Subgroups — #30874354

Order 17152.

812 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569127	C₂	1	2	yes	maximal
2	C₂×#16863514	C₂	1	1	yes	maximal, characteristic
3	#22568817	C₂	1	2	yes	maximal, complemented
4	#22560999	C₂	1	2	yes	maximal, complemented
5	#22560450	C₂	1	4	yes	maximal, complemented
6	#22560381	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560260	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524065	C₂²	1	1	yes	characteristic
10	#16864162	C₂²	1	4	yes	—
11	C₂×#14524049	C₂²	1	2	yes	—
12	#16863519	—	2	4	no	—
13	#16863514	—	2	4	no	—
14	#16863514	C₄	1	2	yes	complemented
15	#16863514	C₄	1	2	yes	complemented
16	#16863514	C₄	1	2	yes	complemented
17	#16863514	C₄	1	2	yes	complemented
18	#16863444	C₂²	1	4	yes	—
19	#16863443	—	2	4	no	—
20	C₂×#14523979	C₂²	1	2	yes	—
21	C₂×#14523979	C₂²	1	2	yes	—
22	C₂×#14523979	C₂²	1	1	yes	characteristic
23	#108×D₆₇	—	2	4	no	—
24	#16863420	C₂²	1	2	yes	—
25	#16863414	C₂²	1	4	yes	—
26	C₂×#14523971	C₂²	1	2	yes	—
27	#16863166	C₂²	1	4	yes	—
28	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
29	#427×C₆₇	C₂²	1	1	yes	characteristic
30	#405×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#339×C₆₇	C₂²	1	2	yes	—
33	C₂×#11761628	C₂³	1	2	yes	—
34	#14524065	C₂×C₄	1	2	yes	—
35	#14524065	C₂×C₄	1	2	yes	—
36	#14524065	D₄	1	2	yes	—
37	#14524065	D₄	1	2	yes	—
38	C₂×Dic₂₆₈	Q₈	1	1	yes	characteristic
39	C₂×#11761617	Q₈	1	2	yes	—
40	C₂×D₅₃₆	Q₈	1	1	yes	characteristic
41	#14524051	—	4	4	no	—
42	#14524049	—	2	8	no	—
43	#14524049	—	2	2	no	—
44	#14524049	—	2	2	no	—
45	#14524049	C₂×C₄	1	4	yes	—
46	#14524049	C₂×C₄	1	4	yes	—
47	C₂×#11761612	C₂³	1	4	yes	—
48	C₂×#11761612	C₂³	1	2	yes	—
49	C₂×#11761611	—	2	2	no	—
50	M₁₆×D₆₇	—	2	8	no	—
51	C₂×C₈×D₆₇	—	2	2	no	—
52	#14523979	—	2	2	no	—
53	#14523979	—	2	2	no	—
54	#14523979	—	2	2	no	—
55	#14523979	—	2	2	no	—
56	#14523979	C₂×C₄	1	2	yes	—
57	#14523979	C₂×C₄	1	2	yes	—
58	#14523978	—	2	8	no	—
59	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
60	#14523971	—	2	8	no	—
61	#14523971	C₂×C₄	1	4	yes	—
62	#14523971	C₂×C₄	1	4	yes	—
63	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	#117×C₆₇	C₂³	1	2	yes	—
67	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
68	#108×C₆₇	D₄	1	2	yes	—
69	#108×C₆₇	D₄	1	2	yes	—
70	#107×C₆₇	—	2	1	no	—
71	#106×C₆₇	—	2	1	no	—
72	#97×C₆₇	—	2	2	no	—
73	C₄×C₈×C₆₇	—	2	2	no	—
74	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
75	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
78	#11761628	—	2	2	no	—
79	#11761628	—	2	2	no	—
80	#11761628	C₂²×C₄	1	4	yes	—
81	#11761628	C₂×D₄	1	4	yes	—
82	C₂×Dic₁₃₄	C₂×Q₈	1	2	yes	—
83	C₂×D₂₆₈	C₂×Q₈	1	2	yes	—
84	C₂×C₂²⋊D₆₇	—	2	2	no	—
85	C₂×C₄×D₆₇	—	2	2	no	—
86	Dic₂₆₈	C₄⋊C₄	1	2	yes	—
87	Dic₂₆₈	C₄⋊C₄	1	2	yes	—
88	#11761617	C₄⋊C₄	1	4	yes	—
89	#11761617	C₄⋊C₄	1	4	yes	—
90	D₅₃₆	C₄⋊C₄	1	2	yes	—
91	D₅₃₆	C₄⋊C₄	1	2	yes	—
92	#11761612	—	2	2	no	—
93	#11761612	—	2	4	no	—
94	#11761612	—	2	4	no	—
95	#11761612	C₂²×C₄	1	4	yes	—
96	#11761611	—	2	4	no	—
97	#11761611	—	2	4	no	—
98	#11761611	—	4	8	no	—
99	C₂×C₆₇⋊C₈	—	2	2	no	—
100	C₂×C₆₇⋊C₈	—	2	4	no	—
101	C₈×D₆₇	—	2	4	no	—
102	C₈×D₆₇	—	2	4	no	—
103	C₈×D₆₇	—	4	8	no	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
106	M₁₆×C₆₇	—	2	2	no	—
107	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
118	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
119	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
120	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₂²⋊C₄×C₆₇	—	2	2	no	—
123	C₄²×C₆₇	—	2	1	no	—
124	C₄²×C₆₇	—	2	2	no	—
125	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂³×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂²×D₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	4	2	no	—
132	C₂×C₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂²⋊D₆₇	—	2	4	no	—
140	C₂²⋊D₆₇	—	2	4	no	—
141	C₂×Dic₆₇	—	2	2	no	—
142	D₂₆₈	—	2	2	no	—
143	D₂₆₈	C₂×C₄⋊C₄	1	4	yes	—
144	C₄×D₆₇	—	2	4	no	—
145	C₄×D₆₇	—	2	4	no	—
146	Dic₁₃₄	—	2	2	no	—
147	Dic₁₃₄	C₂×C₄⋊C₄	1	4	yes	—
148	C₈×C₆₇	—	2	4	no	—
149	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
154	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
155	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
156	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
157	C₆₇⋊C₈	—	2	4	no	—
158	C₆₇⋊C₈	—	4	4	no	—
159	C₂²×C₆₇	—	4	1	no	—
160	C₂²×C₆₇	#520	1	1	yes	characteristic
161	C₂²×C₆₇	#547	1	1	yes	characteristic
162	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
163	C₂×D₆₇	—	2	4	no	—
164	C₂×D₆₇	—	8	2	no	—
165	C₄×C₆₇	—	2	2	no	—
166	C₄×C₆₇	—	4	1	no	—
167	C₄×C₆₇	—	8	2	no	—
168	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
170	C₄×C₆₇	D₄²	1	1	yes	characteristic
171	C₄×C₆₇	#554	1	1	yes	characteristic
172	Dic₆₇	—	2	4	no	—
173	#22694	—	67	1	no	maximal, Sylow
174	C₂×C₆₇	—	2	1	no	—
175	C₂×C₆₇	—	2	1	no	—
176	C₂×C₆₇	—	2	1	no	—
177	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
178	D₆₇	—	4	4	no	—
179	#2957	—	67	2	no	—
180	C₂×#443	—	67	1	no	—
181	#2916	—	67	2	no	—
182	#1953	—	67	2	no	—
183	#1861	—	67	4	no	—
184	#1848	—	67	1	no	—
185	#1814	—	67	2	no	—
186	#1772	D₆₇	1	1	yes	characteristic, complemented
187	C₆₇	#22694	1	1	yes	characteristic, complemented, Sylow
188	C₂×#135	—	67	1	no	—
189	#568	—	67	4	no	—
190	C₂×#131	—	67	2	no	—
191	#444	—	134	4	no	—
192	#443	—	67	2	no	—
193	#443	—	67	2	no	—
194	#443	—	67	2	no	—
195	#443	—	67	2	no	—
196	#443	—	134	4	no	—
197	#430	—	67	4	no	—
198	#430	—	134	4	no	—
199	C₂×#108	—	67	1	no	—
200	C₂×#108	—	67	2	no	—
201	C₂×#108	—	67	2	no	—
202	C₂×#108	—	134	4	no	—
203	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
204	#427	C₂×D₆₇	1	1	yes	characteristic
205	#423	—	67	2	no	—
206	#421	—	67	4	no	—
207	C₂×#104	—	67	2	no	—
208	#405	C₂×D₆₇	1	2	yes	—
209	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
210	#344	—	67	4	no	—
211	#339	C₂×D₆₇	1	2	yes	—
212	C₂×#41	—	67	2	no	—
213	#135	—	67	2	no	—
214	#135	—	67	2	no	—
215	#135	—	67	2	no	—
216	#135	—	67	2	no	—
217	C₂×Q₁₆	—	67	1	no	—
218	C₂×SD₁₆	—	67	2	no	—
219	C₂×D₈	—	67	1	no	—
220	#131	—	268	4	no	—
221	#131	—	67	4	no	—
222	#131	—	67	4	no	—
223	#131	—	134	8	no	—
224	#131	—	134	2	no	—
225	#131	—	134	2	no	—
226	C₂×M₁₆	—	67	2	no	—
227	C₂×M₁₆	—	67	4	no	—
228	C₂×M₁₆	—	134	8	no	—
229	C₂×M₁₆	—	134	2	no	—
230	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
231	C₂²×C₈	—	134	2	no	—
232	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
234	#117	C₂²×D₆₇	1	2	yes	—
235	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
236	#108	—	67	2	no	—
237	#108	—	67	2	no	—
238	#108	—	134	8	no	—
239	#108	—	134	2	no	—
240	#108	—	134	2	no	—
241	#108	—	134	2	no	—
242	#108	—	134	2	no	—
243	#108	D₂₆₈	1	2	yes	—
244	#108	D₂₆₈	1	2	yes	—
245	#107	—	2	1	no	—
246	#106	—	2	1	no	—
247	C₄⋊C₈	—	268	2	no	—
248	#104	—	67	4	no	—
249	#104	—	67	4	no	—
250	#104	—	134	8	no	—
251	#97	—	2	2	no	—
252	C₄×C₈	—	2	2	no	—
253	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
254	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
255	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
256	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
257	#41	—	67	4	no	—
258	#41	—	67	4	no	—
259	#41	—	134	2	no	—
260	#41	—	134	2	no	—
261	C₂×Q₈	—	67	2	no	—
262	C₂×D₄	—	67	2	no	—
263	C₂×D₄	—	134	2	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	134	2	no	—
266	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
267	Q₁₆	—	67	2	no	—
268	Q₁₆	—	67	2	no	—
269	SD₁₆	—	67	4	no	—
270	SD₁₆	—	67	4	no	—
271	D₈	—	67	2	no	—
272	D₈	—	67	2	no	—
273	M₁₆	—	2	2	no	—
274	M₁₆	—	268	8	no	—
275	M₁₆	—	67	4	no	—
276	M₁₆	—	134	4	no	—
277	M₁₆	—	134	4	no	—
278	M₁₆	—	134	2	no	—
279	M₁₆	—	134	4	no	—
280	M₁₆	—	134	4	no	—
281	M₁₆	C₂×D₂₆₈	1	4	yes	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	268	8	no	—
284	C₂×C₈	—	134	4	no	—
285	C₂×C₈	—	134	2	no	—
286	C₂×C₈	—	134	4	no	—
287	C₂×C₈	—	134	4	no	—
288	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
292	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
293	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
294	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
295	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
296	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
297	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
298	C₂×C₈	#11761628	1	1	yes	characteristic
299	C₂×C₈	#11761628	1	1	yes	characteristic
300	C₄⋊C₄	—	2	2	no	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	2	2	no	—
304	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
305	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
306	C₂³	—	134	2	no	—
307	C₂³	#14524090	1	1	yes	characteristic
308	Q₈	—	67	4	no	—
309	Q₈	—	134	2	no	—
310	D₄	—	67	4	no	—
311	D₄	—	134	4	no	—
312	D₄	—	134	4	no	—
313	D₄	—	134	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	134	2	no	—
319	C₂×C₄	—	134	4	no	—
320	C₂×C₄	—	134	4	no	—
321	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
322	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
324	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
325	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
326	C₂×C₄	#14524089	1	1	yes	characteristic
327	C₈	—	2	4	no	—
328	C₈	—	268	4	no	—
329	C₈	—	134	4	no	—
330	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
331	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
332	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
333	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
334	C₈	C₄×D₂₆₈	1	1	yes	characteristic
335	C₈	C₄×D₂₆₈	1	1	yes	characteristic
336	C₈	C₄×D₂₆₈	1	1	yes	characteristic
337	C₈	C₄×D₂₆₈	1	1	yes	characteristic
338	C₂²	—	4	1	no	—
339	C₂²	—	536	2	no	—
340	C₂²	—	134	4	no	—
341	C₂²	#16863876	1	1	yes	characteristic
342	C₂²	#16864051	1	1	yes	characteristic
343	C₂²	Q₈×D₂₆₈	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	4	1	no	—
346	C₄	—	8	2	no	—
347	C₄	—	134	4	no	—
348	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
349	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
350	C₄	D₄×D₂₆₈	1	1	yes	characteristic
351	C₄	#16864106	1	1	yes	characteristic
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	268	4	no	—
356	C₂	C₄⋊C₄×D₂₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1772	C₂
socle	C₂×C₆₇	D₄×C₄⋊C₄
prime radical	C₂×#14524065	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1772	D₆₇
hypocenter	C₆₇	#22694
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772

Complements

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

N	H ≅ G/N	type
3. #22568817	C₂	semidirect
4. #22560999	C₂	semidirect
5. #22560450	C₂	semidirect
6. #22560381	C₂	semidirect
7. #22560260	C₂	semidirect
8. C₆₇×#1772	C₂	semidirect
14. #16863514	C₄	semidirect
15. #16863514	C₄	semidirect
16. #16863514	C₄	semidirect
17. #16863514	C₄	semidirect
186. #1772	D₆₇	semidirect
187. C₆₇	#22694	semidirect

Maximal subgroups

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

subgroup	index
#22569127	2
C₂×#16863514	2
#22568817	2
#22560999	2
#22560450	2
#22560381	2
#22560260	2
C₆₇×#1772	2
#22694	67