	—	2	1	no	—
54	#135×C₇₁	—	2	1	no	—
55	#135×C₇₁	—	2	1	no	—
56	#135×C₇₁	—	2	1	no	—
57	C₂×Q₁₆×C₇₁	Q₈	1	1	yes	characteristic, complemented
58	C₂×SD₁₆×C₇₁	Q₈	1	2	yes	complemented
59	C₂×D₈×C₇₁	Q₈	1	1	yes	characteristic, complemented
60	#131×C₇₁	—	2	2	no	—
61	#131×C₇₁	—	2	2	no	—
62	#131×C₇₁	—	2	2	no	—
63	#131×C₇₁	—	2	2	no	—
64	C₂×M₁₆×C₇₁	—	2	2	no	—
65	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
66	C₂²×C₈×C₇₁	—	2	2	no	—
67	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
70	#108×C₇₁	—	2	1	no	—
71	#108×C₇₁	—	2	1	no	—
72	#104×C₇₁	—	2	2	no	—
73	#104×C₇₁	—	2	2	no	—
74	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
75	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
76	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
77	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
78	C₂×Dic₁₄₂	—	4	2	no	—
79	#11765849	—	2	2	no	—
80	#11765849	—	2	2	no	—
81	C₂×C₇₁⋊C₈	—	2	2	no	—
82	C₄⋊Dic₇₁	—	2	4	no	—
83	C₄⋊Dic₇₁	—	2	2	no	—
84	C₄⋊Dic₇₁	—	4	2	no	—
85	Dic₇₁⋊C₄	—	4	4	no	—
86	C₂²⋊Dic₇₁	—	2	2	no	—
87	C₂²⋊Dic₇₁	—	2	4	no	—
88	C₄×Dic₇₁	—	2	2	no	—
89	C₄×Dic₇₁	—	2	4	no	—
90	#41×C₇₁	—	2	2	no	—
91	#41×C₇₁	—	2	2	no	—
92	#41×C₇₁	—	2	2	no	—
93	#41×C₇₁	—	2	2	no	—
94	C₂×Q₈×C₇₁	C₂×Q₈	1	2	yes	—
95	C₂×D₄×C₇₁	—	2	2	no	—
96	C₂×D₄×C₇₁	C₂×Q₈	1	2	yes	—
97	C₂²×C₄×C₇₁	—	2	1	no	—
98	C₂²×C₄×C₇₁	—	2	2	no	—
99	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
100	Q₁₆×C₇₁	—	2	1	no	—
101	Q₁₆×C₇₁	—	2	1	no	—
102	SD₁₆×C₇₁	—	2	2	no	—
103	SD₁₆×C₇₁	—	2	2	no	—
104	D₈×C₇₁	—	2	1	no	—
105	D₈×C₇₁	—	2	1	no	—
106	M₁₆×C₇₁	—	2	2	no	—
107	M₁₆×C₇₁	—	2	2	no	—
108	M₁₆×C₇₁	—	4	2	no	—
109	M₁₆×C₇₁	—	4	2	no	—
110	C₂×C₈×C₇₁	—	2	2	no	—
111	C₂×C₈×C₇₁	—	4	2	no	—
112	C₂×C₈×C₇₁	—	4	2	no	—
113	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
118	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
119	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
120	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
121	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
122	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
123	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
124	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
125	C₄²×C₇₁	—	2	1	no	—
126	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₇₁	—	2	2	no	—
129	C₂³×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
130	Q₈×C₇₁	—	2	2	no	—
131	Q₈×C₇₁	—	2	2	no	—
132	D₄×C₇₁	—	2	2	no	—
133	D₄×C₇₁	—	2	2	no	—
134	D₄×C₇₁	—	4	2	no	—
135	D₄×C₇₁	—	4	2	no	—
136	C₂×C₄×C₇₁	—	2	1	no	—
137	C₂×C₄×C₇₁	—	2	1	no	—
138	C₂×C₄×C₇₁	—	2	2	no	—
139	C₂×C₄×C₇₁	—	4	2	no	—
140	C₂×C₄×C₇₁	—	4	2	no	—
141	C₂×C₄×C₇₁	—	4	2	no	—
142	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×Dic₇₁	—	4	4	no	—
149	C₂×Dic₇₁	—	4	2	no	—
150	Dic₁₄₂	—	8	4	no	—
151	C₈×C₇₁	—	4	2	no	—
152	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
153	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
154	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
155	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
156	C₈×C₇₁	#124	1	1	yes	characteristic
157	C₈×C₇₁	#124	1	1	yes	characteristic
158	C₈×C₇₁	#124	1	1	yes	characteristic
159	C₈×C₇₁	#124	1	1	yes	characteristic
160	C₇₁⋊C₈	—	4	2	no	—
161	C₂²×C₇₁	—	4	1	no	—
162	C₂²×C₇₁	—	4	2	no	—
163	C₂²×C₇₁	—	8	2	no	—
164	C₂²×C₇₁	#535	1	1	yes	characteristic
165	C₂²×C₇₁	#554	1	1	yes	characteristic
166	C₂²×C₇₁	#555	1	1	yes	characteristic
167	C₄×C₇₁	—	2	2	no	—
168	C₄×C₇₁	—	4	1	no	—
169	C₄×C₇₁	—	4	2	no	—
170	C₄×C₇₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
171	C₄×C₇₁	C₂×#124	1	1	yes	characteristic
172	C₄×C₇₁	#554	1	1	yes	characteristic
173	C₄×C₇₁	#555	1	1	yes	characteristic
174	Dic₇₁	—	8	4	no	—
175	Dic₇₁	—	8	2	no	—
176	#29154	—	71	1	no	maximal, Sylow
177	C₂×C₇₁	—	2	1	no	—
178	C₂×C₇₁	—	2	1	no	—
179	C₂×C₇₁	—	2	1	no	—
180	C₂×C₇₁	—	8	2	no	—
181	C₂×C₇₁	#2563	1	1	yes	characteristic
182	#3160	—	71	1	no	—
183	C₂×#443	D₇₁	1	1	yes	characteristic
184	#2031	—	71	4	no	—
185	#1976	—	71	2	no	—
186	#1833	—	71	2	no	—
187	#1814	—	71	2	no	—
188	#1812	—	71	2	no	—
189	#1772	—	71	1	no	—
190	C₇₁	#29154	1	1	yes	characteristic, complemented, Sylow
191	C₂×#135	C₂×D₇₁	1	1	yes	characteristic
192	C₂×#131	C₂×D₇₁	1	2	yes	—
193	#533	—	71	2	no	—
194	#500	—	142	1	no	—
195	#499	—	142	1	no	—
196	#499	—	142	1	no	—
197	#498	—	142	1	no	—
198	#443	—	2	1	no	—
199	#443	—	2	1	no	—
200	#443	—	2	1	no	—
201	#443	—	2	1	no	—
202	C₂×#108	—	71	1	no	—
203	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
204	#427	—	71	1	no	—
205	#427	—	71	2	no	—
206	#427	—	71	2	no	—
207	#421	—	71	4	no	—
208	C₂×#104	C₂×D₇₁	1	2	yes	—
209	#410	—	71	4	no	—
210	#405	—	71	2	no	—
211	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
212	#344	—	71	4	no	—
213	#339	—	71	2	no	—
214	H(3,ℤ/4)	—	71	4	no	—
215	C₂×#41	C₂²×D₇₁	1	2	yes	—
216	#135	—	2	1	no	—
217	#135	—	2	1	no	—
218	#135	—	2	1	no	—
219	#135	—	2	1	no	—
220	C₂×Q₁₆	Dic₁₄₂	1	1	yes	characteristic, complemented
221	C₂×SD₁₆	Dic₁₄₂	1	2	yes	complemented
222	C₂×D₈	Dic₁₄₂	1	1	yes	characteristic, complemented
223	#131	—	2	2	no	—
224	#131	—	2	2	no	—
225	#131	—	2	2	no	—
226	#131	—	2	2	no	—
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	—	71	2	no	—
229	C₂×M₁₆	C₂²×D₇₁	1	2	yes	—
230	C₂²×C₈	—	2	2	no	—
231	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
233	C₄⋊Q₈	—	142	2	no	—
234	#125	—	142	2	no	—
235	C₂²⋊Q₈	—	284	2	no	—
236	C₄×Q₈	—	284	2	no	—
237	#117	—	71	4	no	—
238	#117	—	71	2	no	—
239	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
240	#108	—	2	1	no	—
241	#108	—	2	1	no	—
242	#108	—	142	1	no	—
243	#108	—	142	1	no	—
244	#107	—	142	2	no	—
245	#107	—	142	2	no	—
246	#107	—	142	1	no	—
247	#106	—	142	2	no	—
248	#106	—	142	2	no	—
249	#106	—	142	1	no	—
250	#104	—	2	2	no	—
251	#104	—	2	2	no	—
252	#104	—	284	4	no	—
253	#99	—	284	4	no	—
254	#97	—	142	2	no	—
255	C₄×C₈	—	142	2	no	—
256	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
257	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
258	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
259	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
260	#41	—	2	2	no	—
261	#41	—	2	2	no	—
262	#41	—	2	2	no	—
263	#41	—	2	2	no	—
264	C₂×Q₈	—	284	2	no	—
265	C₂×Q₈	C₂×Dic₁₄₂	1	2	yes	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	C₂×Dic₁₄₂	1	2	yes	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	2	1	no	—
273	SD₁₆	—	2	2	no	—
274	SD₁₆	—	2	2	no	—
275	D₈	—	2	1	no	—
276	D₈	—	2	1	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	4	2	no	—
280	M₁₆	—	4	2	no	—
281	M₁₆	—	142	2	no	—
282	M₁₆	—	142	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	2	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	—	142	2	no	—
287	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
288	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
289	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
290	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
291	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
292	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
293	C₂×C₈	#11765864	1	1	yes	characteristic
294	C₂×C₈	#11765864	1	1	yes	characteristic
295	C₂×C₈	#11765864	1	1	yes	characteristic
296	C₂×C₈	#11765864	1	1	yes	characteristic
297	C₂×C₈	#11765865	1	1	yes	characteristic
298	C₂×C₈	#11765865	1	1	yes	characteristic
299	C₄⋊C₄	—	284	4	no	—
300	C₄⋊C₄	—	284	2	no	—
301	C₄⋊C₄	—	142	4	no	—
302	C₄⋊C₄	—	142	2	no	—
303	C₂²⋊C₄	—	142	2	no	—
304	C₂²⋊C₄	—	142	4	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	142	2	no	—
307	C₄²	—	142	4	no	—
308	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
309	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
310	C₂³	—	2	2	no	—
311	C₂³	#14556174	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	2	2	no	—
314	Q₈	—	568	4	no	—
315	D₄	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	284	4	no	—
326	C₂×C₄	—	284	2	no	—
327	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
328	C₂×C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
329	C₂×C₄	C₂×#11765864	1	1	yes	characteristic
330	C₂×C₄	C₂×#11765864	1	1	yes	characteristic
331	C₂×C₄	C₂×#11765865	1	1	yes	characteristic
332	C₂×C₄	#14556174	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	284	2	no	—
335	C₈	#14556107	1	1	yes	characteristic
336	C₈	#14556107	1	1	yes	characteristic
337	C₈	#14556107	1	1	yes	characteristic
338	C₈	#14556107	1	1	yes	characteristic
339	C₈	#14556115	1	1	yes	characteristic
340	C₈	#14556115	1	1	yes	characteristic
341	C₈	#14556115	1	1	yes	characteristic
342	C₈	#14556115	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	#17986563	1	1	yes	characteristic
347	C₂²	#17986706	1	1	yes	characteristic
348	C₂²	#17986714	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	568	4	no	—
353	C₄	—	568	2	no	—
354	C₄	C₂×#14556107	1	1	yes	characteristic
355	C₄	C₂×#14556115	1	1	yes	characteristic
356	C₄	#17986706	1	1	yes	characteristic
357	C₄	#17986714	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	8	2	no	—
362	C₂	#22593513	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17986714
commutator, cosocle	C₂²×C₄×C₇₁	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₂×#443×C₇₁	C₂
socle	C₂×C₇₁	#2563
prime radical	C₂×#135×C₇₁	C₂²
hypercenter	C₂×#443	D₇₁
hypocenter	C₇₁	#29154
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. #22598854	C₂	semidirect
2. #22589512	C₂	semidirect
4. #22588465	C₂	semidirect
5. #22588394	C₂	semidirect
6. #22588386	C₂	semidirect
7. #22588219	C₂	semidirect
57. C₂×Q₁₆×C₇₁	Q₈	semidirect
58. C₂×SD₁₆×C₇₁	Q₈	semidirect
59. C₂×D₈×C₇₁	Q₈	semidirect
190. C₇₁	#29154	semidirect
220. C₂×Q₁₆	Dic₁₄₂	semidirect
221. C₂×SD₁₆	Dic₁₄₂	semidirect
222. C₂×D₈	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
#22598854	2
#22589512	2
#22589239	2
#22588465	2
#22588394	2
#22588386	2
#22588219	2
C₂×#443×C₇₁	2
#29154	71