Subgroups — #30827574

Order 17152.

718 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	#22569244	C₂	1	2	yes	maximal, complemented
2	#22559123	C₂	1	2	yes	maximal, complemented
3	#22559039	C₂	1	4	yes	maximal, complemented
4	#22559033	C₂	1	2	yes	maximal, complemented
5	#22558972	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558652	C₂	1	2	yes	maximal, complemented
7	#22558498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#16863474	C₂²	1	4	yes	complemented
10	#16863468	—	2	4	no	—
11	D₄×C₆₇⋊C₈	C₂²	1	4	yes	complemented
12	#16863453	C₂²	1	2	yes	—
13	#16863449	C₂²	1	1	yes	characteristic
14	#16863360	—	2	2	no	—
15	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
16	C₄×#11761612	C₂²	1	1	yes	characteristic
17	#16863149	C₂²	1	2	yes	complemented
18	#16863147	C₂²	1	2	yes	complemented
19	D₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	complemented
20	D₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	complemented
21	#16863118	C₂²	1	2	yes	complemented
22	#16863117	C₂²	1	2	yes	complemented
23	C₂²×D₈×C₆₇	C₄	1	1	yes	characteristic
24	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
25	C₄×D₈×C₆₇	C₄	1	1	yes	characteristic
26	C₄×D₈×C₆₇	C₄	1	2	yes	—
27	C₄×D₈×C₆₇	C₄	1	1	yes	characteristic
28	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×D₈×C₆₇	C₂²	1	2	yes	—
30	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
32	C₂×#102×C₆₇	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#11761612	—	2	4	no	—
35	C₂×#11761612	—	2	1	no	—
36	C₂²×C₆₇⋊C₈	—	2	4	no	—
37	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
38	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
39	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
40	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
41	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
42	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
43	#14523954	—	2	2	no	—
44	#14523954	—	2	1	no	—
45	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
46	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
47	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
48	C₂³×C₄×C₆₇	—	2	2	no	—
49	C₂×D₈×C₆₇	—	2	4	no	—
50	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₄×D₄×C₆₇	—	2	2	no	—
56	C₄×D₄×C₆₇	—	2	2	no	—
57	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
58	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
59	C₄×D₄×C₆₇	C₂³	1	2	yes	—
60	C₄×D₄×C₆₇	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	#107×C₆₇	C₂×C₄	1	2	yes	—
65	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
66	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
67	#102×C₆₇	C₂×C₄	1	2	yes	—
68	#102×C₆₇	C₂×C₄	1	2	yes	—
69	#102×C₆₇	C₂×C₄	1	2	yes	—
70	#102×C₆₇	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	#11761612	—	4	2	no	—
76	#11761612	—	8	4	no	—
77	C₂×C₆₇⋊C₈	—	2	4	no	—
78	C₂×C₆₇⋊C₈	—	2	2	no	—
79	C₂×C₆₇⋊C₈	—	8	4	no	—
80	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
81	C₂⁴×C₆₇	—	2	2	no	—
82	C₂×D₄×C₆₇	—	2	4	no	—
83	C₂×D₄×C₆₇	—	2	2	no	—
84	C₂×D₄×C₆₇	—	4	2	no	—
85	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	2	no	—
90	C₂²×C₄×C₆₇	—	2	2	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	8	4	no	—
93	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
96	D₈×C₆₇	—	2	4	no	—
97	D₈×C₆₇	—	2	2	no	—
98	D₈×C₆₇	M₁₆	1	4	yes	complemented
99	C₂×C₈×C₆₇	—	2	2	no	—
100	C₂×C₈×C₆₇	—	4	1	no	—
101	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
107	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
108	C₂²⋊C₄×C₆₇	—	2	2	no	—
109	C₂²⋊C₄×C₆₇	—	2	2	no	—
110	C₂²⋊C₄×C₆₇	—	2	2	no	—
111	C₂²⋊C₄×C₆₇	—	2	2	no	—
112	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₇	#41	1	1	yes	characteristic
115	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
116	C₂³×C₆₇	—	2	2	no	—
117	C₂³×C₆₇	—	2	2	no	—
118	C₂³×C₆₇	—	4	4	no	—
119	C₂³×C₆₇	—	4	4	no	—
120	C₂³×C₆₇	—	8	2	no	—
121	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
122	D₄×C₆₇	—	2	2	no	—
123	D₄×C₆₇	—	2	2	no	—
124	D₄×C₆₇	—	2	2	no	—
125	D₄×C₆₇	—	4	4	no	—
126	D₄×C₆₇	C₂×M₁₆	1	4	yes	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	4	4	no	—
134	C₂×C₄×C₆₇	—	4	4	no	—
135	C₂×C₄×C₆₇	—	8	2	no	—
136	C₂×C₄×C₆₇	—	8	2	no	—
137	C₂×C₄×C₆₇	—	8	2	no	—
138	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
144	C₈×C₆₇	—	2	1	no	—
145	C₈×C₆₇	—	2	1	no	—
146	C₈×C₆₇	—	2	1	no	—
147	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
148	C₆₇⋊C₈	—	2	4	no	—
149	C₆₇⋊C₈	—	4	2	no	—
150	C₆₇⋊C₈	—	4	4	no	—
151	C₂²×C₆₇	—	2	1	no	—
152	C₂²×C₆₇	—	2	1	no	—
153	C₂²×C₆₇	—	2	4	no	—
154	C₂²×C₆₇	—	4	2	no	—
155	C₂²×C₆₇	—	4	2	no	—
156	C₂²×C₆₇	—	4	2	no	—
157	C₂²×C₆₇	—	4	2	no	—
158	C₂²×C₆₇	—	4	2	no	—
159	C₂²×C₆₇	—	8	8	no	—
160	C₂²×C₆₇	#444	1	2	yes	—
161	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
162	C₄×C₆₇	—	2	1	no	—
163	C₄×C₆₇	—	2	1	no	—
164	C₄×C₆₇	—	2	1	no	—
165	C₄×C₆₇	—	2	1	no	—
166	C₄×C₆₇	—	4	2	no	—
167	C₄×C₆₇	—	4	2	no	—
168	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
169	C₄×C₆₇	#568	1	1	yes	characteristic
170	C₄×C₆₇	#576	1	2	yes	—
171	#18064	—	67	1	no	maximal, Sylow
172	C₂×C₆₇	—	2	2	no	—
173	C₂×C₆₇	—	4	4	no	—
174	C₂×C₆₇	—	8	2	no	—
175	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
176	C₂×C₆₇	#2948	1	1	yes	characteristic
177	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
178	#2974	—	67	2	no	—
179	C₂×C₄×D₈	D₆₇	1	1	yes	characteristic
180	#1594	—	67	2	no	—
181	#1581	—	67	4	no	—
182	#1580	—	67	2	no	—
183	#1568	—	67	1	no	—
184	#1490	—	67	2	no	—
185	#1451	—	67	1	no	—
186	C₆₇	#18064	1	1	yes	characteristic, complemented, Sylow
187	C₂²×D₈	Dic₆₇	1	1	yes	characteristic
188	C₂×C₄×D₄	C₂×D₆₇	1	2	yes	—
189	C₄×D₈	Dic₆₇	1	2	yes	—
190	C₄×D₈	Dic₆₇	1	1	yes	characteristic
191	C₄×D₈	Dic₆₇	1	1	yes	characteristic
192	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
193	C₄×D₈	C₂×D₆₇	1	2	yes	—
194	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
195	#436	—	67	4	no	—
196	#435	—	134	4	no	—
197	C₈×D₄	—	67	4	no	—
198	#432	—	67	2	no	—
199	#431	—	67	1	no	—
200	C₂×#107	Dic₆₇	1	1	yes	characteristic
201	C₂×#102	Dic₆₇	1	2	yes	—
202	#407	—	134	2	no	—
203	C₂×C₂²⋊C₈	—	134	2	no	—
204	C₄×M₁₆	—	67	1	no	—
205	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
206	#335	—	67	2	no	—
207	#334	—	67	2	no	—
208	D₄⋊C₈	—	67	4	no	—
209	D₄⋊C₈	—	67	4	no	—
210	#322	—	67	2	no	—
211	#322	—	67	2	no	—
212	C₂²×D₄	C₂×Dic₆₇	1	2	yes	—
213	C₂³×C₄	—	2	2	no	—
214	C₂×D₈	—	2	4	no	—
215	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
216	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic
217	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic
218	C₂×M₁₆	—	134	4	no	—
219	C₂×M₁₆	—	134	1	no	—
220	C₂²×C₈	—	134	4	no	—
221	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
222	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
223	C₄×D₄	—	2	2	no	—
224	C₄×D₄	—	2	2	no	—
225	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
226	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
227	C₄×D₄	C₂²×D₆₇	1	2	yes	—
228	C₄×D₄	C₂²×D₆₇	1	2	yes	—
229	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
232	#107	C₂×Dic₆₇	1	2	yes	—
233	#107	C₂×Dic₆₇	1	1	yes	characteristic
234	#107	C₂×Dic₆₇	1	1	yes	characteristic
235	C₄⋊C₈	—	67	2	no	—
236	C₄⋊C₈	—	67	4	no	—
237	#102	C₂×Dic₆₇	1	2	yes	—
238	#102	C₂×Dic₆₇	1	2	yes	—
239	#102	C₂×Dic₆₇	1	2	yes	—
240	#102	C₂×Dic₆₇	1	2	yes	—
241	C₂²⋊C₈	—	67	2	no	—
242	C₂²⋊C₈	—	134	4	no	—
243	C₂²⋊C₈	—	134	4	no	—
244	C₂²⋊C₈	—	134	2	no	—
245	#97	—	134	2	no	—
246	#97	—	134	1	no	—
247	C₄×C₈	—	67	2	no	—
248	C₄×C₈	—	67	2	no	—
249	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
250	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
251	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
252	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
253	C₂⁴	—	2	2	no	—
254	C₂×D₄	—	2	4	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
258	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	8	4	no	—
265	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
266	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
267	C₂²×C₄	#11761627	1	1	yes	characteristic
268	D₈	—	2	4	no	—
269	D₈	—	2	2	no	—
270	D₈	#11761612	1	4	yes	complemented
271	M₁₆	—	268	2	no	—
272	M₁₆	—	536	4	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	—	536	4	no	—
276	C₂×C₈	—	67	4	no	—
277	C₂×C₈	—	134	4	no	—
278	C₂×C₈	—	134	2	no	—
279	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
280	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
281	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
282	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
285	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
286	C₂²⋊C₄	—	2	2	no	—
287	C₂²⋊C₄	—	2	2	no	—
288	C₂²⋊C₄	—	2	2	no	—
289	C₂²⋊C₄	—	2	2	no	—
290	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
291	C₄²	D₄×D₆₇	1	1	yes	characteristic
292	C₄²	#11761627	1	1	yes	characteristic
293	C₄²	C₂³×D₆₇	1	1	yes	characteristic
294	C₂³	—	2	2	no	—
295	C₂³	—	2	2	no	—
296	C₂³	—	4	4	no	—
297	C₂³	—	4	4	no	—
298	C₂³	—	8	2	no	—
299	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
300	D₄	—	2	2	no	—
301	D₄	—	2	2	no	—
302	D₄	—	2	2	no	—
303	D₄	—	4	4	no	—
304	D₄	C₂×#11761612	1	4	yes	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
317	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
318	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
319	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
320	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
321	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	268	2	no	—
326	C₈	—	268	4	no	—
327	C₈	—	134	4	no	—
328	C₈	C₂×#11761612	1	2	yes	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	4	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	4	2	no	—
335	C₂²	—	4	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	8	8	no	—
338	C₂²	#16863518	1	2	yes	—
339	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	2	no	—
346	C₄	C₂²×#11761612	1	1	yes	characteristic
347	C₄	#16864163	1	1	yes	characteristic
348	C₄	#16864197	1	2	yes	—
349	C₂	—	2	2	no	—
350	C₂	—	4	4	no	—
351	C₂	—	8	2	no	—
352	C₂	D₄×#11761612	1	1	yes	characteristic
353	C₂	#22569061	1	1	yes	characteristic
354	C₂	C₂×#16863518	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×C₄×D₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₈×C₆₇	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×C₄×D₈	D₆₇
hypocenter	C₆₇	#18064
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #22569244	C₂	semidirect
2. #22559123	C₂	semidirect
3. #22559039	C₂	semidirect
4. #22559033	C₂	semidirect
5. #22558972	C₂	semidirect
6. #22558652	C₂	semidirect
7. #22558498	C₂	semidirect
9. #16863474	C₂²	semidirect
11. D₄×C₆₇⋊C₈	C₂²	semidirect
17. #16863149	C₂²	semidirect
18. #16863147	C₂²	semidirect
19. D₄⋊(C₆₇⋊C₈)	C₂²	semidirect
20. D₄⋊(C₆₇⋊C₈)	C₂²	semidirect
21. #16863118	C₂²	semidirect
22. #16863117	C₂²	semidirect
98. D₈×C₆₇	M₁₆	semidirect
186. C₆₇	#18064	semidirect
270. D₈	#11761612	semidirect

Maximal subgroups

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

subgroup	index
#22569244	2
#22559123	2
#22559039	2
#22559033	2
#22558972	2
#22558652	2
#22558498	2
C₂×C₄×D₈×C₆₇	2
#18064	67