Subgroups — #30745771

Order 17152.

522 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	#22568842	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863453	C₂	1	1	yes	maximal, characteristic
3	#22558081	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2920	C₂	1	1	yes	maximal, characteristic
5	#16863453	—	2	1	no	—
6	#16863453	—	2	1	no	—
7	#16863453	—	2	1	no	—
8	#16863453	C₄	1	2	yes	—
9	#16863453	C₂²	1	2	yes	—
10	#16863453	C₂²	1	2	yes	—
11	#16863449	—	2	1	no	—
12	#16863449	C₄	1	2	yes	—
13	#16863420	C₄	1	2	yes	—
14	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
15	#16863360	—	2	1	no	—
16	C₂×C₂²⋊(C₆₇⋊C₈)	C₄	1	2	yes	—
17	C₄×#11761612	—	2	1	no	—
18	C₂×#14523954	—	2	1	no	—
19	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	#432×C₆₇	—	2	1	no	—
21	#432×C₆₇	—	2	1	no	—
22	#431×C₆₇	C₄	1	2	yes	—
23	#431×C₆₇	C₄	1	2	yes	—
24	#423×C₆₇	—	2	1	no	—
25	#407×C₆₇	—	2	1	no	—
26	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
27	C₂×#11761612	—	2	2	no	—
28	C₂²×C₆₇⋊C₈	—	2	2	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
33	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
34	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
35	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
36	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
37	#14523954	—	2	2	no	—
38	#14523954	—	2	2	no	—
39	#14523954	—	2	2	no	—
40	C₄×C₆₇⋊C₈	—	2	2	no	—
41	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
44	C₂×M₁₆×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₆₇	—	2	2	no	—
58	C₄⋊C₈×C₆₇	—	2	2	no	—
59	C₂²⋊C₈×C₆₇	—	2	2	no	—
60	C₂²⋊C₈×C₆₇	—	2	2	no	—
61	#97×C₆₇	—	2	2	no	—
62	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
63	#11761612	—	4	4	no	—
64	C₂×C₆₇⋊C₈	—	2	4	no	—
65	C₂×C₆₇⋊C₈	—	2	4	no	—
66	C₂×C₆₇⋊C₈	—	4	4	no	—
67	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	—	4	2	no	—
69	C₂²×C₄×C₆₇	—	4	2	no	—
70	C₂²×C₄×C₆₇	—	4	2	no	—
71	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
89	M₁₆×C₆₇	—	4	4	no	—
90	C₂×C₈×C₆₇	—	2	4	no	—
91	C₄²×C₆₇	C₄²	1	1	yes	characteristic
92	C₄²×C₆₇	C₄²	1	1	yes	characteristic
93	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
95	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
100	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₆₇	—	2	2	no	—
108	C₂³×C₆₇	—	4	1	no	—
109	C₂³×C₆₇	—	4	1	no	—
110	C₂³×C₆₇	#95	1	1	yes	characteristic
111	C₂³×C₆₇	#95	1	1	yes	characteristic
112	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	4	1	no	—
122	C₂×C₄×C₆₇	—	4	1	no	—
123	C₂×C₄×C₆₇	—	4	1	no	—
124	C₂×C₄×C₆₇	—	4	1	no	—
125	C₂×C₄×C₆₇	—	4	1	no	—
126	C₂×C₄×C₆₇	—	4	1	no	—
127	C₂×C₄×C₆₇	—	4	1	no	—
128	C₂×C₄×C₆₇	—	4	1	no	—
129	C₂×C₄×C₆₇	—	4	1	no	—
130	C₂×C₄×C₆₇	—	4	1	no	—
131	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
132	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
133	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
134	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
135	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
136	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₈×C₆₇	—	4	4	no	—
144	C₆₇⋊C₈	—	4	4	no	—
145	C₆₇⋊C₈	—	4	4	no	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	—	2	1	no	—
148	C₂²×C₆₇	—	2	1	no	—
149	C₂²×C₆₇	—	2	1	no	—
150	C₂²×C₆₇	—	2	1	no	—
151	C₂²×C₆₇	—	2	1	no	—
152	C₂²×C₆₇	—	2	1	no	—
153	C₂²×C₆₇	—	2	1	no	—
154	C₂²×C₆₇	—	4	4	no	—
155	C₂²×C₆₇	#338	1	2	yes	—
156	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
157	C₄×C₆₇	—	2	1	no	—
158	C₄×C₆₇	—	2	1	no	—
159	C₄×C₆₇	—	2	1	no	—
160	C₄×C₆₇	—	2	1	no	—
161	C₄×C₆₇	—	2	1	no	—
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	—	2	1	no	—
167	C₄×C₆₇	#344	1	2	yes	—
168	C₄×C₆₇	#421	1	2	yes	—
169	#7998	—	67	1	no	maximal, Sylow
170	C₂×C₆₇	—	2	2	no	—
171	C₂×C₆₇	—	4	1	no	—
172	C₂×C₆₇	—	4	1	no	—
173	C₂×C₆₇	C₂×#338	1	1	yes	characteristic
174	C₂×C₆₇	#1733	1	1	yes	characteristic
175	C₂×C₆₇	C₂×#344	1	1	yes	characteristic
176	#2920	—	67	1	no	—
177	#2920	D₆₇	1	1	yes	characteristic
178	C₂×#432	—	67	1	no	—
179	#1291	—	67	4	no	—
180	C₆₇	#7998	1	1	yes	characteristic, complemented, Sylow
181	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
182	#432	—	2	1	no	—
183	#432	—	2	1	no	—
184	#432	—	67	2	no	—
185	#432	—	67	2	no	—
186	#432	—	67	2	no	—
187	#432	—	134	1	no	—
188	#432	—	134	1	no	—
189	#432	—	134	1	no	—
190	#431	—	67	2	no	—
191	#431	—	134	1	no	—
192	#431	Dic₆₇	1	2	yes	—
193	#431	Dic₆₇	1	2	yes	—
194	#423	—	2	1	no	—
195	#423	—	67	2	no	—
196	C₂×C₄⋊C₈	—	134	1	no	—
197	#407	—	2	1	no	—
198	#407	—	134	1	no	—
199	C₂×C₂²⋊C₈	—	67	2	no	—
200	C₄×M₁₆	—	134	1	no	—
201	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
202	C₂×#97	—	134	1	no	—
203	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
204	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
205	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
206	C₂×M₁₆	—	2	2	no	—
207	C₂×M₁₆	—	134	2	no	—
208	C₂²×C₈	—	134	2	no	—
209	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
211	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
212	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
213	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
214	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
215	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
216	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
219	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
220	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	—	2	2	no	—
223	C₄⋊C₈	—	67	4	no	—
224	C₄⋊C₈	—	134	2	no	—
225	C₄⋊C₈	—	134	2	no	—
226	C₄⋊C₈	—	134	2	no	—
227	C₂²⋊C₈	—	2	2	no	—
228	C₂²⋊C₈	—	2	2	no	—
229	C₂²⋊C₈	—	67	4	no	—
230	C₂²⋊C₈	—	134	2	no	—
231	C₂²⋊C₈	—	134	2	no	—
232	C₂²⋊C₈	—	134	2	no	—
233	#97	—	2	2	no	—
234	#97	—	134	2	no	—
235	#97	—	134	2	no	—
236	#97	—	134	2	no	—
237	C₄×C₈	—	134	2	no	—
238	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
239	C₂⁴	Dic₆₇⋊C₄	1	1	yes	characteristic
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
244	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
245	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
246	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
247	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
248	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
249	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
251	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
252	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
253	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
254	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
255	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
256	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
257	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
260	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
261	M₁₆	—	4	4	no	—
262	M₁₆	—	268	4	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	268	4	no	—
265	C₂×C₈	—	134	4	no	—
266	C₂×C₈	—	134	4	no	—
267	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
268	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
269	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
270	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
271	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
272	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
273	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
274	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
275	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
276	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
277	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
278	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
279	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
280	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
281	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
282	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
283	C₂³	—	2	2	no	—
284	C₂³	—	4	1	no	—
285	C₂³	—	4	1	no	—
286	C₂³	C₆₇⋊#95	1	1	yes	characteristic
287	C₂³	C₆₇⋊#95	1	1	yes	characteristic
288	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	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	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
308	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
309	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
310	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
311	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
312	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
314	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
315	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
316	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
317	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
318	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
319	C₈	—	4	4	no	—
320	C₈	—	268	4	no	—
321	C₈	—	268	4	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	4	4	no	—
331	C₂²	C₆₇⋊#338	1	2	yes	—
332	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	#16863167	1	2	yes	—
344	C₄	#16863414	1	2	yes	—
345	C₂	—	2	2	no	—
346	C₂	—	4	1	no	—
347	C₂	—	4	1	no	—
348	C₂	C₂×C₆₇⋊#338	1	1	yes	characteristic
349	C₂	#22559919	1	1	yes	characteristic
350	C₂	C₂×#16863167	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₆₇⋊#95
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2920	C₂
socle	C₂²×C₆₇	C₂×#95
prime radical	C₂⁴×C₆₇	C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2920	D₆₇
hypocenter	C₆₇	#7998
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
3. #22558081	C₂	semidirect
180. C₆₇	#7998	semidirect

Maximal subgroups

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

subgroup	index
#22568842	2
C₂×#16863453	2
#22558081	2
C₆₇×#2920	2
#7998	67