Subgroups — #30875577

Order 17152.

934 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22563158	C₂	1	2	yes	maximal, complemented
2	#22561076	C₂	1	2	yes	maximal
3	#22560831	C₂	1	4	yes	maximal, complemented
4	#22560369	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560209	C₂	1	2	yes	maximal, complemented
6	#107×Dic₆₇	C₂	1	2	yes	maximal, complemented
7	C₈×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14524002	—	2	2	no	—
10	C₂×#14523977	C₂²	1	2	yes	—
11	C₂×#14523977	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#14523964	C₂²	1	4	yes	—
14	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
15	#16863299	C₂²	1	4	yes	—
16	#16863289	—	2	4	no	—
17	#16863266	C₂²	1	2	yes	—
18	#16863256	—	2	2	no	—
19	C₄⋊C₄×Dic₆₇	C₂²	1	4	yes	—
20	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
21	#16863158	C₂²	1	4	yes	—
22	#16863150	C₂²	1	2	yes	—
23	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
25	C₄×D₈×C₆₇	C₄	1	8	yes	complemented
26	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#102×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
30	#14524002	—	4	8	no	—
31	C₂×C₄⋊Dic₆₇	—	2	2	no	—
32	C₂×C₄⋊Dic₆₇	—	2	2	no	—
33	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
36	C₂×C₂²⋊Dic₆₇	—	2	4	no	—
37	C₂×C₂²⋊Dic₆₇	—	2	4	no	—
38	C₂×C₄×Dic₆₇	—	2	4	no	—
39	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
40	#14523977	—	2	4	no	—
41	#14523977	—	4	2	no	—
42	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
43	#14523964	—	4	4	no	—
44	C₈×Dic₆₇	—	2	4	no	—
45	C₆₇⋊#95	—	2	4	no	—
46	C₆₇⋊#95	D₄	1	2	yes	—
47	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
48	C₂³×C₄×C₆₇	—	2	2	no	—
49	C₂×D₈×C₆₇	—	2	2	no	—
50	C₂×D₈×C₆₇	—	2	2	no	—
51	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₄×D₄×C₆₇	—	2	4	no	—
55	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
56	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
57	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	#107×C₆₇	C₂×C₄	1	4	yes	—
60	#102×C₆₇	C₂×C₄	1	8	yes	—
61	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
62	C₂²×Dic₆₇	—	2	4	no	—
63	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
64	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
65	C₂×C₆₇⋊C₈	—	2	2	no	—
66	C₂×C₆₇⋊C₈	—	2	2	no	—
67	C₂×C₆₇⋊C₈	—	2	2	no	—
68	C₄⋊Dic₆₇	—	2	2	no	—
69	C₄⋊Dic₆₇	—	2	4	no	—
70	C₄⋊Dic₆₇	—	2	4	no	—
71	C₄⋊Dic₆₇	—	4	4	no	—
72	C₄⋊Dic₆₇	—	8	2	no	—
73	C₄⋊Dic₆₇	D₈	1	4	yes	complemented
74	C₂²⋊Dic₆₇	—	4	8	no	—
75	C₂²⋊Dic₆₇	—	8	4	no	—
76	C₄×Dic₆₇	—	2	2	no	—
77	C₄×Dic₆₇	—	2	2	no	—
78	C₄×Dic₆₇	—	4	8	no	—
79	C₂⁴×C₆₇	—	2	2	no	—
80	C₂×D₄×C₆₇	—	2	2	no	—
81	C₂×D₄×C₆₇	—	2	2	no	—
82	C₂×D₄×C₆₇	—	2	2	no	—
83	C₂×D₄×C₆₇	—	2	2	no	—
84	C₂×D₄×C₆₇	—	2	2	no	—
85	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	2	8	no	—
88	C₂²×C₄×C₆₇	—	4	4	no	—
89	C₂²×C₄×C₆₇	—	4	4	no	—
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
93	D₈×C₆₇	—	2	8	no	—
94	C₂×C₈×C₆₇	—	2	1	no	—
95	C₂×C₈×C₆₇	—	2	1	no	—
96	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₆₇	#41	1	2	yes	—
99	C₂×C₈×C₆₇	#41	1	2	yes	—
100	C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
102	C₂²⋊C₄×C₆₇	—	2	8	no	—
103	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
104	C₄²×C₆₇	#41	1	2	yes	—
105	C₂³×C₆₇	—	2	4	no	—
106	C₂³×C₆₇	—	4	2	no	—
107	C₂³×C₆₇	—	4	2	no	—
108	C₂³×C₆₇	—	4	2	no	—
109	C₂³×C₆₇	—	4	2	no	—
110	C₂³×C₆₇	—	4	2	no	—
111	C₂³×C₆₇	—	4	2	no	—
112	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
113	D₄×C₆₇	—	2	4	no	—
114	D₄×C₆₇	—	2	4	no	—
115	D₄×C₆₇	—	4	2	no	—
116	D₄×C₆₇	—	4	2	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	4	no	—
120	C₂×C₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	4	16	no	—
123	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₆₇	#135	1	2	yes	—
127	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×Dic₆₇	—	2	2	no	—
134	C₂×Dic₆₇	—	2	2	no	—
135	C₂×Dic₆₇	—	2	2	no	—
136	C₂×Dic₆₇	—	2	2	no	—
137	C₂×Dic₆₇	—	2	2	no	—
138	C₂×Dic₆₇	—	4	4	no	—
139	C₂×Dic₆₇	—	4	4	no	—
140	C₂×Dic₆₇	—	4	4	no	—
141	C₂×Dic₆₇	C₂×D₈	1	4	yes	—
142	C₈×C₆₇	—	2	2	no	—
143	C₈×C₆₇	#117	1	4	yes	complemented
144	C₆₇⋊C₈	—	4	2	no	—
145	C₂²×C₆₇	—	4	4	no	—
146	C₂²×C₆₇	—	4	4	no	—
147	C₂²×C₆₇	—	4	4	no	—
148	C₂²×C₆₇	—	8	2	no	—
149	C₂²×C₆₇	—	8	2	no	—
150	C₂²×C₆₇	—	8	2	no	—
151	C₂²×C₆₇	—	8	2	no	—
152	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂²×C₆₇	#547	1	1	yes	characteristic
154	C₂²×C₆₇	#550	1	1	yes	characteristic
155	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
156	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
157	C₂²×C₆₇	#576	1	1	yes	characteristic
158	C₂²×C₆₇	#578	1	1	yes	characteristic
159	C₄×C₆₇	—	2	2	no	—
160	C₄×C₆₇	—	4	4	no	—
161	C₄×C₆₇	C₄×D₈	1	4	yes	—
162	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
163	C₄×C₆₇	#520	1	1	yes	characteristic
164	C₄×C₆₇	#555	1	1	yes	characteristic
165	C₄×C₆₇	#556	1	1	yes	characteristic
166	Dic₆₇	—	2	4	no	—
167	Dic₆₇	—	4	2	no	—
168	Dic₆₇	—	8	4	no	—
169	#22807	—	67	1	no	maximal, Sylow
170	C₂×C₆₇	—	8	4	no	—
171	C₂×C₆₇	#2354	1	1	yes	characteristic
172	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
173	C₂×C₆₇	#2951	1	1	yes	characteristic
174	C₂×C₆₇	#3334	1	1	yes	characteristic
175	C₂×C₆₇	#3341	1	1	yes	characteristic
176	C₂×C₆₇	#3391	1	1	yes	characteristic
177	C₂×C₆₇	#3393	1	1	yes	characteristic
178	C₂×C₄×D₈	D₆₇	1	1	yes	characteristic
179	#2354	—	67	2	no	—
180	#1970	—	67	2	no	—
181	#1925	—	67	4	no	—
182	#1845	—	67	1	no	—
183	#1802	—	67	2	no	—
184	C₄×#107	—	67	2	no	—
185	C₈×C₄⋊C₄	—	67	1	no	—
186	C₆₇	#22807	1	1	yes	characteristic, complemented, Sylow
187	C₂²×D₈	C₂×D₆₇	1	1	yes	characteristic
188	C₂×C₄×D₄	C₂×D₆₇	1	2	yes	—
189	C₂×#117	—	134	2	no	—
190	C₄×D₈	Dic₆₇	1	8	yes	complemented
191	C₂×#107	—	67	2	no	—
192	C₂×#107	—	67	2	no	—
193	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
194	C₂×C₄⋊C₈	—	67	1	no	—
195	C₂×#102	—	67	4	no	—
196	C₂×#102	C₂×D₆₇	1	2	yes	—
197	C₂×C₄×C₈	—	67	2	no	—
198	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
199	#390	—	67	4	no	—
200	#388	—	134	4	no	—
201	#384	—	67	2	no	—
202	#380	—	134	2	no	—
203	C₄×C₄⋊C₄	—	67	1	no	—
204	C₄×C₄⋊C₄	—	67	4	no	—
205	#340	—	67	4	no	—
206	#336	—	67	2	no	—
207	C₂²×D₄	C₂²×D₆₇	1	2	yes	—
208	C₂³×C₄	—	2	2	no	—
209	C₂×D₈	—	2	2	no	—
210	C₂×D₈	—	2	2	no	—
211	C₂×D₈	C₂×Dic₆₇	1	4	yes	—
212	C₂²×C₈	—	67	2	no	—
213	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
215	C₄×D₄	—	2	4	no	—
216	C₄×D₄	C₂×Dic₆₇	1	8	yes	—
217	#117	—	268	8	no	—
218	C₂×C₄⋊C₄	—	67	1	no	—
219	C₂×C₄⋊C₄	—	67	1	no	—
220	C₂×C₄⋊C₄	—	67	4	no	—
221	C₂×C₄⋊C₄	—	134	2	no	—
222	C₂×C₄⋊C₄	—	134	2	no	—
223	C₂×C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
224	C₂×C₂²⋊C₄	—	2	2	no	—
225	C₂×C₂²⋊C₄	—	134	4	no	—
226	C₂×C₂²⋊C₄	—	134	4	no	—
227	C₂×C₄²	—	67	2	no	—
228	C₂×C₄²	—	134	4	no	—
229	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
230	#107	—	268	2	no	—
231	#107	—	134	4	no	—
232	#107	C₂×Dic₆₇	1	4	yes	—
233	C₄⋊C₈	—	134	2	no	—
234	#102	—	268	4	no	—
235	#102	C₂×Dic₆₇	1	8	yes	—
236	C₄×C₈	—	134	4	no	—
237	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
238	#95	—	67	2	no	—
239	#95	—	134	4	no	—
240	C₂⁴	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	C₂²×Dic₆₇	1	4	yes	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	8	no	—
249	C₂²×C₄	—	4	4	no	—
250	C₂²×C₄	—	4	4	no	—
251	C₂²×C₄	—	67	2	no	—
252	C₂²×C₄	—	67	2	no	—
253	C₂²×C₄	—	134	4	no	—
254	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
255	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
256	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
257	D₈	—	2	8	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	134	2	no	—
261	C₂×C₈	—	134	2	no	—
262	C₂×C₈	—	134	2	no	—
263	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
264	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
265	C₂×C₈	#11761628	1	2	yes	—
266	C₂×C₈	#11761628	1	2	yes	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	268	4	no	—
269	C₄⋊C₄	—	536	2	no	—
270	C₄⋊C₄	—	67	4	no	—
271	C₄⋊C₄	—	134	4	no	—
272	C₄⋊C₄	—	134	4	no	—
273	C₄⋊C₄	—	134	2	no	—
274	C₄⋊C₄	C₂²×Dic₆₇	1	4	yes	—
275	C₂²⋊C₄	—	2	8	no	—
276	C₂²⋊C₄	—	268	8	no	—
277	C₂²⋊C₄	—	536	4	no	—
278	C₄²	—	268	8	no	—
279	C₄²	—	134	2	no	—
280	C₄²	—	134	2	no	—
281	C₄²	C₂²×Dic₆₇	1	2	yes	—
282	C₄²	#11761627	1	2	yes	—
283	C₂³	—	2	4	no	—
284	C₂³	—	4	2	no	—
285	C₂³	—	4	2	no	—
286	C₂³	—	4	2	no	—
287	C₂³	—	4	2	no	—
288	C₂³	—	4	2	no	—
289	C₂³	—	4	2	no	—
290	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
291	D₄	—	2	4	no	—
292	D₄	—	2	4	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	16	no	—
301	C₂×C₄	—	268	4	no	—
302	C₂×C₄	—	268	4	no	—
303	C₂×C₄	—	268	4	no	—
304	C₂×C₄	—	67	4	no	—
305	C₂×C₄	—	134	2	no	—
306	C₂×C₄	—	134	2	no	—
307	C₂×C₄	—	134	2	no	—
308	C₂×C₄	—	134	2	no	—
309	C₂×C₄	—	134	2	no	—
310	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
311	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
312	C₂×C₄	D₈×D₆₇	1	2	yes	—
313	C₂×C₄	#14524062	1	2	yes	—
314	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
315	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
316	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
317	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
318	C₂×C₄	#14524089	1	1	yes	characteristic
319	C₂×C₄	#14524090	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	268	2	no	—
322	C₈	#14524002	1	4	yes	complemented
323	C₂²	—	4	4	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	4	4	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
331	C₂²	#16864051	1	1	yes	characteristic
332	C₂²	#16864067	1	1	yes	characteristic
333	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
334	C₂²	C₂×#14524062	1	1	yes	characteristic
335	C₂²	#16864198	1	1	yes	characteristic
336	C₂²	#16864210	1	1	yes	characteristic
337	C₄	—	2	2	no	—
338	C₄	—	4	4	no	—
339	C₄	—	268	2	no	—
340	C₄	—	536	4	no	—
341	C₄	—	134	4	no	—
342	C₄	D₈×Dic₆₇	1	4	yes	—
343	C₄	C₂×#14524002	1	1	yes	characteristic
344	C₄	#16863879	1	1	yes	characteristic
345	C₄	#16864111	1	1	yes	characteristic
346	C₄	#16864116	1	1	yes	characteristic
347	C₂	—	8	4	no	—
348	C₂	#22563158	1	1	yes	characteristic
349	C₂	C₂×D₈×Dic₆₇	1	1	yes	characteristic
350	C₂	#22569085	1	1	yes	characteristic
351	C₂	#22572558	1	1	yes	characteristic
352	C₂	#22572633	1	1	yes	characteristic
353	C₂	#22573183	1	1	yes	characteristic
354	C₂	#22573195	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×C₄×D₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂²×D₈×C₆₇	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₄×D₈	D₆₇
hypocenter	C₆₇	#22807
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #22563158	C₂	semidirect
3. #22560831	C₂	semidirect
4. #22560369	C₂	semidirect
5. #22560209	C₂	semidirect
6. #107×Dic₆₇	C₂	semidirect
7. C₈×C₄⋊Dic₆₇	C₂	semidirect
25. C₄×D₈×C₆₇	C₄	semidirect
73. C₄⋊Dic₆₇	D₈	semidirect
143. C₈×C₆₇	#117	semidirect
186. C₆₇	#22807	semidirect
190. C₄×D₈	Dic₆₇	semidirect
322. C₈	#14524002	semidirect

Maximal subgroups

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

subgroup	index
#22563158	2
#22561076	2
#22560831	2
#22560369	2
#22560209	2
#107×Dic₆₇	2
C₈×C₄⋊Dic₆₇	2
C₂×C₄×D₈×C₆₇	2
#22807	67