Subgroups — #30755949

Order 17152.

538 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560361	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#22558092	C₂	1	4	yes	maximal
4	C₆₇×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
10	#16863151	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
12	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₄	1	2	yes	—
14	#384×C₆₇	—	2	1	no	—
15	#384×C₆₇	—	2	1	no	—
16	#384×C₆₇	C₄	1	2	yes	—
17	#384×C₆₇	C₂²	1	2	yes	—
18	C₄³×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₆₇⋊C₈	—	2	2	no	—
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
23	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
25	C₄⋊(C₆₇⋊C₈)	Q₈	1	4	yes	complemented
26	C₄×C₆₇⋊C₈	—	4	2	no	—
27	C₂²×Q₈×C₆₇	—	2	2	no	—
28	C₄⋊Q₈×C₆₇	—	4	4	no	—
29	C₄⋊Q₈×C₆₇	—	4	4	no	—
30	C₄⋊Q₈×C₆₇	—	4	2	no	—
31	C₄⋊Q₈×C₆₇	—	4	2	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₆₇⋊C₈	—	2	4	no	—
46	C₂×C₆₇⋊C₈	—	4	2	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₂×C₆₇⋊C₈	—	4	2	no	—
49	C₂×C₆₇⋊C₈	—	4	2	no	—
50	C₂×C₆₇⋊C₈	—	4	2	no	—
51	C₂×Q₈×C₆₇	—	4	4	no	—
52	C₂×Q₈×C₆₇	—	4	4	no	—
53	C₂×Q₈×C₆₇	—	4	4	no	—
54	C₂²×C₄×C₆₇	—	2	4	no	—
55	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₆₇	—	4	4	no	—
63	C₄⋊C₄×C₆₇	—	4	4	no	—
64	C₄⋊C₄×C₆₇	—	4	4	no	—
65	C₄⋊C₄×C₆₇	—	4	4	no	—
66	C₄⋊C₄×C₆₇	—	4	2	no	—
67	C₄⋊C₄×C₆₇	—	4	2	no	—
68	C₄⋊C₄×C₆₇	—	4	4	no	—
69	C₄²×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	—	2	2	no	—
72	C₄²×C₆₇	D₈	1	2	yes	—
73	C₄²×C₆₇	SD₁₆	1	2	yes	—
74	C₄²×C₆₇	SD₁₆	1	2	yes	—
75	C₄²×C₆₇	SD₁₆	1	2	yes	—
76	C₄²×C₆₇	Q₁₆	1	2	yes	—
77	C₄²×C₆₇	Q₁₆	1	2	yes	—
78	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
79	C₄²×C₆₇	#41	1	2	yes	—
80	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	Q₈×C₆₇	—	8	8	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	1	no	—
87	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	4	4	no	—
89	C₂×C₄×C₆₇	—	4	4	no	—
90	C₂×C₄×C₆₇	—	4	4	no	—
91	C₂×C₄×C₆₇	#100	1	2	yes	—
92	C₂×C₄×C₆₇	#101	1	2	yes	—
93	C₂×C₄×C₆₇	#102	1	2	yes	—
94	C₂×C₄×C₆₇	#102	1	2	yes	—
95	C₂×C₄×C₆₇	#103	1	2	yes	—
96	C₂×C₄×C₆₇	#103	1	2	yes	—
97	C₂×C₄×C₆₇	#103	1	2	yes	—
98	C₂×C₄×C₆₇	#103	1	2	yes	—
99	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
101	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
110	C₆₇⋊C₈	—	4	4	no	—
111	C₆₇⋊C₈	—	8	2	no	—
112	C₂²×C₆₇	#391	1	1	yes	characteristic
113	C₂²×C₆₇	C₂×#100	1	1	yes	characteristic
114	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
115	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
116	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
117	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
118	C₂²×C₆₇	#418	1	1	yes	characteristic
119	C₄×C₆₇	—	2	2	no	—
120	C₄×C₆₇	—	2	2	no	—
121	C₄×C₆₇	—	2	2	no	—
122	C₄×C₆₇	—	8	4	no	—
123	C₄×C₆₇	#332	1	4	yes	—
124	C₄×C₆₇	#333	1	4	yes	—
125	C₄×C₆₇	#476	1	2	yes	—
126	C₄×C₆₇	#477	1	2	yes	—
127	C₄×C₆₇	#486	1	1	yes	characteristic
128	C₄×C₆₇	#487	1	1	yes	characteristic
129	C₄×C₆₇	#498	1	1	yes	characteristic
130	C₄×C₆₇	#500	1	1	yes	characteristic
131	#9896	—	67	1	no	maximal, Sylow
132	C₂×C₆₇	C₂×#332	1	1	yes	characteristic
133	C₂×C₆₇	C₂×#333	1	1	yes	characteristic
134	C₂×C₆₇	#1545	1	1	yes	characteristic
135	C₂×C₆₇	#1549	1	1	yes	characteristic
136	C₂×C₆₇	#1987	1	1	yes	characteristic
137	C₂×C₆₇	#1990	1	1	yes	characteristic
138	C₂×C₆₇	#1997	1	1	yes	characteristic
139	#2868	D₆₇	1	1	yes	characteristic
140	#1842	—	67	1	no	—
141	C₄×C₄⋊C₈	—	67	1	no	—
142	#1296	—	67	4	no	—
143	C₆₇	#9896	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄⋊Q₈	—	2	1	no	—
145	C₂×C₄⋊Q₈	—	2	1	no	—
146	C₂×C₄⋊Q₈	Dic₆₇	1	2	yes	—
147	C₂×C₄⋊C₈	—	67	2	no	—
148	C₂×C₄⋊C₈	—	67	2	no	—
149	C₂×C₄⋊C₈	—	134	1	no	—
150	C₂×C₄⋊C₈	—	134	1	no	—
151	C₂×C₄×C₈	—	134	1	no	—
152	#384	—	2	1	no	—
153	#384	—	2	1	no	—
154	#384	Dic₆₇	1	2	yes	—
155	#384	C₂×D₆₇	1	2	yes	—
156	C₄³	C₂×D₆₇	1	1	yes	characteristic
157	#336	—	134	1	no	—
158	C₂²×Q₈	—	2	2	no	—
159	C₂²×C₈	—	134	2	no	—
160	C₂²×C₈	—	134	2	no	—
161	C₄⋊Q₈	—	4	4	no	—
162	C₄⋊Q₈	—	4	4	no	—
163	C₄⋊Q₈	—	4	2	no	—
164	C₄⋊Q₈	—	4	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
171	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
172	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
174	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
175	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
177	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
178	C₄⋊C₈	—	268	2	no	—
179	C₄⋊C₈	—	268	2	no	—
180	C₄⋊C₈	—	67	4	no	—
181	C₄⋊C₈	—	134	4	no	—
182	C₄⋊C₈	—	134	2	no	—
183	C₄×C₈	—	268	2	no	—
184	C₂×Q₈	—	4	4	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
192	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
195	C₂×C₈	—	268	2	no	—
196	C₂×C₈	—	268	2	no	—
197	C₂×C₈	—	268	2	no	—
198	C₂×C₈	—	268	2	no	—
199	C₂×C₈	—	268	2	no	—
200	C₂×C₈	—	134	4	no	—
201	C₄⋊C₄	—	4	4	no	—
202	C₄⋊C₄	—	4	4	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	#11761613	1	2	yes	—
212	C₄²	#11761615	1	2	yes	—
213	C₄²	#11761615	1	2	yes	—
214	C₄²	#11761616	1	2	yes	—
215	C₄²	#11761618	1	2	yes	—
216	C₄²	#11761618	1	2	yes	—
217	C₄²	Q₈×D₆₇	1	2	yes	—
218	C₄²	#11761629	1	2	yes	—
219	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
220	Q₈	—	8	8	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	#14523960	1	2	yes	—
231	C₂×C₄	#14523962	1	2	yes	—
232	C₂×C₄	#14523964	1	2	yes	—
233	C₂×C₄	#14523964	1	2	yes	—
234	C₂×C₄	#14523967	1	2	yes	—
235	C₂×C₄	#14523967	1	2	yes	—
236	C₂×C₄	#14523967	1	2	yes	—
237	C₂×C₄	#14523967	1	2	yes	—
238	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
239	C₂×C₄	#14524019	1	2	yes	—
240	C₂×C₄	#14524032	1	1	yes	characteristic
241	C₂×C₄	#14524041	1	1	yes	characteristic
242	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
243	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
244	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
245	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
246	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
247	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
248	C₂×C₄	#14524069	1	2	yes	—
249	C₈	—	268	4	no	—
250	C₈	—	536	2	no	—
251	C₂²	#16863303	1	1	yes	characteristic
252	C₂²	C₂×#14523960	1	1	yes	characteristic
253	C₂²	C₂×#14523962	1	1	yes	characteristic
254	C₂²	C₂×#14523964	1	1	yes	characteristic
255	C₂²	C₂×#14523967	1	1	yes	characteristic
256	C₂²	C₂×#14523967	1	1	yes	characteristic
257	C₂²	#16863401	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	8	4	no	—
262	C₄	#16863142	1	4	yes	—
263	C₄	#16863145	1	4	yes	—
264	C₄	#16863683	1	2	yes	—
265	C₄	#16863688	1	2	yes	—
266	C₄	#16863744	1	1	yes	characteristic
267	C₄	#16863748	1	1	yes	characteristic
268	C₄	#16863792	1	1	yes	characteristic
269	C₄	#16863799	1	1	yes	characteristic
270	C₂	C₂×#16863142	1	1	yes	characteristic
271	C₂	C₂×#16863145	1	1	yes	characteristic
272	C₂	#22558877	1	1	yes	characteristic
273	C₂	#22558894	1	1	yes	characteristic
274	C₂	#22561158	1	1	yes	characteristic
275	C₂	#22561171	1	1	yes	characteristic
276	C₂	#22561199	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2868	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2868	D₆₇
hypocenter	C₆₇	#9896
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
25. C₄⋊(C₆₇⋊C₈)	Q₈	semidirect
143. C₆₇	#9896	semidirect

Maximal subgroups

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

subgroup	index
#22560361	2
C₄×C₄⋊(C₆₇⋊C₈)	2
#22558092	2
C₆₇×#2868	2
#9896	67