Subgroups — #30854887

Order 17152.

826 conjugacy classes of subgroups.

The subgroup lattice has 344 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	#22560080	C₂	1	2	yes	maximal
5	#22560076	C₂	1	1	yes	maximal, characteristic
6	C₈×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
7	#22560070	C₂	1	2	yes	maximal
8	#22560065	C₂	1	2	yes	maximal
9	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
10	#22559959	C₂	1	2	yes	maximal
11	C₆₇×#1744	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×#14523953	C₂²	1	2	yes	—
20	C₂×#14523953	C₂²	1	2	yes	—
21	C₂×C₄×C₆₇⋊C₈	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	C₄³×C₆₇	C₂²	1	1	yes	characteristic
36	#336×C₆₇	C₂²	1	1	yes	characteristic
37	#336×C₆₇	C₂²	1	1	yes	characteristic
38	#336×C₆₇	C₂²	1	1	yes	characteristic
39	#336×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	#14523953	—	4	2	no	—
56	#14523953	C₂×C₄	1	8	yes	—
57	C₄×C₆₇⋊C₈	—	4	2	no	—
58	C₈×Dic₆₇	—	4	2	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	C₂×C₄	1	4	yes	—
74	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
75	C₂×Dic₁₃₄	—	2	4	no	—
76	C₂×Dic₁₃₄	C₄²	1	4	yes	—
77	C₂²×Dic₆₇	C₂²×C₄	1	4	yes	—
78	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
79	C₂²×Dic₆₇	C₂²×C₄	1	2	yes	—
80	C₂×C₆₇⋊C₈	—	2	2	no	—
81	C₂×C₆₇⋊C₈	—	2	2	no	—
82	C₂×C₆₇⋊C₈	—	2	2	no	—
83	C₂×C₆₇⋊C₈	—	2	2	no	—
84	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
85	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
86	C₄⋊Dic₆₇	—	2	4	no	—
87	C₄⋊Dic₆₇	C₄²	1	4	yes	—
88	Dic₆₇⋊C₄	—	2	8	no	—
89	Dic₆₇⋊C₄	C₄²	1	8	yes	—
90	C₄×Dic₆₇	—	2	2	no	—
91	C₄×Dic₆₇	—	2	2	no	—
92	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
93	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
94	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
99	C₂×C₈×C₆₇	—	2	1	no	—
100	C₂×C₈×C₆₇	—	2	1	no	—
101	C₂×C₈×C₆₇	—	2	1	no	—
102	C₂×C₈×C₆₇	—	2	1	no	—
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	C₄²×C₆₇	—	2	4	no	—
112	C₄²×C₆₇	—	2	4	no	—
113	C₄²×C₆₇	—	2	1	no	—
114	C₄²×C₆₇	—	2	1	no	—
115	C₄²×C₆₇	—	2	1	no	—
116	C₄²×C₆₇	—	2	1	no	—
117	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
118	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₆₇	#131	1	4	yes	—
129	C₂×C₄×C₆₇	#131	1	4	yes	—
130	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
134	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
135	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
136	C₂×Dic₆₇	—	2	4	no	—
137	C₂×Dic₆₇	—	2	4	no	—
138	C₂×Dic₆₇	—	2	4	no	—
139	C₂×Dic₆₇	—	2	4	no	—
140	C₂×Dic₆₇	—	2	4	no	—
141	C₂×Dic₆₇	C₂×C₄²	1	4	yes	—
142	C₂×Dic₆₇	C₂×C₄²	1	4	yes	—
143	Dic₁₃₄	—	2	8	no	—
144	C₈×C₆₇	—	2	2	no	—
145	C₈×C₆₇	—	2	2	no	—
146	C₈×C₆₇	—	2	2	no	—
147	C₈×C₆₇	—	2	2	no	—
148	C₆₇⋊C₈	—	2	4	no	—
149	C₆₇⋊C₈	—	2	4	no	—
150	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
151	C₂²×C₆₇	#519	1	1	yes	characteristic
152	C₂²×C₆₇	#520	1	1	yes	characteristic
153	C₂²×C₆₇	C₂×#131	1	2	yes	—
154	C₂²×C₆₇	C₂×#131	1	2	yes	—
155	C₄×C₆₇	—	2	2	no	—
156	C₄×C₆₇	—	2	4	no	—
157	C₄×C₆₇	—	4	2	no	—
158	C₄×C₆₇	#405	1	4	yes	—
159	C₄×C₆₇	#519	1	1	yes	characteristic
160	C₄×C₆₇	#519	1	1	yes	characteristic
161	C₄×C₆₇	#520	1	1	yes	characteristic
162	C₄×C₆₇	#520	1	1	yes	characteristic
163	Dic₆₇	—	2	4	no	—
164	Dic₆₇	—	2	4	no	—
165	Dic₆₇	—	4	4	no	—
166	#20571	—	67	1	no	maximal, Sylow
167	C₂×C₆₇	#2276	1	1	yes	characteristic
168	C₂×C₆₇	C₂×#405	1	1	yes	characteristic
169	C₂×C₆₇	C₄×#131	1	1	yes	characteristic
170	C₂×C₆₇	#2986	1	2	yes	—
171	C₂×C₆₇	#2994	1	2	yes	—
172	C₂×#446	—	67	1	no	—
173	C₂×C₈×Q₈	—	67	1	no	—
174	#2295	—	67	1	no	—
175	#1771	—	67	2	no	—
176	#1770	—	67	1	no	—
177	C₈×C₄⋊C₄	—	67	1	no	—
178	C₈×C₄⋊C₄	—	67	2	no	—
179	#1768	—	67	2	no	—
180	C₄×C₄⋊C₈	—	67	1	no	—
181	#1746	—	67	2	no	—
182	#1744	D₆₇	1	1	yes	characteristic
183	C₆₇	#20571	1	1	yes	characteristic, complemented, Sylow
184	C₂×C₄×Q₈	—	67	1	no	—
185	#446	—	67	8	no	—
186	C₈×Q₈	—	67	8	no	—
187	C₂×C₄⋊C₈	—	67	2	no	—
188	C₂×C₄⋊C₈	—	67	2	no	—
189	C₂×C₄⋊C₈	—	67	1	no	—
190	C₂×C₄⋊C₈	—	67	1	no	—
191	C₂×#97	—	67	2	no	—
192	C₂×#97	—	67	2	no	—
193	C₂×C₄×C₈	—	67	2	no	—
194	C₂×C₄×C₈	—	67	2	no	—
195	C₂×C₄×C₈	—	67	2	no	—
196	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
198	#391	—	67	2	no	—
199	#389	—	67	2	no	—
200	#387	—	67	4	no	—
201	C₄×C₄⋊C₄	—	67	1	no	—
202	C₄×C₄⋊C₄	—	67	2	no	—
203	#376	—	67	2	no	—
204	C₄³	C₂×D₆₇	1	1	yes	characteristic
205	#336	—	67	2	no	—
206	#336	—	67	2	no	—
207	#336	—	67	2	no	—
208	#336	C₂×D₆₇	1	1	yes	characteristic
209	#336	C₂×D₆₇	1	1	yes	characteristic
210	#336	C₂×D₆₇	1	1	yes	characteristic
211	#336	C₂×D₆₇	1	1	yes	characteristic
212	C₂²×Q₈	—	67	1	no	—
213	C₂²×C₈	—	67	2	no	—
214	C₂²×C₈	—	67	2	no	—
215	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
216	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
217	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
219	C₄×Q₈	—	67	4	no	—
220	C₄×Q₈	—	67	4	no	—
221	C₂×C₄⋊C₄	—	67	2	no	—
222	C₂×C₄⋊C₄	—	67	1	no	—
223	C₂×C₄⋊C₄	—	67	4	no	—
224	C₂×C₄⋊C₄	—	67	2	no	—
225	C₂×C₄²	—	67	2	no	—
226	C₂×C₄²	—	67	2	no	—
227	C₂×C₄²	C₄×D₆₇	1	2	yes	—
228	C₂×C₄²	C₄×D₆₇	1	2	yes	—
229	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
232	C₄⋊C₈	—	67	8	no	—
233	C₄⋊C₈	—	67	8	no	—
234	C₄⋊C₈	—	67	4	no	—
235	C₄⋊C₈	—	67	4	no	—
236	#97	—	268	2	no	—
237	#97	—	67	8	no	—
238	C₄×C₈	—	268	2	no	—
239	C₄×C₈	—	268	2	no	—
240	C₄×C₈	—	67	8	no	—
241	C₄×C₈	C₄×D₆₇	1	4	yes	—
242	C₄×C₈	C₄×D₆₇	1	4	yes	—
243	#95	—	67	4	no	—
244	#95	—	67	4	no	—
245	#95	—	67	2	no	—
246	#95	—	67	4	no	—
247	C₂×Q₈	—	67	4	no	—
248	C₂×Q₈	—	134	4	no	—
249	C₂²×C₄	—	67	2	no	—
250	C₂²×C₄	—	67	2	no	—
251	C₂²×C₄	—	67	4	no	—
252	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
253	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
255	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
256	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	67	4	no	—
266	C₂×C₈	—	67	4	no	—
267	C₂×C₈	—	134	2	no	—
268	C₂×C₈	—	134	2	no	—
269	C₂×C₈	—	134	2	no	—
270	C₂×C₈	—	134	2	no	—
271	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
272	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
273	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
274	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
275	C₄⋊C₄	—	67	8	no	—
276	C₄⋊C₄	—	67	4	no	—
277	C₄⋊C₄	—	134	8	no	—
278	C₄⋊C₄	—	134	4	no	—
279	C₄²	—	2	4	no	—
280	C₄²	—	2	4	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	67	4	no	—
286	C₄²	—	67	4	no	—
287	C₄²	—	134	2	no	—
288	C₄²	—	134	2	no	—
289	C₄²	C₂×C₄×D₆₇	1	2	yes	—
290	C₄²	C₂×C₄×D₆₇	1	2	yes	—
291	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
292	Q₈	—	134	8	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	67	4	no	—
300	C₂×C₄	—	67	4	no	—
301	C₂×C₄	—	134	4	no	—
302	C₂×C₄	—	134	4	no	—
303	C₂×C₄	—	134	4	no	—
304	C₂×C₄	—	134	4	no	—
305	C₂×C₄	—	134	4	no	—
306	C₂×C₄	C₄²×D₆₇	1	2	yes	—
307	C₂×C₄	C₄²×D₆₇	1	2	yes	—
308	C₂×C₄	#14524049	1	4	yes	—
309	C₂×C₄	#14524051	1	4	yes	—
310	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
312	C₂×C₄	#14524089	1	1	yes	characteristic
313	C₂×C₄	#14524090	1	1	yes	characteristic
314	C₂×C₄	#14524090	1	1	yes	characteristic
315	C₂×C₄	#14524090	1	1	yes	characteristic
316	C₈	—	2	2	no	—
317	C₈	—	2	2	no	—
318	C₈	—	2	2	no	—
319	C₈	—	2	2	no	—
320	C₈	—	134	4	no	—
321	C₈	—	134	4	no	—
322	C₂²	C₂×C₄²×D₆₇	1	1	yes	characteristic
323	C₂²	#16863873	1	1	yes	characteristic
324	C₂²	#16863876	1	1	yes	characteristic
325	C₂²	C₂×#14524049	1	2	yes	—
326	C₂²	C₂×#14524051	1	2	yes	—
327	C₄	—	2	4	no	—
328	C₄	—	2	2	no	—
329	C₄	—	4	2	no	—
330	C₄	—	268	4	no	—
331	C₄	—	134	4	no	—
332	C₄	—	134	4	no	—
333	C₄	#16863353	1	4	yes	—
334	C₄	#16863873	1	1	yes	characteristic
335	C₄	#16863873	1	1	yes	characteristic
336	C₄	#16863876	1	1	yes	characteristic
337	C₄	#16863876	1	1	yes	characteristic
338	C₂	#22562572	1	1	yes	characteristic
339	C₂	C₂×#16863353	1	1	yes	characteristic
340	C₂	C₄×#14524049	1	1	yes	characteristic
341	C₂	#22569310	1	2	yes	—
342	C₂	#22569371	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂²×C₄²
Fitting	C₆₇×#1744	C₂
socle, prime radical	C₂³×C₆₇	C₂³×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1744	D₆₇
hypocenter	C₆₇	#20571
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1744

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
183. C₆₇	#20571	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
#22560080	2
#22560076	2
C₈×C₄⋊Dic₆₇	2
#22560070	2
#22560065	2
C₄×C₄⋊(C₆₇⋊C₈)	2
#22559959	2
C₆₇×#1744	2
#20571	67