Subgroups — #30854447

Order 17152.

886 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863526	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
3	#22562686	C₂	1	1	yes	maximal, characteristic
4	#22560083	C₂	1	2	yes	maximal
5	#22560076	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₆₇⋊C₄	C₂	1	2	yes	maximal
7	C₈×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
8	#22560067	C₂	1	2	yes	maximal
9	#22560062	C₂	1	1	yes	maximal, characteristic
10	#22559895	C₂	1	2	yes	maximal
11	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
13	#16863526	C₄	1	8	yes	complemented
14	C₈×Dic₁₃₄	C₄	1	8	yes	complemented
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
18	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
19	C₂×#14523954	C₂²	1	2	yes	—
20	C₂×#14523953	C₂²	1	2	yes	—
21	C₂×#14523953	C₂²	1	2	yes	—
22	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
23	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
24	#16863302	C₄	1	2	yes	—
25	#16863294	C₄	1	2	yes	—
26	#16863280	C₄	1	4	yes	—
27	C₄×Dic₆₇⋊C₄	C₂²	1	2	yes	—
28	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
29	#16863242	C₂²	1	2	yes	—
30	#16863151	C₂²	1	2	yes	—
31	#16863150	C₂²	1	2	yes	—
32	#16863150	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₄³×C₆₇	C₂²	1	1	yes	characteristic
40	C₂²×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
42	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
43	C₄×Dic₁₃₄	C₂×C₄	1	4	yes	—
44	C₄×Dic₁₃₄	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×Dic₆₇⋊C₄	C₂×C₄	1	4	yes	—
48	C₂×Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
50	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
51	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
52	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
53	Dic₆₇⋊C₈	C₂×C₄	1	8	yes	—
54	Dic₆₇⋊C₈	C₂×C₄	1	8	yes	—
55	#14523954	—	2	4	no	—
56	#14523953	—	2	4	no	—
57	#14523953	C₂×C₄	1	8	yes	—
58	C₈×Dic₆₇	—	2	4	no	—
59	C₈×Dic₆₇	C₂×C₄	1	8	yes	—
60	C₆₇⋊#95	C₂×C₄	1	4	yes	—
61	C₆₇⋊#95	C₂×C₄	1	4	yes	—
62	C₆₇⋊#95	C₂×C₄	1	2	yes	—
63	C₆₇⋊#95	C₂×C₄	1	4	yes	—
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₆₇	—	2	2	no	—
74	C₄×C₈×C₆₇	—	2	2	no	—
75	C₄×C₈×C₆₇	—	2	2	no	—
76	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
77	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
78	C₄×C₈×C₆₇	Q₈	1	4	yes	complemented
79	C₂×Dic₁₃₄	—	2	4	no	—
80	C₂×Dic₁₃₄	C₄²	1	4	yes	—
81	C₂²×Dic₆₇	C₂²×C₄	1	4	yes	—
82	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
83	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
84	C₂×C₆₇⋊C₈	—	2	2	no	—
85	C₂×C₆₇⋊C₈	—	2	2	no	—
86	C₂×C₆₇⋊C₈	—	2	2	no	—
87	C₂×C₆₇⋊C₈	—	2	2	no	—
88	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
89	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
90	C₄⋊Dic₆₇	—	2	4	no	—
91	C₄⋊Dic₆₇	C₄²	1	4	yes	—
92	Dic₆₇⋊C₄	—	2	8	no	—
93	Dic₆₇⋊C₄	C₄²	1	8	yes	—
94	C₄×Dic₆₇	—	2	2	no	—
95	C₄×Dic₆₇	—	2	2	no	—
96	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
97	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
103	C₂×C₈×C₆₇	—	2	1	no	—
104	C₂×C₈×C₆₇	—	2	1	no	—
105	C₂×C₈×C₆₇	—	2	1	no	—
106	C₂×C₈×C₆₇	—	2	1	no	—
107	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
112	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
113	C₂×C₈×C₆₇	#41	1	2	yes	—
114	C₂×C₈×C₆₇	#41	1	2	yes	—
115	C₄²×C₆₇	—	2	4	no	—
116	C₄²×C₆₇	—	2	4	no	—
117	C₄²×C₆₇	—	2	1	no	—
118	C₄²×C₆₇	—	2	1	no	—
119	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
120	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
121	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
122	C₄²×C₆₇	#41	1	2	yes	—
123	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
130	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
132	C₂×C₄×C₆₇	#131	1	4	yes	—
133	C₂×C₄×C₆₇	#131	1	4	yes	—
134	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂³×C₄	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	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
140	C₂×Dic₆₇	—	2	4	no	—
141	C₂×Dic₆₇	—	2	4	no	—
142	C₂×Dic₆₇	—	2	4	no	—
143	C₂×Dic₆₇	—	2	4	no	—
144	C₂×Dic₆₇	—	2	4	no	—
145	C₂×Dic₆₇	C₂×C₄²	1	4	yes	—
146	C₂×Dic₆₇	C₂×C₄²	1	4	yes	—
147	Dic₁₃₄	—	2	8	no	—
148	C₈×C₆₇	—	2	2	no	—
149	C₈×C₆₇	—	2	2	no	—
150	C₈×C₆₇	C₄×Q₈	1	4	yes	—
151	C₈×C₆₇	C₄×Q₈	1	4	yes	—
152	C₆₇⋊C₈	—	2	4	no	—
153	C₆₇⋊C₈	—	2	4	no	—
154	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
155	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
156	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
157	C₂²×C₆₇	C₂×#131	1	2	yes	—
158	C₂²×C₆₇	C₂×#131	1	2	yes	—
159	C₄×C₆₇	—	2	2	no	—
160	C₄×C₆₇	—	2	4	no	—
161	C₄×C₆₇	—	2	4	no	—
162	C₄×C₆₇	#405	1	4	yes	—
163	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
164	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
165	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
166	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
167	Dic₆₇	—	2	4	no	—
168	Dic₆₇	—	2	4	no	—
169	Dic₆₇	—	4	4	no	—
170	#20516	—	67	1	no	maximal, Sylow
171	C₂×C₆₇	C₄²×Q₈	1	1	yes	characteristic
172	C₂×C₆₇	C₂×#405	1	1	yes	characteristic
173	C₂×C₆₇	C₄×#131	1	1	yes	characteristic
174	C₂×C₆₇	#2960	1	2	yes	—
175	C₂×C₆₇	#2966	1	2	yes	—
176	C₂×#446	—	67	1	no	—
177	C₂×C₈×Q₈	—	67	1	no	—
178	#2295	—	67	1	no	—
179	#1771	—	67	2	no	—
180	#1770	—	67	1	no	—
181	C₈×C₄⋊C₄	—	67	1	no	—
182	C₈×C₄⋊C₄	—	67	2	no	—
183	#1768	—	67	2	no	—
184	#1767	—	67	1	no	—
185	#1726	—	67	2	no	—
186	C₄²×C₈	D₆₇	1	1	yes	characteristic
187	C₆₇	#20516	1	1	yes	characteristic, complemented, Sylow
188	C₂×C₄×Q₈	—	67	1	no	—
189	#446	—	67	8	no	—
190	C₈×Q₈	—	67	8	no	—
191	C₂×C₄⋊C₈	—	67	2	no	—
192	C₂×C₄⋊C₈	—	67	2	no	—
193	C₂×C₄⋊C₈	—	67	1	no	—
194	C₂×C₄⋊C₈	—	67	1	no	—
195	C₂×#97	—	67	2	no	—
196	C₂×#97	—	67	2	no	—
197	C₂×#97	—	67	2	no	—
198	C₂×C₄×C₈	—	67	2	no	—
199	C₂×C₄×C₈	—	67	2	no	—
200	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
201	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
202	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
203	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
204	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
205	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
206	#391	—	67	2	no	—
207	#389	—	67	2	no	—
208	#387	—	67	4	no	—
209	C₄×C₄⋊C₄	—	67	1	no	—
210	C₄×C₄⋊C₄	—	67	2	no	—
211	#376	—	67	2	no	—
212	C₄³	C₂×D₆₇	1	1	yes	characteristic
213	#336	—	67	2	no	—
214	#336	—	67	2	no	—
215	#336	—	67	2	no	—
216	C₂²×Q₈	—	67	1	no	—
217	C₂²×C₈	—	67	2	no	—
218	C₂²×C₈	—	67	2	no	—
219	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
221	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
222	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
223	C₄×Q₈	—	67	4	no	—
224	C₄×Q₈	—	67	4	no	—
225	C₂×C₄⋊C₄	—	67	2	no	—
226	C₂×C₄⋊C₄	—	67	1	no	—
227	C₂×C₄⋊C₄	—	67	4	no	—
228	C₂×C₄⋊C₄	—	67	2	no	—
229	C₂×C₄²	—	67	2	no	—
230	C₂×C₄²	—	67	2	no	—
231	C₂×C₄²	C₄×D₆₇	1	2	yes	—
232	C₂×C₄²	C₄×D₆₇	1	2	yes	—
233	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
234	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
236	C₄⋊C₈	—	67	8	no	—
237	C₄⋊C₈	—	67	8	no	—
238	C₄⋊C₈	—	67	4	no	—
239	C₄⋊C₈	—	67	4	no	—
240	#97	—	67	8	no	—
241	#97	—	134	4	no	—
242	#97	—	134	4	no	—
243	C₄×C₈	—	2	2	no	—
244	C₄×C₈	—	2	2	no	—
245	C₄×C₈	—	2	2	no	—
246	C₄×C₈	—	67	8	no	—
247	C₄×C₈	—	134	4	no	—
248	C₄×C₈	Dic₁₃₄	1	4	yes	complemented
249	C₄×C₈	C₄×D₆₇	1	4	yes	—
250	C₄×C₈	C₄×D₆₇	1	4	yes	—
251	#95	—	67	4	no	—
252	#95	—	67	4	no	—
253	#95	—	67	2	no	—
254	#95	—	67	4	no	—
255	C₂×Q₈	—	67	4	no	—
256	C₂×Q₈	—	134	4	no	—
257	C₂²×C₄	—	67	2	no	—
258	C₂²×C₄	—	67	2	no	—
259	C₂²×C₄	—	67	4	no	—
260	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
262	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
263	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
264	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	67	4	no	—
270	C₂×C₈	—	67	4	no	—
271	C₂×C₈	—	134	2	no	—
272	C₂×C₈	—	134	2	no	—
273	C₂×C₈	—	134	2	no	—
274	C₂×C₈	—	134	2	no	—
275	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
276	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
277	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
278	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
279	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
280	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
281	C₂×C₈	#11761628	1	2	yes	—
282	C₂×C₈	#11761628	1	2	yes	—
283	C₄⋊C₄	—	67	8	no	—
284	C₄⋊C₄	—	67	4	no	—
285	C₄⋊C₄	—	134	8	no	—
286	C₄⋊C₄	—	134	4	no	—
287	C₄²	—	2	4	no	—
288	C₄²	—	2	4	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	67	4	no	—
292	C₄²	—	67	4	no	—
293	C₄²	—	134	2	no	—
294	C₄²	—	134	2	no	—
295	C₄²	C₂×C₄×D₆₇	1	2	yes	—
296	C₄²	C₂×C₄×D₆₇	1	2	yes	—
297	C₄²	C₂×Dic₁₃₄	1	2	yes	—
298	C₄²	#11761628	1	2	yes	—
299	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
300	Q₈	—	134	8	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	67	4	no	—
307	C₂×C₄	—	67	4	no	—
308	C₂×C₄	—	134	4	no	—
309	C₂×C₄	—	134	4	no	—
310	C₂×C₄	—	134	4	no	—
311	C₂×C₄	—	134	4	no	—
312	C₂×C₄	—	134	4	no	—
313	C₂×C₄	C₄²×D₆₇	1	2	yes	—
314	C₂×C₄	C₄²×D₆₇	1	2	yes	—
315	C₂×C₄	C₄×Dic₁₃₄	1	4	yes	—
316	C₂×C₄	#14524049	1	4	yes	—
317	C₂×C₄	#14524049	1	4	yes	—
318	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
319	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
320	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
321	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
322	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
323	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	2	no	—
326	C₈	—	134	4	no	—
327	C₈	—	134	4	no	—
328	C₈	C₄×Dic₁₃₄	1	4	yes	—
329	C₈	C₄×Dic₁₃₄	1	4	yes	—
330	C₂²	C₂×C₄²×D₆₇	1	1	yes	characteristic
331	C₂²	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
332	C₂²	C₄×#11761628	1	1	yes	characteristic
333	C₂²	C₂×#14524049	1	2	yes	—
334	C₂²	C₂×#14524049	1	2	yes	—
335	C₄	—	2	2	no	—
336	C₄	—	2	4	no	—
337	C₄	—	2	4	no	—
338	C₄	—	268	4	no	—
339	C₄	—	134	4	no	—
340	C₄	—	134	4	no	—
341	C₄	#16863350	1	4	yes	—
342	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
343	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
344	C₄	C₄×#11761628	1	1	yes	characteristic
345	C₄	C₄×#11761628	1	1	yes	characteristic
346	C₂	C₄²×Dic₁₃₄	1	1	yes	characteristic
347	C₂	C₂×#16863350	1	1	yes	characteristic
348	C₂	C₄×#14524049	1	1	yes	characteristic
349	C₂	#22569143	1	2	yes	—
350	C₂	#22569184	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄²×C₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂³×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄²×C₈	D₆₇
hypocenter	C₆₇	#20516
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
13. #16863526	C₄	semidirect
14. C₈×Dic₁₃₄	C₄	semidirect
78. C₄×C₈×C₆₇	Q₈	semidirect
187. C₆₇	#20516	semidirect
248. C₄×C₈	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863526	2
C₂×C₈×Dic₁₃₄	2
#22562686	2
#22560083	2
#22560076	2
C₈×Dic₆₇⋊C₄	2
C₈×C₄⋊Dic₆₇	2
#22560067	2
#22560062	2
#22559895	2
C₄²×C₈×C₆₇	2
#20516	67