Subgroups — #30825075

Order 17152.

610 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568875	C₂	1	2	yes	maximal
2	#22558989	C₂	1	4	yes	maximal, complemented
3	#22558980	C₂	1	2	yes	maximal, complemented
4	#22558972	C₂	1	2	yes	maximal, complemented
5	#22558968	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558957	C₂	1	2	yes	maximal, complemented
7	C₂×#16863149	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#16863457	—	2	4	no	—
10	#16863453	C₂²	1	4	yes	—
11	#16863449	C₂²	1	4	yes	—
12	#16863420	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
14	#16863363	—	2	2	no	—
15	#16863352	—	2	2	no	—
16	#16863149	C₂²	1	2	yes	—
17	#16863149	C₂²	1	2	yes	—
18	#16863149	C₂²	1	1	yes	characteristic
19	#16863149	C₂²	1	1	yes	characteristic
20	#16863149	C₂²	1	2	yes	—
21	#16863147	C₂²	1	4	yes	—
22	#16863147	C₂²	1	4	yes	—
23	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
24	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic
25	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic
26	C₈×D₄×C₆₇	C₄	1	1	yes	characteristic
27	C₈×D₄×C₆₇	C₄	1	2	yes	—
28	C₈×D₄×C₆₇	C₄	1	1	yes	characteristic
29	C₈×D₄×C₆₇	C₄	1	2	yes	—
30	#431×C₆₇	C₂²	1	1	yes	characteristic
31	#431×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
34	C₂×#11761612	—	2	2	no	—
35	C₂²×C₆₇⋊C₈	—	2	2	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
38	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
39	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
40	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
41	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
42	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
43	#14523954	—	2	4	no	—
44	C₄×C₆₇⋊C₈	—	2	4	no	—
45	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
50	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄×Q₈×C₆₇	Q₈	1	1	yes	characteristic
52	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
53	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₄×D₄×C₆₇	Q₈	1	2	yes	—
55	C₄×D₄×C₆₇	Q₈	1	1	yes	characteristic
56	#117×C₆₇	D₄	1	1	yes	characteristic
57	#117×C₆₇	D₄	1	2	yes	—
58	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
60	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
63	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
64	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
65	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
66	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₆₇	C₂³	1	2	yes	—
72	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	#11761612	—	4	4	no	—
75	C₂×C₆₇⋊C₈	—	2	2	no	—
76	C₂×C₆₇⋊C₈	—	2	4	no	—
77	C₂×C₆₇⋊C₈	—	2	2	no	—
78	C₂×C₆₇⋊C₈	—	4	4	no	—
79	#41×C₆₇	—	2	2	no	—
80	#41×C₆₇	—	2	2	no	—
81	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
82	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
83	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	4	no	—
92	C₂×C₈×C₆₇	—	4	1	no	—
93	C₂×C₈×C₆₇	—	4	2	no	—
94	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
101	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
102	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
103	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
105	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
106	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
107	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
108	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₆₇	C₂×D₄	1	2	yes	—
110	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
112	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
113	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	Q₈×C₆₇	—	2	2	no	—
117	D₄×C₆₇	—	2	4	no	—
118	D₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	2	1	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	—	4	1	no	—
126	C₂×C₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	#135	1	2	yes	—
134	C₂×C₄×C₆₇	#135	1	2	yes	—
135	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
138	C₈×C₆₇	—	2	2	no	—
139	C₈×C₆₇	—	2	1	no	—
140	C₈×C₆₇	—	2	1	no	—
141	C₈×C₆₇	—	2	2	no	—
142	C₈×C₆₇	—	2	1	no	—
143	C₈×C₆₇	#131	1	2	yes	—
144	C₆₇⋊C₈	—	4	2	no	—
145	C₆₇⋊C₈	—	4	2	no	—
146	C₆₇⋊C₈	—	4	4	no	—
147	C₂²×C₆₇	—	2	1	no	—
148	C₂²×C₆₇	—	2	1	no	—
149	C₂²×C₆₇	—	2	2	no	—
150	C₂²×C₆₇	—	2	2	no	—
151	C₂²×C₆₇	—	2	2	no	—
152	C₂²×C₆₇	#427	1	2	yes	—
153	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	2	2	no	—
156	C₄×C₆₇	—	2	2	no	—
157	C₄×C₆₇	—	2	1	no	—
158	C₄×C₆₇	—	2	1	no	—
159	C₄×C₆₇	—	2	2	no	—
160	C₄×C₆₇	#427	1	2	yes	—
161	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
162	C₄×C₆₇	#568	1	1	yes	characteristic
163	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
164	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
165	#17829	—	67	1	no	maximal, Sylow
166	C₂×C₆₇	—	2	2	no	—
167	C₂×C₆₇	—	4	2	no	—
168	C₂×C₆₇	#2906	1	1	yes	characteristic
169	C₂×C₆₇	C₂×#427	1	1	yes	characteristic
170	C₂×C₆₇	#2916	1	1	yes	characteristic
171	C₈×#41	D₆₇	1	1	yes	characteristic
172	#2925	—	67	2	no	—
173	#1572	—	67	4	no	—
174	#1570	—	67	2	no	—
175	#1568	—	67	2	no	—
176	#1567	—	67	1	no	—
177	#1564	—	67	2	no	—
178	C₂×#335	—	67	1	no	—
179	C₆₇	#17829	1	1	yes	characteristic, complemented, Sylow
180	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
181	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
182	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
183	C₈×D₄	Dic₆₇	1	1	yes	characteristic
184	C₈×D₄	Dic₆₇	1	2	yes	—
185	C₈×D₄	Dic₆₇	1	1	yes	characteristic
186	C₈×D₄	Dic₆₇	1	2	yes	—
187	#433	—	134	4	no	—
188	#432	—	67	4	no	—
189	#431	—	67	4	no	—
190	#431	C₂×D₆₇	1	1	yes	characteristic
191	#431	C₂×D₆₇	1	2	yes	—
192	#423	—	67	2	no	—
193	C₂×C₄⋊C₈	—	67	2	no	—
194	#408	—	134	2	no	—
195	#405	—	134	2	no	—
196	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
198	#335	—	67	1	no	—
199	#335	—	67	1	no	—
200	#335	—	67	2	no	—
201	#335	—	67	2	no	—
202	#335	—	67	2	no	—
203	#334	—	67	4	no	—
204	#334	—	67	4	no	—
205	C₂×#41	D₂₆₈	1	1	yes	characteristic
206	C₂×M₁₆	—	134	2	no	—
207	C₂²×C₈	—	134	2	no	—
208	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
209	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
210	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
211	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
212	C₄×Q₈	Dic₁₃₄	1	1	yes	characteristic
213	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
214	C₄×D₄	Dic₁₃₄	1	2	yes	—
215	C₄×D₄	Dic₁₃₄	1	1	yes	characteristic
216	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
217	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
218	#117	D₂₆₈	1	1	yes	characteristic
219	#117	D₂₆₈	1	2	yes	—
220	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
222	C₄⋊C₈	—	67	2	no	—
223	C₄⋊C₈	—	67	2	no	—
224	C₄⋊C₈	—	67	4	no	—
225	C₄⋊C₈	—	134	2	no	—
226	C₄⋊C₈	—	134	2	no	—
227	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
228	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
229	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
230	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
231	C₂²⋊C₈	—	134	4	no	—
232	C₂²⋊C₈	—	134	4	no	—
233	C₂²⋊C₈	C₂×Dic₆₇	1	2	yes	—
234	C₂²⋊C₈	C₂×Dic₆₇	1	2	yes	—
235	C₂²⋊C₈	C₂×Dic₆₇	1	2	yes	—
236	#97	—	134	4	no	—
237	C₄×C₈	—	134	4	no	—
238	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
239	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
240	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
241	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
242	C₄×C₈	C₂²×D₆₇	1	2	yes	—
243	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
245	#41	—	2	2	no	—
246	#41	—	2	2	no	—
247	C₂×Q₈	C₄⋊Dic₆₇	1	1	yes	characteristic
248	C₂×D₄	C₄⋊Dic₆₇	1	2	yes	—
249	C₂×D₄	C₄⋊Dic₆₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
251	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
253	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
254	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
255	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
256	M₁₆	—	268	4	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	—	268	4	no	—
262	C₂×C₈	—	134	2	no	—
263	C₂×C₈	—	134	4	no	—
264	C₂×C₈	—	134	2	no	—
265	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
266	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
267	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
268	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
269	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
270	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
271	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
272	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
273	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
274	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
275	C₂²⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
276	C₂²⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
277	C₂²⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
278	C₄²	C₂²×Dic₆₇	1	2	yes	—
279	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
280	C₄²	C₂×D₂₆₈	1	2	yes	—
281	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
282	C₄²	C₂×Dic₁₃₄	1	2	yes	—
283	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
284	C₄²	C₂³×D₆₇	1	1	yes	characteristic
285	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
286	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
287	Q₈	—	2	2	no	—
288	D₄	—	2	4	no	—
289	D₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
299	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
300	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
301	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
302	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
304	C₂×C₄	#14524065	1	2	yes	—
305	C₂×C₄	#14524065	1	2	yes	—
306	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
308	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	2	no	—
313	C₈	—	2	1	no	—
314	C₈	—	268	2	no	—
315	C₈	—	268	2	no	—
316	C₈	—	268	4	no	—
317	C₈	#14524050	1	2	yes	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	#16863434	1	2	yes	—
324	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	#16863434	1	2	yes	—
332	C₄	C₂×#14524050	1	1	yes	characteristic
333	C₄	#16864163	1	1	yes	characteristic
334	C₄	C₂×#14524065	1	1	yes	characteristic
335	C₄	C₂×#14524065	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	4	2	no	—
338	C₂	#22568777	1	1	yes	characteristic
339	C₂	C₂×#16863434	1	1	yes	characteristic
340	C₂	#22568820	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₈×#41×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₆₇	D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₈×#41	D₆₇
hypocenter	C₆₇	#17829
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #22558989	C₂	semidirect
3. #22558980	C₂	semidirect
4. #22558972	C₂	semidirect
5. #22558968	C₂	semidirect
6. #22558957	C₂	semidirect
7. C₂×#16863149	C₂	semidirect
179. C₆₇	#17829	semidirect

Maximal subgroups

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

subgroup	index
#22568875	2
#22558989	2
#22558980	2
#22558972	2
#22558968	2
#22558957	2
C₂×#16863149	2
C₈×#41×C₆₇	2
#17829	67