Subgroups — #30746802

Order 17152.

528 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560062	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863145	C₂	1	2	yes	maximal
4	#22558092	C₂	1	2	yes	maximal
5	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×#14523954	—	2	1	no	—
9	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
10	#16863151	—	2	1	no	—
11	#16863151	—	2	1	no	—
12	#16863145	C₄	1	8	yes	complemented
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₇	C₄	1	2	yes	—
15	#391×C₆₇	C₄	1	2	yes	—
16	#389×C₆₇	—	2	1	no	—
17	#389×C₆₇	—	2	1	no	—
18	#384×C₆₇	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
20	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
21	C₄³×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₆₇⋊C₈	—	2	2	no	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
27	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
28	#14523954	—	4	2	no	—
29	C₄×C₆₇⋊C₈	—	4	2	no	—
30	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
31	C₄⋊Q₈×C₆₇	—	2	2	no	—
32	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
33	C₄×Q₈×C₆₇	—	4	4	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₇	—	2	2	no	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	#95×C₆₇	—	2	2	no	—
49	#95×C₆₇	—	2	2	no	—
50	C₂×C₆₇⋊C₈	—	2	4	no	—
51	C₂×C₆₇⋊C₈	—	2	4	no	—
52	C₂×C₆₇⋊C₈	—	4	2	no	—
53	C₂×C₆₇⋊C₈	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	4	2	no	—
55	C₂×C₆₇⋊C₈	—	4	2	no	—
56	C₂×Q₈×C₆₇	—	2	2	no	—
57	C₂×Q₈×C₆₇	—	4	2	no	—
58	C₂×Q₈×C₆₇	—	4	2	no	—
59	C₂×Q₈×C₆₇	C₄²	1	4	yes	—
60	C₂²×C₄×C₆₇	—	2	2	no	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₆₇	—	2	4	no	—
70	C₄⋊C₄×C₆₇	—	2	4	no	—
71	C₄⋊C₄×C₆₇	—	4	2	no	—
72	C₄⋊C₄×C₆₇	—	4	4	no	—
73	C₄⋊C₄×C₆₇	—	4	2	no	—
74	C₄⋊C₄×C₆₇	—	4	1	no	—
75	C₄⋊C₄×C₆₇	—	4	1	no	—
76	C₄²×C₆₇	—	2	2	no	—
77	C₄²×C₆₇	—	2	2	no	—
78	C₄²×C₆₇	—	2	1	no	—
79	C₄²×C₆₇	—	2	1	no	—
80	C₄²×C₆₇	—	2	2	no	—
81	C₄²×C₆₇	—	4	1	no	—
82	C₄²×C₆₇	—	4	4	no	—
83	C₄²×C₆₇	SD₁₆	1	2	yes	—
84	C₄²×C₆₇	Q₁₆	1	2	yes	—
85	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
86	C₄²×C₆₇	#41	1	2	yes	—
87	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	Q₈×C₆₇	—	4	4	no	—
89	Q₈×C₆₇	—	4	4	no	—
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	2	1	no	—
92	C₂×C₄×C₆₇	—	2	1	no	—
93	C₂×C₄×C₆₇	—	2	1	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	1	no	—
96	C₂×C₄×C₆₇	—	2	1	no	—
97	C₂×C₄×C₆₇	—	2	1	no	—
98	C₂×C₄×C₆₇	—	2	4	no	—
99	C₂×C₄×C₆₇	—	2	1	no	—
100	C₂×C₄×C₆₇	—	2	1	no	—
101	C₂×C₄×C₆₇	—	2	1	no	—
102	C₂×C₄×C₆₇	—	4	2	no	—
103	C₂×C₄×C₆₇	—	4	2	no	—
104	C₂×C₄×C₆₇	—	4	2	no	—
105	C₂×C₄×C₆₇	—	4	2	no	—
106	C₂×C₄×C₆₇	—	4	2	no	—
107	C₂×C₄×C₆₇	#100	1	2	yes	—
108	C₂×C₄×C₆₇	#103	1	2	yes	—
109	C₂×C₄×C₆₇	#103	1	2	yes	—
110	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
116	C₂×C₄×C₆₇	#135	1	2	yes	—
117	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
119	C₂×C₄×C₆₇	#137	1	2	yes	—
120	C₆₇⋊C₈	—	4	4	no	—
121	C₆₇⋊C₈	—	4	4	no	—
122	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
123	C₂²×C₆₇	C₂×#100	1	1	yes	characteristic
124	C₂²×C₆₇	#413	1	1	yes	characteristic
125	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
126	C₂²×C₆₇	#416	1	1	yes	characteristic
127	C₂²×C₆₇	#417	1	1	yes	characteristic
128	C₂²×C₆₇	#419	1	1	yes	characteristic
129	C₄×C₆₇	—	2	2	no	—
130	C₄×C₆₇	—	2	2	no	—
131	C₄×C₆₇	—	2	2	no	—
132	C₄×C₆₇	—	2	1	no	—
133	C₄×C₆₇	—	2	1	no	—
134	C₄×C₆₇	—	4	1	no	—
135	C₄×C₆₇	—	4	4	no	—
136	C₄×C₆₇	—	8	2	no	—
137	C₄×C₆₇	C₄×SD₁₆	1	2	yes	—
138	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
139	C₄×C₆₇	#440	1	2	yes	—
140	C₄×C₆₇	#441	1	2	yes	—
141	#8235	—	67	1	no	maximal, Sylow
142	C₂×C₆₇	#1548	1	1	yes	characteristic
143	C₂×C₆₇	#1550	1	1	yes	characteristic
144	C₂×C₆₇	#1551	1	1	yes	characteristic
145	C₂×C₆₇	#1554	1	1	yes	characteristic
146	C₂×C₆₇	C₄×#100	1	1	yes	characteristic
147	C₂×C₆₇	C₄×#103	1	1	yes	characteristic
148	C₂×C₆₇	#1763	1	1	yes	characteristic
149	C₄×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
150	#1767	—	67	1	no	—
151	C₄×C₄⋊C₈	—	67	1	no	—
152	C₂×#333	—	67	2	no	—
153	#1296	—	67	2	no	—
154	C₆₇	#8235	1	1	yes	characteristic, complemented, Sylow
155	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
156	C₂×C₄×Q₈	Dic₆₇	1	2	yes	—
157	C₂×C₄⋊C₈	—	67	2	no	—
158	C₂×C₄⋊C₈	—	67	2	no	—
159	C₂×#97	—	134	1	no	—
160	C₂×C₄×C₈	—	134	1	no	—
161	#391	Dic₆₇	1	2	yes	—
162	#389	—	2	1	no	—
163	#389	—	2	1	no	—
164	#384	C₂×D₆₇	1	1	yes	characteristic
165	C₄×C₄⋊C₄	—	2	1	no	—
166	C₄×C₄⋊C₄	—	2	1	no	—
167	C₄³	C₂×D₆₇	1	1	yes	characteristic
168	#336	—	134	1	no	—
169	#336	—	134	1	no	—
170	#333	—	67	8	no	—
171	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
172	C₂²×C₈	—	134	2	no	—
173	C₂²×C₈	—	134	2	no	—
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	C₄×D₆₇	1	4	yes	—
176	C₄×Q₈	—	4	4	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
183	C₂×C₄²	—	2	2	no	—
184	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
185	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
186	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
188	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
191	C₄⋊C₈	—	67	4	no	—
192	C₄⋊C₈	—	67	4	no	—
193	C₄⋊C₈	—	134	2	no	—
194	C₄⋊C₈	—	134	2	no	—
195	#97	—	268	2	no	—
196	C₄×C₈	—	268	2	no	—
197	#95	—	2	2	no	—
198	#95	—	2	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×Q₈	—	4	2	no	—
202	C₂×Q₈	C₄×Dic₆₇	1	4	yes	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
206	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
212	C₂×C₈	—	268	2	no	—
213	C₂×C₈	—	268	2	no	—
214	C₂×C₈	—	268	2	no	—
215	C₂×C₈	—	268	2	no	—
216	C₂×C₈	—	134	4	no	—
217	C₂×C₈	—	134	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	4	1	no	—
231	C₄²	—	4	4	no	—
232	C₄²	#11761615	1	2	yes	—
233	C₄²	#11761618	1	2	yes	—
234	C₄²	C₂×C₄×D₆₇	1	2	yes	—
235	C₄²	#11761628	1	2	yes	—
236	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
237	Q₈	—	4	4	no	—
238	Q₈	—	4	4	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	#14523960	1	2	yes	—
257	C₂×C₄	#14523967	1	2	yes	—
258	C₂×C₄	#14523967	1	2	yes	—
259	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
260	C₂×C₄	#14524002	1	1	yes	characteristic
261	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
262	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
263	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
264	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
265	C₂×C₄	#14524063	1	2	yes	—
266	C₂×C₄	#14524068	1	1	yes	characteristic
267	C₂×C₄	#14524073	1	1	yes	characteristic
268	C₂×C₄	#14524074	1	2	yes	—
269	C₈	—	268	4	no	—
270	C₈	—	268	4	no	—
271	C₂²	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
272	C₂²	C₂×#14523960	1	1	yes	characteristic
273	C₂²	#16863380	1	1	yes	characteristic
274	C₂²	C₂×#14523967	1	1	yes	characteristic
275	C₂²	#16863392	1	1	yes	characteristic
276	C₂²	#16863397	1	1	yes	characteristic
277	C₂²	#16863405	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	4	no	—
284	C₄	—	4	1	no	—
285	C₄	—	8	2	no	—
286	C₄	C₄×#11761615	1	2	yes	—
287	C₄	C₄×#11761618	1	2	yes	—
288	C₄	#16863494	1	2	yes	—
289	C₄	#16863501	1	2	yes	—
290	C₂	#22558890	1	1	yes	characteristic
291	C₂	#22558899	1	1	yes	characteristic
292	C₂	#22558904	1	1	yes	characteristic
293	C₂	#22558919	1	1	yes	characteristic
294	C₂	C₄×#14523960	1	1	yes	characteristic
295	C₂	C₄×#14523967	1	1	yes	characteristic
296	C₂	#22560039	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×C₄⋊Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#8235
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
12. #16863145	C₄	semidirect
154. C₆₇	#8235	semidirect

Maximal subgroups

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

subgroup	index
#22560062	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₂×#16863145	2
#22558092	2
C₄×C₄⋊Q₈×C₆₇	2
#8235	67