Subgroups — #30746807

Order 17152.

528 conjugacy classes of subgroups.

The subgroup lattice has 304 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×#16863142	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	#16863142	C₄	1	8	yes	complemented
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
15	#391×C₆₇	—	2	1	no	—
16	#389×C₆₇	—	2	1	no	—
17	#389×C₆₇	C₄	1	2	yes	—
18	#384×C₆₇	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
20	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
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	—	2	1	no	—
31	C₄⋊Q₈×C₆₇	—	2	2	no	—
32	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
33	C₄×Q₈×C₆₇	—	4	2	no	—
34	C₄×Q₈×C₆₇	—	4	2	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×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₇	—	2	2	no	—
42	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂×C₄	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	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	#95×C₆₇	—	2	2	no	—
50	#95×C₆₇	—	2	2	no	—
51	C₂×C₆₇⋊C₈	—	2	4	no	—
52	C₂×C₆₇⋊C₈	—	2	4	no	—
53	C₂×C₆₇⋊C₈	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	4	2	no	—
55	C₂×C₆₇⋊C₈	—	4	2	no	—
56	C₂×C₆₇⋊C₈	—	4	2	no	—
57	C₂×Q₈×C₆₇	—	2	4	no	—
58	C₂×Q₈×C₆₇	—	4	1	no	—
59	C₂×Q₈×C₆₇	—	4	1	no	—
60	C₂×Q₈×C₆₇	—	4	2	no	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₄⋊C₄×C₆₇	—	2	2	no	—
71	C₄⋊C₄×C₆₇	—	2	4	no	—
72	C₄⋊C₄×C₆₇	—	4	2	no	—
73	C₄⋊C₄×C₆₇	—	4	4	no	—
74	C₄⋊C₄×C₆₇	—	4	1	no	—
75	C₄⋊C₄×C₆₇	—	4	1	no	—
76	C₄⋊C₄×C₆₇	—	4	1	no	—
77	C₄⋊C₄×C₆₇	—	4	1	no	—
78	C₄⋊C₄×C₆₇	C₄²	1	4	yes	—
79	C₄²×C₆₇	—	2	2	no	—
80	C₄²×C₆₇	—	2	2	no	—
81	C₄²×C₆₇	—	2	1	no	—
82	C₄²×C₆₇	—	2	1	no	—
83	C₄²×C₆₇	—	2	2	no	—
84	C₄²×C₆₇	—	4	1	no	—
85	C₄²×C₆₇	—	4	4	no	—
86	C₄²×C₆₇	D₈	1	2	yes	—
87	C₄²×C₆₇	SD₁₆	1	2	yes	—
88	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
89	C₄²×C₆₇	#41	1	2	yes	—
90	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	Q₈×C₆₇	—	4	4	no	—
92	Q₈×C₆₇	—	4	4	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	1	no	—
99	C₂×C₄×C₆₇	—	2	1	no	—
100	C₂×C₄×C₆₇	—	2	1	no	—
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	2	1	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	—	4	2	no	—
106	C₂×C₄×C₆₇	—	4	2	no	—
107	C₂×C₄×C₆₇	—	4	2	no	—
108	C₂×C₄×C₆₇	—	4	2	no	—
109	C₂×C₄×C₆₇	—	4	2	no	—
110	C₂×C₄×C₆₇	#101	1	2	yes	—
111	C₂×C₄×C₆₇	#102	1	2	yes	—
112	C₂×C₄×C₆₇	#102	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₆₇	#135	1	2	yes	—
120	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#137	1	2	yes	—
123	C₆₇⋊C₈	—	4	4	no	—
124	C₆₇⋊C₈	—	4	4	no	—
125	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
127	C₂²×C₆₇	#413	1	1	yes	characteristic
128	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
129	C₂²×C₆₇	#416	1	1	yes	characteristic
130	C₂²×C₆₇	#417	1	1	yes	characteristic
131	C₂²×C₆₇	#419	1	1	yes	characteristic
132	C₄×C₆₇	—	2	2	no	—
133	C₄×C₆₇	—	2	2	no	—
134	C₄×C₆₇	—	2	2	no	—
135	C₄×C₆₇	—	2	1	no	—
136	C₄×C₆₇	—	2	1	no	—
137	C₄×C₆₇	—	4	1	no	—
138	C₄×C₆₇	—	4	4	no	—
139	C₄×C₆₇	—	8	2	no	—
140	C₄×C₆₇	C₄×D₈	1	2	yes	—
141	C₄×C₆₇	C₄×SD₁₆	1	2	yes	—
142	C₄×C₆₇	#440	1	2	yes	—
143	C₄×C₆₇	#441	1	2	yes	—
144	#8236	—	67	1	no	maximal, Sylow
145	C₂×C₆₇	#1546	1	1	yes	characteristic
146	C₂×C₆₇	#1547	1	1	yes	characteristic
147	C₂×C₆₇	#1552	1	1	yes	characteristic
148	C₂×C₆₇	#1554	1	1	yes	characteristic
149	C₂×C₆₇	C₄×#101	1	1	yes	characteristic
150	C₂×C₆₇	C₄×#102	1	1	yes	characteristic
151	C₂×C₆₇	#1763	1	1	yes	characteristic
152	C₄×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
153	#1767	—	67	1	no	—
154	C₄×C₄⋊C₈	—	67	1	no	—
155	C₂×#332	—	67	2	no	—
156	#1296	—	67	2	no	—
157	C₆₇	#8236	1	1	yes	characteristic, complemented, Sylow
158	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
159	C₂×C₄×Q₈	—	2	1	no	—
160	C₂×C₄⋊C₈	—	67	2	no	—
161	C₂×C₄⋊C₈	—	67	2	no	—
162	C₂×#97	—	134	1	no	—
163	C₂×C₄×C₈	—	134	1	no	—
164	#391	—	2	1	no	—
165	#389	—	2	1	no	—
166	#389	Dic₆₇	1	2	yes	—
167	#384	C₂×D₆₇	1	1	yes	characteristic
168	C₄×C₄⋊C₄	—	2	1	no	—
169	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
170	C₄³	C₂×D₆₇	1	1	yes	characteristic
171	#336	—	134	1	no	—
172	#336	—	134	1	no	—
173	#332	—	67	8	no	—
174	C₂²×Q₈	—	2	1	no	—
175	C₂²×C₈	—	134	2	no	—
176	C₂²×C₈	—	134	2	no	—
177	C₄⋊Q₈	—	2	2	no	—
178	C₄⋊Q₈	C₄×D₆₇	1	4	yes	—
179	C₄×Q₈	—	4	2	no	—
180	C₄×Q₈	—	4	2	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
186	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
189	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
192	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
193	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
194	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
195	C₄⋊C₈	—	67	4	no	—
196	C₄⋊C₈	—	67	4	no	—
197	C₄⋊C₈	—	134	2	no	—
198	C₄⋊C₈	—	134	2	no	—
199	#97	—	268	2	no	—
200	C₄×C₈	—	268	2	no	—
201	#95	—	2	2	no	—
202	#95	—	2	2	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	4	1	no	—
205	C₂×Q₈	—	4	1	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
211	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
213	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
216	C₂×C₈	—	268	2	no	—
217	C₂×C₈	—	268	2	no	—
218	C₂×C₈	—	268	2	no	—
219	C₂×C₈	—	268	2	no	—
220	C₂×C₈	—	134	4	no	—
221	C₂×C₈	—	134	4	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	C₄×Dic₆₇	1	4	yes	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	1	no	—
237	C₄²	—	4	4	no	—
238	C₄²	#11761613	1	2	yes	—
239	C₄²	#11761616	1	2	yes	—
240	C₄²	C₂×C₄×D₆₇	1	2	yes	—
241	C₄²	#11761628	1	2	yes	—
242	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
243	Q₈	—	4	4	no	—
244	Q₈	—	4	4	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	#14523962	1	2	yes	—
263	C₂×C₄	#14523964	1	2	yes	—
264	C₂×C₄	#14523964	1	2	yes	—
265	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
266	C₂×C₄	#14524002	1	1	yes	characteristic
267	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
268	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
269	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
270	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
271	C₂×C₄	#14524063	1	2	yes	—
272	C₂×C₄	#14524068	1	1	yes	characteristic
273	C₂×C₄	#14524073	1	1	yes	characteristic
274	C₂×C₄	#14524074	1	2	yes	—
275	C₈	—	268	4	no	—
276	C₈	—	268	4	no	—
277	C₂²	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
278	C₂²	C₂×#14523962	1	1	yes	characteristic
279	C₂²	#16863380	1	1	yes	characteristic
280	C₂²	C₂×#14523964	1	1	yes	characteristic
281	C₂²	#16863392	1	1	yes	characteristic
282	C₂²	#16863397	1	1	yes	characteristic
283	C₂²	#16863405	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	4	4	no	—
290	C₄	—	4	1	no	—
291	C₄	—	8	2	no	—
292	C₄	C₄×#11761613	1	2	yes	—
293	C₄	C₄×#11761616	1	2	yes	—
294	C₄	#16863494	1	2	yes	—
295	C₄	#16863501	1	2	yes	—
296	C₂	#22558881	1	1	yes	characteristic
297	C₂	#22558886	1	1	yes	characteristic
298	C₂	#22558909	1	1	yes	characteristic
299	C₂	#22558919	1	1	yes	characteristic
300	C₂	C₄×#14523962	1	1	yes	characteristic
301	C₂	C₄×#14523964	1	1	yes	characteristic
302	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	#8236
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. #16863142	C₄	semidirect
157. C₆₇	#8236	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×#16863142	2
#22558092	2
C₄×C₄⋊Q₈×C₆₇	2
#8236	67