Subgroups — #30745368

Order 17152.

802 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863449	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863449	C₂	1	1	yes	maximal, characteristic
3	#22558050	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863449	—	2	2	no	—
6	#16863449	—	2	2	no	—
7	#16863449	C₄	1	4	yes	complemented
8	#16863449	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
12	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
13	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
16	C₂²×C₄²×C₆₇	—	2	1	no	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
19	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
20	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂×#95×C₆₇	—	2	1	no	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	complemented
28	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
29	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
30	C₄×C₆₇⋊C₈	—	2	4	no	—
31	C₄×C₆₇⋊C₈	—	2	2	no	—
32	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	complemented
33	C₂³×C₄×C₆₇	—	2	2	no	—
34	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₆₇	C₈	1	8	yes	complemented
40	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₆₇	—	2	2	no	—
42	C₂×C₄²×C₆₇	—	2	2	no	—
43	C₂×C₄²×C₆₇	—	4	4	no	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	D₄	1	2	yes	—
48	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
51	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	#95×C₆₇	—	2	2	no	—
53	#95×C₆₇	—	2	2	no	—
54	C₂×C₆₇⋊C₈	—	2	4	no	—
55	C₂×C₆₇⋊C₈	—	2	4	no	—
56	C₂×C₆₇⋊C₈	—	4	4	no	—
57	C₂×C₆₇⋊C₈	—	4	2	no	—
58	C₂×C₆₇⋊C₈	—	4	2	no	—
59	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	—	2	2	no	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	—	2	8	no	—
63	C₂²×C₄×C₆₇	—	2	4	no	—
64	C₂²×C₄×C₆₇	—	4	2	no	—
65	C₂²×C₄×C₆₇	—	4	4	no	—
66	C₂²×C₄×C₆₇	—	4	4	no	—
67	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
77	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₆₇	—	2	8	no	—
82	C₄⋊C₄×C₆₇	—	2	4	no	—
83	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
84	C₄²×C₆₇	—	4	4	no	—
85	C₄²×C₆₇	—	4	4	no	—
86	C₄²×C₆₇	C₄²	1	2	yes	—
87	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
90	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
91	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
92	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
96	C₂³×C₆₇	—	2	1	no	—
97	C₂³×C₆₇	—	2	1	no	—
98	C₂³×C₆₇	—	2	2	no	—
99	C₂³×C₆₇	#95	1	2	yes	—
100	C₂³×C₆₇	C₂²⋊C₈	1	4	yes	—
101	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	—	2	1	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	2	8	no	—
109	C₂×C₄×C₆₇	—	4	2	no	—
110	C₂×C₄×C₆₇	—	4	2	no	—
111	C₂×C₄×C₆₇	—	4	16	no	—
112	C₂×C₄×C₆₇	—	4	4	no	—
113	C₂×C₄×C₆₇	—	4	4	no	—
114	C₂×C₄×C₆₇	#95	1	2	yes	—
115	C₂×C₄×C₆₇	#95	1	2	yes	—
116	C₂×C₄×C₆₇	#95	1	2	yes	—
117	C₂×C₄×C₆₇	C₄×C₈	1	4	yes	—
118	C₂×C₄×C₆₇	#97	1	4	yes	—
119	C₂×C₄×C₆₇	C₂²⋊C₈	1	4	yes	—
120	C₂×C₄×C₆₇	C₄⋊C₈	1	8	yes	—
121	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
130	C₂×C₄×C₆₇	#137	1	2	yes	—
131	C₆₇⋊C₈	—	4	4	no	—
132	C₆₇⋊C₈	—	4	4	no	—
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	—	2	4	no	—
135	C₂²×C₆₇	—	2	4	no	—
136	C₂²×C₆₇	—	2	2	no	—
137	C₂²×C₆₇	#336	1	4	yes	—
138	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
139	C₂²×C₆₇	C₂×C₂²⋊C₈	1	2	yes	—
140	C₂²×C₆₇	#418	1	1	yes	characteristic
141	C₂²×C₆₇	#419	1	1	yes	characteristic
142	C₂²×C₆₇	#421	1	2	yes	—
143	C₄×C₆₇	—	2	1	no	—
144	C₄×C₆₇	—	2	2	no	—
145	C₄×C₆₇	—	2	1	no	—
146	C₄×C₆₇	—	2	2	no	—
147	C₄×C₆₇	—	4	8	no	—
148	C₄×C₆₇	#336	1	4	yes	—
149	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
150	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
151	C₄×C₆₇	#417	1	2	yes	—
152	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
153	C₄×C₆₇	#428	1	2	yes	—
154	#7892	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	—	2	4	no	—
156	C₂×C₆₇	#1478	1	2	yes	—
157	C₂×C₆₇	#1479	1	2	yes	—
158	C₂×C₆₇	C₂×#336	1	1	yes	characteristic
159	C₂×C₆₇	#1733	1	1	yes	characteristic
160	C₂×C₆₇	#1736	1	1	yes	characteristic
161	C₂×#431	—	67	1	no	—
162	C₂×#431	—	67	1	no	—
163	C₂×C₄×C₄⋊C₄	D₆₇	1	1	yes	characteristic
164	#1279	—	67	4	no	—
165	C₆₇	#7892	1	1	yes	characteristic, complemented, Sylow
166	C₂²×C₄⋊C₄	Dic₆₇	1	2	yes	—
167	C₂²×C₄²	—	2	1	no	—
168	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
169	#431	—	67	4	no	—
170	#431	—	67	4	no	—
171	#431	—	134	2	no	—
172	#431	—	134	2	no	—
173	C₂×C₄⋊C₈	—	67	2	no	—
174	C₂×C₄⋊C₈	—	134	1	no	—
175	C₂×C₂²⋊C₈	—	134	1	no	—
176	C₂×C₂²⋊C₈	—	134	1	no	—
177	C₂×C₄×C₈	—	67	2	no	—
178	C₂×C₄×C₈	—	134	1	no	—
179	C₄×C₄⋊C₄	—	2	1	no	—
180	C₄×C₄⋊C₄	—	2	1	no	—
181	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
182	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
183	C₂×#95	—	2	1	no	—
184	C₂³×C₄	—	2	2	no	—
185	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
186	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
187	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
188	C₂²×C₈	—	134	2	no	—
189	C₂²×C₈	—	134	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	C₆₇⋊C₈	1	8	yes	complemented
193	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	4	4	no	—
197	C₂×C₄²	Dic₁₃₄	1	2	yes	—
198	C₂×C₄²	C₄×D₆₇	1	2	yes	—
199	C₂×C₄²	C₄×D₆₇	1	2	yes	—
200	C₂×C₄²	D₂₆₈	1	2	yes	—
201	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
202	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
204	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
205	C₄⋊C₈	—	67	4	no	—
206	C₄⋊C₈	—	134	4	no	—
207	C₄⋊C₈	—	134	2	no	—
208	C₂²⋊C₈	—	134	4	no	—
209	C₂²⋊C₈	—	134	4	no	—
210	C₄×C₈	—	67	4	no	—
211	C₄×C₈	—	134	4	no	—
212	C₄×C₈	—	134	2	no	—
213	#95	—	2	2	no	—
214	#95	—	2	2	no	—
215	C₂⁴	C₂²⋊Dic₆₇	1	1	yes	characteristic
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	8	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
224	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
225	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
226	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
229	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
230	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
231	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
232	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
233	C₂²×C₄	#11761612	1	4	yes	—
234	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
237	C₂×C₈	—	268	4	no	—
238	C₂×C₈	—	268	2	no	—
239	C₂×C₈	—	268	2	no	—
240	C₂×C₈	—	134	4	no	—
241	C₂×C₈	—	134	4	no	—
242	C₄⋊C₄	—	2	8	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	C₂×C₆₇⋊C₈	1	8	yes	—
245	C₄²	—	4	4	no	—
246	C₄²	—	4	4	no	—
247	C₄²	C₄×Dic₆₇	1	2	yes	—
248	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
249	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
250	C₄²	Dic₆₇⋊C₄	1	2	yes	—
251	C₄²	Dic₆₇⋊C₄	1	2	yes	—
252	C₄²	C₄⋊Dic₆₇	1	2	yes	—
253	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
254	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
255	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
256	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
257	C₂³	—	2	1	no	—
258	C₂³	—	2	1	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₆₇⋊#95	1	2	yes	—
261	C₂³	C₂²⋊(C₆₇⋊C₈)	1	4	yes	—
262	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	16	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	C₆₇⋊#95	1	2	yes	—
276	C₂×C₄	C₆₇⋊#95	1	2	yes	—
277	C₂×C₄	C₆₇⋊#95	1	2	yes	—
278	C₂×C₄	C₄×C₆₇⋊C₈	1	4	yes	—
279	C₂×C₄	#14523954	1	4	yes	—
280	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	4	yes	—
281	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	8	yes	—
282	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
283	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
285	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
286	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
289	C₂×C₄	C₂×#11761612	1	2	yes	—
290	C₂×C₄	#14524069	1	2	yes	—
291	C₂×C₄	#14524074	1	2	yes	—
292	C₈	—	268	4	no	—
293	C₈	—	268	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	#16863151	1	4	yes	—
299	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
300	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
301	C₂²	#16863401	1	1	yes	characteristic
302	C₂²	#16863405	1	1	yes	characteristic
303	C₂²	#16863413	1	2	yes	—
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	8	no	—
309	C₄	#16863151	1	4	yes	—
310	C₄	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
311	C₄	C₂×#14523954	1	1	yes	characteristic
312	C₄	#16863395	1	2	yes	—
313	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	2	yes	—
314	C₄	#16863436	1	2	yes	—
315	C₂	—	2	4	no	—
316	C₂	#22558600	1	2	yes	—
317	C₂	#22558604	1	2	yes	—
318	C₂	C₂×#16863151	1	1	yes	characteristic
319	C₂	#22559918	1	1	yes	characteristic
320	C₂	#22559928	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₄×C₄⋊C₄×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₄×C₄⋊C₄	D₆₇
hypocenter	C₆₇	#7892
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #22558050	C₂	semidirect
7. #16863449	C₄	semidirect
8. #16863449	C₄	semidirect
27. C₄⋊(C₆₇⋊C₈)	C₂×C₄	semidirect
32. C₄×C₆₇⋊C₈	C₂×C₄	semidirect
39. C₂×C₄⋊C₄×C₆₇	C₈	semidirect
165. C₆₇	#7892	semidirect
192. C₂×C₄⋊C₄	C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863449	2
#22558050	2
C₂×C₄×C₄⋊C₄×C₆₇	2
#7892	67