Subgroups — #32097413

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×#17986121	C₂	1	1	yes	maximal, characteristic
3	#22589514	C₂	1	2	yes	maximal, complemented
4	#22589510	C₂	1	2	yes	maximal, complemented
5	#22589239	C₂	1	1	yes	maximal, characteristic
6	#22588464	C₂	1	1	yes	maximal, characteristic, complemented
7	#22588463	C₂	1	1	yes	maximal, characteristic, complemented
8	#22588392	C₂	1	2	yes	maximal, complemented
9	#22588384	C₂	1	2	yes	maximal, complemented
10	#22588225	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₁×#1976	C₂	1	1	yes	maximal, characteristic
12	C₂×#14556135	C₂²	1	2	yes	—
13	#17986552	C₂²	1	2	yes	—
14	#17986407	—	2	2	no	—
15	#17986406	—	2	2	no	—
16	#17986121	—	2	2	no	—
17	#17986121	—	2	2	no	—
18	C₂×#14556064	C₂²	1	1	yes	characteristic
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	#17986016	C₂²	1	2	yes	—
24	C₂×#14556055	C₂²	1	2	yes	—
25	#17985973	C₂²	1	2	yes	—
26	#17985954	C₂²	1	1	yes	characteristic
27	C₂×#14556038	C₂²	1	1	yes	characteristic
28	#17985770	C₂²	1	2	yes	—
29	#17985769	C₂²	1	2	yes	—
30	#17985760	C₂²	1	1	yes	characteristic
31	#17985758	C₂²	1	1	yes	characteristic
32	#17985755	C₂²	1	2	yes	—
33	#17985755	C₂²	1	2	yes	—
34	C₂×#135×C₇₁	C₂²	1	1	yes	characteristic
35	#427×C₇₁	C₂²	1	1	yes	characteristic
36	#421×C₇₁	C₂²	1	2	yes	—
37	#410×C₇₁	C₂²	1	2	yes	—
38	#405×C₇₁	C₂²	1	1	yes	characteristic
39	#14556135	—	2	2	no	—
40	#14556135	—	2	2	no	—
41	#14556135	—	2	2	no	—
42	#14556135	—	2	2	no	—
43	C₂×#11765849	—	2	2	no	—
44	C₂×#11765849	C₂³	1	1	yes	characteristic
45	C₂×#11765849	C₂³	1	2	yes	—
46	C₂²×C₇₁⋊C₈	—	2	2	no	—
47	C₂²×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
48	#14556127	—	2	2	no	—
49	#14556119	—	2	2	no	—
50	#14556105	—	4	2	no	—
51	C₄×Dic₁₄₂	—	4	2	no	—
52	#14556087	C₂³	1	1	yes	characteristic
53	#14556087	C₂³	1	2	yes	—
54	C₂×C₄×Dic₇₁	C₂³	1	1	yes	characteristic
55	#14556064	—	2	1	no	—
56	#14556064	—	2	1	no	—
57	#14556063	—	2	2	no	—
58	#14556062	—	2	1	no	—
59	#14556061	—	2	2	no	—
60	#14556061	—	2	1	no	—
61	#14556060	—	2	1	no	—
62	#14556059	—	2	2	no	—
63	#14556059	—	2	1	no	—
64	#14556055	—	2	4	no	—
65	#14556055	—	4	2	no	—
66	#14556043	—	4	2	no	—
67	#14556038	—	2	1	no	—
68	#14556038	D₄	1	2	yes	—
69	#14556038	D₄	1	2	yes	—
70	C₈×Dic₇₁	—	2	1	no	—
71	C₂×#41×C₇₁	C₂³	1	2	yes	—
72	#135×C₇₁	—	2	1	no	—
73	#135×C₇₁	—	2	1	no	—
74	#135×C₇₁	—	2	1	no	—
75	#135×C₇₁	—	2	1	no	—
76	C₂×Q₁₆×C₇₁	Q₈	1	1	yes	characteristic, complemented
77	C₂×SD₁₆×C₇₁	Q₈	1	2	yes	complemented
78	C₂×D₈×C₇₁	Q₈	1	1	yes	characteristic, complemented
79	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
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	#107×C₇₁	—	2	1	no	—
84	#106×C₇₁	—	2	1	no	—
85	#104×C₇₁	—	4	2	no	—
86	#99×C₇₁	—	4	2	no	—
87	#97×C₇₁	—	2	1	no	—
88	C₄×C₈×C₇₁	—	2	1	no	—
89	C₂×Dic₁₄₂	—	4	2	no	—
90	C₂²×Dic₇₁	—	2	1	no	—
91	#11765849	—	2	2	no	—
92	#11765849	—	2	1	no	—
93	#11765849	—	2	2	no	—
94	#11765849	—	2	1	no	—
95	#11765849	—	4	2	no	—
96	#11765849	—	4	2	no	—
97	C₂×C₇₁⋊C₈	—	2	1	no	—
98	C₂×C₇₁⋊C₈	—	2	2	no	—
99	C₂×C₇₁⋊C₈	—	4	2	no	—
100	C₂×C₇₁⋊C₈	—	4	2	no	—
101	C₂×C₇₁⋊C₈	C₂×D₄	1	2	yes	—
102	C₂×C₇₁⋊C₈	#41	1	1	yes	characteristic
103	C₂×C₇₁⋊C₈	#41	1	1	yes	characteristic
104	C₂×C₇₁⋊C₈	#41	1	2	yes	—
105	C₄⋊Dic₇₁	—	2	2	no	—
106	C₄⋊Dic₇₁	—	2	1	no	—
107	C₄⋊Dic₇₁	—	4	2	no	—
108	Dic₇₁⋊C₄	—	4	2	no	—
109	Dic₇₁⋊C₄	—	4	2	no	—
110	C₂²⋊Dic₇₁	—	2	1	no	—
111	C₂²⋊Dic₇₁	—	2	2	no	—
112	C₄×Dic₇₁	—	2	1	no	—
113	C₄×Dic₇₁	—	2	1	no	—
114	C₄×Dic₇₁	—	2	2	no	—
115	C₄×Dic₇₁	C₂×D₄	1	2	yes	—
116	#41×C₇₁	—	2	2	no	—
117	#41×C₇₁	—	2	2	no	—
118	#41×C₇₁	—	2	2	no	—
119	#41×C₇₁	—	2	2	no	—
120	C₂×Q₈×C₇₁	C₂×Q₈	1	2	yes	—
121	C₂×D₄×C₇₁	—	2	2	no	—
122	C₂×D₄×C₇₁	C₂×Q₈	1	2	yes	—
123	C₂²×C₄×C₇₁	—	2	2	no	—
124	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
125	Q₁₆×C₇₁	—	2	1	no	—
126	Q₁₆×C₇₁	—	2	1	no	—
127	SD₁₆×C₇₁	—	2	2	no	—
128	SD₁₆×C₇₁	—	2	2	no	—
129	D₈×C₇₁	—	2	1	no	—
130	D₈×C₇₁	—	2	1	no	—
131	M₁₆×C₇₁	—	2	1	no	—
132	M₁₆×C₇₁	—	2	1	no	—
133	C₂×C₈×C₇₁	—	2	1	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	—	2	2	no	—
147	C₂³×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
148	Q₈×C₇₁	—	2	2	no	—
149	Q₈×C₇₁	—	2	2	no	—
150	D₄×C₇₁	—	2	2	no	—
151	D₄×C₇₁	—	2	2	no	—
152	D₄×C₇₁	—	4	2	no	—
153	D₄×C₇₁	—	4	2	no	—
154	C₂×C₄×C₇₁	—	2	2	no	—
155	C₂×C₄×C₇₁	—	4	2	no	—
156	C₂×C₄×C₇₁	—	4	2	no	—
157	C₂×C₄×C₇₁	—	4	2	no	—
158	C₂×C₄×C₇₁	—	4	1	no	—
159	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
161	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
162	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
163	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
164	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
165	C₂×Dic₇₁	—	2	2	no	—
166	C₂×Dic₇₁	—	4	2	no	—
167	C₂×Dic₇₁	—	4	1	no	—
168	C₂×Dic₇₁	—	8	1	no	—
169	Dic₁₄₂	—	8	4	no	—
170	C₈×C₇₁	—	4	1	no	—
171	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
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	C₇₁⋊C₈	—	4	2	no	—
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	—	4	2	no	—
181	C₂²×C₇₁	—	8	2	no	—
182	C₂²×C₇₁	#535	1	1	yes	characteristic
183	C₂²×C₇₁	#554	1	1	yes	characteristic
184	C₂²×C₇₁	#555	1	1	yes	characteristic
185	C₄×C₇₁	—	4	2	no	—
186	C₄×C₇₁	—	8	2	no	—
187	C₄×C₇₁	—	8	1	no	—
188	C₄×C₇₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
189	C₄×C₇₁	C₂×#124	1	1	yes	characteristic
190	C₄×C₇₁	#554	1	1	yes	characteristic
191	C₄×C₇₁	#555	1	1	yes	characteristic
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	—	8	2	no	—
200	C₂×C₇₁	#2563	1	1	yes	characteristic
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	D₇₁	1	1	yes	characteristic
207	#1833	—	71	1	no	—
208	#1833	—	71	1	no	—
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	C₂×D₇₁	1	1	yes	characteristic
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	—	71	1	no	—
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	C₂×D₇₁	1	2	yes	—
228	C₂×#104	—	71	2	no	—
229	#410	—	71	2	no	—
230	#410	C₂×D₇₁	1	2	yes	—
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	—	71	2	no	—
236	#339	—	71	1	no	—
237	#339	—	71	1	no	—
238	H(3,ℤ/4)	—	71	2	no	—
239	H(3,ℤ/4)	—	71	2	no	—
240	C₂×#41	C₂²×D₇₁	1	2	yes	—
241	#135	—	2	1	no	—
242	#135	—	2	1	no	—
243	#135	—	2	1	no	—
244	#135	—	2	1	no	—
245	C₂×Q₁₆	Dic₁₄₂	1	1	yes	characteristic, complemented
246	C₂×SD₁₆	Dic₁₄₂	1	2	yes	complemented
247	C₂×D₈	Dic₁₄₂	1	1	yes	characteristic, complemented
248	#131	—	142	2	no	—
249	#131	—	142	2	no	—
250	#131	—	142	2	no	—
251	#131	—	142	2	no	—
252	C₂×M₁₆	—	71	2	no	—
253	C₂×M₁₆	—	71	1	no	—
254	C₂×M₁₆	—	142	2	no	—
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	—	142	2	no	—
269	#108	—	142	1	no	—
270	#108	—	142	1	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	—	4	2	no	—
280	#104	—	284	2	no	—
281	#104	—	142	4	no	—
282	#99	—	4	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	—	2	2	no	—
291	#41	—	2	2	no	—
292	#41	—	2	2	no	—
293	#41	—	2	2	no	—
294	C₂×Q₈	—	284	2	no	—
295	C₂×Q₈	C₂×Dic₁₄₂	1	2	yes	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	C₂×Dic₁₄₂	1	2	yes	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	142	1	no	—
300	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
301	Q₁₆	—	2	1	no	—
302	Q₁₆	—	2	1	no	—
303	SD₁₆	—	2	2	no	—
304	SD₁₆	—	2	2	no	—
305	D₈	—	2	1	no	—
306	D₈	—	2	1	no	—
307	M₁₆	—	2	1	no	—
308	M₁₆	—	2	1	no	—
309	M₁₆	—	284	2	no	—
310	M₁₆	—	284	2	no	—
311	M₁₆	—	142	2	no	—
312	M₁₆	—	142	1	no	—
313	M₁₆	—	142	2	no	—
314	M₁₆	—	142	1	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	284	2	no	—
317	C₂×C₈	—	284	2	no	—
318	C₂×C₈	—	71	2	no	—
319	C₂×C₈	—	71	1	no	—
320	C₂×C₈	—	71	1	no	—
321	C₂×C₈	—	71	2	no	—
322	C₂×C₈	—	142	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	C₂×Dic₁₄₂	1	1	yes	characteristic
327	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
328	C₂×C₈	#11765864	1	1	yes	characteristic
329	C₂×C₈	#11765864	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	—	2	2	no	—
348	C₂³	#14556173	1	1	yes	characteristic
349	Q₈	—	2	2	no	—
350	Q₈	—	2	2	no	—
351	Q₈	—	568	4	no	—
352	D₄	—	2	2	no	—
353	D₄	—	2	2	no	—
354	D₄	—	4	2	no	—
355	D₄	—	4	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	284	2	no	—
362	C₂×C₄	—	284	1	no	—
363	C₂×C₄	—	568	1	no	—
364	C₂×C₄	—	142	2	no	—
365	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
366	C₂×C₄	C₂²×Dic₁₄₂	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×#11765864	1	1	yes	characteristic
370	C₂×C₄	#14556173	1	1	yes	characteristic
371	C₈	—	4	1	no	—
372	C₈	—	284	1	no	—
373	C₈	—	284	2	no	—
374	C₈	—	71	2	no	—
375	C₈	—	71	2	no	—
376	C₈	#14556105	1	1	yes	characteristic
377	C₈	#14556105	1	1	yes	characteristic
378	C₈	#14556105	1	1	yes	characteristic
379	C₈	#14556105	1	1	yes	characteristic
380	C₂²	—	4	1	no	—
381	C₂²	—	4	2	no	—
382	C₂²	—	8	2	no	—
383	C₂²	#17986562	1	1	yes	characteristic
384	C₂²	#17986708	1	1	yes	characteristic
385	C₂²	#17986713	1	1	yes	characteristic
386	C₄	—	4	2	no	—
387	C₄	—	8	2	no	—
388	C₄	—	8	1	no	—
389	C₄	—	284	2	no	—
390	C₄	—	568	2	no	—
391	C₄	—	568	1	no	—
392	C₄	C₂×#14556105	1	1	yes	characteristic
393	C₄	#124×D₇₁	1	1	yes	characteristic
394	C₄	#17986708	1	1	yes	characteristic
395	C₄	#17986713	1	1	yes	characteristic
396	C₂	—	2	1	no	—
397	C₂	—	2	1	no	—
398	C₂	—	2	1	no	—
399	C₂	—	8	2	no	—
400	C₂	#22593511	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17986713
commutator, cosocle	C₂²×C₄×C₇₁	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₇₁×#1976	C₂
socle	C₂×C₇₁	#2563
prime radical	C₂×#135×C₇₁	C₂²
hypercenter	#1976	D₇₁
hypocenter	C₇₁	#29155
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1976

Complements

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

N	H ≅ G/N	type
1. #22598863	C₂	semidirect
3. #22589514	C₂	semidirect
4. #22589510	C₂	semidirect
6. #22588464	C₂	semidirect
7. #22588463	C₂	semidirect
8. #22588392	C₂	semidirect
9. #22588384	C₂	semidirect
10. #22588225	C₂	semidirect
76. C₂×Q₁₆×C₇₁	Q₈	semidirect
77. C₂×SD₁₆×C₇₁	Q₈	semidirect
78. C₂×D₈×C₇₁	Q₈	semidirect
212. C₇₁	#29155	semidirect
245. C₂×Q₁₆	Dic₁₄₂	semidirect
246. C₂×SD₁₆	Dic₁₄₂	semidirect
247. 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
#22598863	2
C₂×#17986121	2
#22589514	2
#22589510	2
#22589239	2
#22588464	2
#22588463	2
#22588392	2
#22588384	2
#22588225	2
C₇₁×#1976	2
#29155	71