Subgroups — #30745360

Order 17152.

830 conjugacy classes of subgroups.

The subgroup lattice has 302 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	C₂²×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22558050	C₂	1	4	yes	maximal, complemented
4	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₆₇⋊C₈	—	2	1	no	—
6	#16863449	—	2	2	no	—
7	#16863449	C₄	1	4	yes	complemented
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
10	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
11	C₂×C₄×C₆₇⋊C₈	—	2	2	no	—
12	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
13	C₂×C₄×C₆₇⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
15	C₂×#117×C₆₇	C₄	1	2	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
19	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
20	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
21	#376×C₆₇	—	2	1	no	—
22	C₄³×C₆₇	—	2	1	no	—
23	C₂²×C₆₇⋊C₈	—	2	8	no	—
24	C₂²×C₆₇⋊C₈	—	2	4	no	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₂²×C₆₇⋊C₈	—	2	2	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
29	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
30	C₄×C₆₇⋊C₈	—	2	4	no	—
31	C₄×C₆₇⋊C₈	—	2	4	no	—
32	C₄×C₆₇⋊C₈	—	2	2	no	—
33	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
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	#117×C₆₇	—	2	4	no	—
38	#117×C₆₇	C₈	1	8	yes	complemented
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄²×C₆₇	—	2	2	no	—
42	C₂×C₄²×C₆₇	—	2	2	no	—
43	C₂×C₄²×C₆₇	—	2	2	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	8	no	—
55	C₂×C₆₇⋊C₈	—	2	16	no	—
56	C₂×C₆₇⋊C₈	—	2	4	no	—
57	C₂×C₆₇⋊C₈	—	2	4	no	—
58	C₂×C₆₇⋊C₈	—	4	4	no	—
59	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	—	2	4	no	—
61	C₂²×C₄×C₆₇	—	2	4	no	—
62	C₂²×C₄×C₆₇	—	2	4	no	—
63	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
73	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₆₇	—	2	4	no	—
78	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	complemented
79	C₂²⋊C₄×C₆₇	—	2	8	no	—
80	C₄²×C₆₇	—	2	8	no	—
81	C₄²×C₆₇	—	4	4	no	—
82	C₄²×C₆₇	—	4	4	no	—
83	C₄²×C₆₇	C₄²	1	2	yes	—
84	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
87	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
88	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
89	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
93	C₂³×C₆₇	—	2	2	no	—
94	C₂³×C₆₇	—	2	2	no	—
95	C₂³×C₆₇	#95	1	2	yes	—
96	C₂³×C₆₇	C₂²⋊C₈	1	4	yes	—
97	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	2	4	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	8	no	—
105	C₂×C₄×C₆₇	—	4	4	no	—
106	C₂×C₄×C₆₇	—	4	4	no	—
107	C₂×C₄×C₆₇	#95	1	2	yes	—
108	C₂×C₄×C₆₇	#95	1	2	yes	—
109	C₂×C₄×C₆₇	#95	1	2	yes	—
110	C₂×C₄×C₆₇	C₄×C₈	1	4	yes	—
111	C₂×C₄×C₆₇	#97	1	4	yes	—
112	C₂×C₄×C₆₇	C₂²⋊C₈	1	4	yes	—
113	C₂×C₄×C₆₇	#104	1	4	yes	—
114	C₂×C₄×C₆₇	C₄⋊C₈	1	8	yes	—
115	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₆₇	#135	1	4	yes	—
124	C₆₇⋊C₈	—	2	8	no	—
125	C₆₇⋊C₈	—	4	4	no	—
126	C₂²×C₆₇	—	2	8	no	—
127	C₂²×C₆₇	—	2	2	no	—
128	C₂²×C₆₇	—	2	2	no	—
129	C₂²×C₆₇	#336	1	4	yes	—
130	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
131	C₂²×C₆₇	C₂×C₂²⋊C₈	1	2	yes	—
132	C₂²×C₆₇	#416	1	2	yes	—
133	C₂²×C₆₇	C₂×#104	1	2	yes	—
134	C₄×C₆₇	—	2	1	no	—
135	C₄×C₆₇	—	2	1	no	—
136	C₄×C₆₇	—	2	2	no	—
137	C₄×C₆₇	—	4	8	no	—
138	C₄×C₆₇	#336	1	4	yes	—
139	C₄×C₆₇	#339	1	4	yes	—
140	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
141	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
142	C₄×C₆₇	#416	1	2	yes	—
143	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
144	C₄×C₆₇	#427	1	2	yes	—
145	#7890	—	67	1	no	maximal, Sylow
146	C₂×C₆₇	—	2	4	no	—
147	C₂×C₆₇	#1476	1	4	yes	—
148	C₂×C₆₇	C₂×#336	1	1	yes	characteristic
149	C₂×C₆₇	C₂×#339	1	1	yes	characteristic
150	C₂×C₆₇	#1735	1	1	yes	characteristic
151	C₂×#431	—	67	1	no	—
152	C₂²×C₄×C₈	—	67	1	no	—
153	C₄×#117	D₆₇	1	1	yes	characteristic
154	#1279	—	67	4	no	—
155	C₆₇	#7890	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₈	—	134	1	no	—
157	C₂×#117	Dic₆₇	1	2	yes	—
158	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
159	#431	—	67	4	no	—
160	#431	—	134	2	no	—
161	C₂×C₄⋊C₈	—	67	2	no	—
162	C₂×C₂²⋊C₈	—	134	1	no	—
163	C₂×C₄×C₈	—	67	4	no	—
164	C₂×C₄×C₈	—	67	2	no	—
165	C₂×C₄×C₈	—	134	2	no	—
166	C₂×C₄×C₈	—	134	1	no	—
167	C₂×C₄×C₈	—	134	1	no	—
168	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
169	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
170	C₄×C₂²⋊C₄	—	2	1	no	—
171	C₄×C₂²⋊C₄	—	2	1	no	—
172	#376	—	2	1	no	—
173	C₄³	—	2	1	no	—
174	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
175	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
176	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
177	C₂²×C₈	—	134	2	no	—
178	C₂²×C₈	—	134	8	no	—
179	C₂²×C₈	—	134	4	no	—
180	C₂²×C₈	—	134	2	no	—
181	#117	—	2	4	no	—
182	#117	C₆₇⋊C₈	1	8	yes	complemented
183	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	Dic₁₃₄	1	2	yes	—
189	C₂×C₄²	C₄×D₆₇	1	2	yes	—
190	C₂×C₄²	C₄×D₆₇	1	2	yes	—
191	C₂×C₄²	D₂₆₈	1	2	yes	—
192	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
193	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
194	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
195	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
196	C₄⋊C₈	—	67	4	no	—
197	C₄⋊C₈	—	134	2	no	—
198	C₂²⋊C₈	—	134	4	no	—
199	C₄×C₈	—	67	4	no	—
200	C₄×C₈	—	134	4	no	—
201	C₄×C₈	—	134	4	no	—
202	C₄×C₈	—	134	2	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	C₂⁴	C₂²⋊Dic₆₇	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
210	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
211	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
212	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
213	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
214	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
215	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
216	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
217	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
218	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
219	C₂²×C₄	#11761612	1	4	yes	—
220	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
223	C₂×C₈	—	268	4	no	—
224	C₂×C₈	—	134	4	no	—
225	C₂×C₈	—	134	8	no	—
226	C₂×C₈	—	134	16	no	—
227	C₂×C₈	—	134	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	C₂×C₆₇⋊C₈	1	8	yes	complemented
230	C₂²⋊C₄	—	2	8	no	—
231	C₄²	—	2	8	no	—
232	C₄²	—	4	4	no	—
233	C₄²	—	4	4	no	—
234	C₄²	C₄×Dic₆₇	1	2	yes	—
235	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
236	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
237	C₄²	Dic₆₇⋊C₄	1	2	yes	—
238	C₄²	Dic₆₇⋊C₄	1	2	yes	—
239	C₄²	C₄⋊Dic₆₇	1	2	yes	—
240	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
241	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
242	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
243	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	C₆₇⋊#95	1	2	yes	—
247	C₂³	C₂²⋊(C₆₇⋊C₈)	1	4	yes	—
248	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	C₆₇⋊#95	1	2	yes	—
259	C₂×C₄	C₆₇⋊#95	1	2	yes	—
260	C₂×C₄	C₆₇⋊#95	1	2	yes	—
261	C₂×C₄	C₄×C₆₇⋊C₈	1	4	yes	—
262	C₂×C₄	#14523954	1	4	yes	—
263	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	4	yes	—
264	C₂×C₄	#14523970	1	4	yes	—
265	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	8	yes	—
266	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
267	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
269	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
270	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
273	C₂×C₄	C₂×#11761612	1	2	yes	—
274	C₂×C₄	#14524063	1	4	yes	—
275	C₈	—	268	4	no	—
276	C₈	—	134	8	no	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	#16863151	1	4	yes	—
281	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
282	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
283	C₂²	#16863392	1	2	yes	—
284	C₂²	C₂×#14523970	1	2	yes	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	8	no	—
289	C₄	#16863151	1	4	yes	—
290	C₄	#16863155	1	4	yes	—
291	C₄	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
292	C₄	C₂×#14523954	1	1	yes	characteristic
293	C₄	#16863390	1	2	yes	—
294	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	2	yes	—
295	C₄	#16863432	1	2	yes	—
296	C₂	—	2	4	no	—
297	C₂	#22558591	1	4	yes	—
298	C₂	C₂×#16863151	1	1	yes	characteristic
299	C₂	C₂×#16863155	1	1	yes	characteristic
300	C₂	#22559925	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#117×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#117	D₆₇
hypocenter	C₆₇	#7890
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #22558050	C₂	semidirect
7. #16863449	C₄	semidirect
13. C₂×C₄×C₆₇⋊C₈	C₄	semidirect
38. #117×C₆₇	C₈	semidirect
78. C₄⋊C₄×C₆₇	C₂×C₈	semidirect
155. C₆₇	#7890	semidirect
182. #117	C₆₇⋊C₈	semidirect
229. C₄⋊C₄	C₂×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
C₂²×C₄×C₆₇⋊C₈	2
#22558050	2
C₄×#117×C₆₇	2
#7890	67