Subgroups — #30743809

Order 17152.

516 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#22558032	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₆₇	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	—	2	1	no	—
8	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
9	#16863151	C₂²	1	2	yes	—
10	#16863151	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
13	#391×C₆₇	—	2	1	no	—
14	#389×C₆₇	—	2	1	no	—
15	#389×C₆₇	C₂²	1	2	yes	—
16	#384×C₆₇	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
18	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
19	C₄³×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	C₂²×C₆₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
24	C₄×C₆₇⋊C₈	—	4	2	no	—
25	C₄×C₆₇⋊C₈	—	4	2	no	—
26	C₂²×Q₈×C₆₇	—	2	1	no	—
27	C₄⋊Q₈×C₆₇	—	2	2	no	—
28	C₄⋊Q₈×C₆₇	C₈	1	4	yes	complemented
29	C₄×Q₈×C₆₇	—	4	2	no	—
30	C₄×Q₈×C₆₇	—	4	2	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₇	—	2	2	no	—
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	#95×C₆₇	—	2	2	no	—
46	#95×C₆₇	—	2	2	no	—
47	C₂×C₆₇⋊C₈	—	2	4	no	—
48	C₂×C₆₇⋊C₈	—	2	4	no	—
49	C₂×C₆₇⋊C₈	—	4	2	no	—
50	C₂×C₆₇⋊C₈	—	4	2	no	—
51	C₂×C₆₇⋊C₈	—	4	2	no	—
52	C₂×C₆₇⋊C₈	—	4	2	no	—
53	C₂×Q₈×C₆₇	—	2	4	no	—
54	C₂×Q₈×C₆₇	—	4	2	no	—
55	C₂×Q₈×C₆₇	—	4	2	no	—
56	C₂²×C₄×C₆₇	—	2	2	no	—
57	C₂²×C₄×C₆₇	—	2	2	no	—
58	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₆₇	—	2	4	no	—
66	C₄⋊C₄×C₆₇	—	2	2	no	—
67	C₄⋊C₄×C₆₇	—	4	2	no	—
68	C₄⋊C₄×C₆₇	—	4	2	no	—
69	C₄⋊C₄×C₆₇	—	4	2	no	—
70	C₄⋊C₄×C₆₇	—	4	2	no	—
71	C₄⋊C₄×C₆₇	—	4	2	no	—
72	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
73	C₄²×C₆₇	—	2	2	no	—
74	C₄²×C₆₇	—	2	2	no	—
75	C₄²×C₆₇	—	2	2	no	—
76	C₄²×C₆₇	—	4	1	no	—
77	C₄²×C₆₇	—	4	4	no	—
78	C₄²×C₆₇	C₂×C₈	1	2	yes	—
79	C₄²×C₆₇	M₁₆	1	2	yes	—
80	C₄²×C₆₇	D₈	1	2	yes	—
81	C₄²×C₆₇	SD₁₆	1	2	yes	—
82	C₄²×C₆₇	SD₁₆	1	2	yes	—
83	C₄²×C₆₇	Q₁₆	1	2	yes	—
84	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	Q₈×C₆₇	—	4	8	no	—
86	C₂×C₄×C₆₇	—	2	1	no	—
87	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	2	1	no	—
92	C₂×C₄×C₆₇	—	2	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	—	4	2	no	—
97	C₂×C₄×C₆₇	—	4	2	no	—
98	C₂×C₄×C₆₇	—	4	2	no	—
99	C₂×C₄×C₆₇	—	4	2	no	—
100	C₂×C₄×C₆₇	—	4	2	no	—
101	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
102	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₆₇	#101	1	2	yes	—
104	C₂×C₄×C₆₇	#102	1	2	yes	—
105	C₂×C₄×C₆₇	#102	1	2	yes	—
106	C₂×C₄×C₆₇	#103	1	2	yes	—
107	C₂×C₄×C₆₇	#103	1	2	yes	—
108	C₂×C₄×C₆₇	#104	1	2	yes	—
109	C₂×C₄×C₆₇	#104	1	2	yes	—
110	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
116	C₆₇⋊C₈	—	4	4	no	—
117	C₆₇⋊C₈	—	4	4	no	—
118	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₆₇	#410	1	1	yes	characteristic
120	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
121	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
122	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
123	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
124	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
125	C₄×C₆₇	—	2	1	no	—
126	C₄×C₆₇	—	2	1	no	—
127	C₄×C₆₇	—	2	1	no	—
128	C₄×C₆₇	—	2	1	no	—
129	C₄×C₆₇	—	2	1	no	—
130	C₄×C₆₇	—	2	1	no	—
131	C₄×C₆₇	—	4	4	no	—
132	C₄×C₆₇	—	4	1	no	—
133	C₄×C₆₇	—	8	2	no	—
134	C₄×C₆₇	D₄⋊C₈	1	4	yes	—
135	C₄×C₆₇	Q₈⋊C₈	1	4	yes	—
136	C₄×C₆₇	#332	1	4	yes	—
137	#7570	—	67	1	no	maximal, Sylow
138	C₂×C₆₇	#1461	1	1	yes	characteristic
139	C₂×C₆₇	C₂×D₄⋊C₈	1	1	yes	characteristic
140	C₂×C₆₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
141	C₂×C₆₇	#1502	1	1	yes	characteristic
142	C₂×C₆₇	#1503	1	1	yes	characteristic
143	C₂×C₆₇	#1526	1	1	yes	characteristic
144	C₂×C₆₇	C₂×#332	1	1	yes	characteristic
145	C₄×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
146	C₄×C₄⋊C₈	—	67	1	no	—
147	C₄×C₄⋊C₈	—	67	1	no	—
148	#1271	—	67	4	no	—
149	C₆₇	#7570	1	1	yes	characteristic, complemented, Sylow
150	C₂×C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic
151	C₂×C₄×Q₈	—	2	1	no	—
152	C₂×C₄⋊C₈	—	134	1	no	—
153	C₂×C₄⋊C₈	—	134	1	no	—
154	C₂×C₄×C₈	—	134	1	no	—
155	C₂×C₄×C₈	—	134	1	no	—
156	#391	—	2	1	no	—
157	#389	—	2	1	no	—
158	#389	C₂×D₆₇	1	2	yes	—
159	#384	Dic₆₇	1	1	yes	characteristic
160	C₄×C₄⋊C₄	—	2	1	no	—
161	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
162	C₄³	C₂×D₆₇	1	1	yes	characteristic
163	#336	—	67	2	no	—
164	#336	—	67	2	no	—
165	C₂²×Q₈	—	2	1	no	—
166	C₂²×C₈	—	134	2	no	—
167	C₂²×C₈	—	134	2	no	—
168	C₄⋊Q₈	—	2	2	no	—
169	C₄⋊Q₈	C₆₇⋊C₈	1	4	yes	complemented
170	C₄×Q₈	—	4	2	no	—
171	C₄×Q₈	—	4	2	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
177	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
180	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
183	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
184	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
186	C₄⋊C₈	—	134	4	no	—
187	C₄⋊C₈	—	134	4	no	—
188	C₄×C₈	—	268	2	no	—
189	C₄×C₈	—	268	2	no	—
190	#95	—	2	2	no	—
191	#95	—	2	2	no	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×Q₈	—	4	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
198	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
204	C₂×C₈	—	268	2	no	—
205	C₂×C₈	—	268	2	no	—
206	C₂×C₈	—	268	2	no	—
207	C₂×C₈	—	268	2	no	—
208	C₂×C₈	—	134	4	no	—
209	C₂×C₈	—	134	4	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	4	1	no	—
222	C₄²	—	4	4	no	—
223	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
224	C₄²	#11761612	1	2	yes	—
225	C₄²	#11761613	1	2	yes	—
226	C₄²	#11761615	1	2	yes	—
227	C₄²	#11761616	1	2	yes	—
228	C₄²	#11761618	1	2	yes	—
229	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
230	Q₈	—	4	8	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
247	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
248	C₂×C₄	#14523962	1	2	yes	—
249	C₂×C₄	#14523964	1	2	yes	—
250	C₂×C₄	#14523964	1	2	yes	—
251	C₂×C₄	#14523967	1	2	yes	—
252	C₂×C₄	#14523967	1	2	yes	—
253	C₂×C₄	#14523970	1	2	yes	—
254	C₂×C₄	#14523970	1	2	yes	—
255	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
256	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
257	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
258	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
259	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
260	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
261	C₈	—	268	4	no	—
262	C₈	—	268	4	no	—
263	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
264	C₂²	#16863370	1	1	yes	characteristic
265	C₂²	C₂×#14523962	1	1	yes	characteristic
266	C₂²	C₂×#14523964	1	1	yes	characteristic
267	C₂²	C₂×#14523967	1	1	yes	characteristic
268	C₂²	C₂×#14523970	1	1	yes	characteristic
269	C₂²	C₂×#14523970	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	4	no	—
278	C₄	—	8	2	no	—
279	C₄	D₄⋊(C₆₇⋊C₈)	1	4	yes	—
280	C₄	Q₈⋊(C₆₇⋊C₈)	1	4	yes	—
281	C₄	#16863142	1	4	yes	—
282	C₂	#22558533	1	1	yes	characteristic
283	C₂	C₂×D₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
284	C₂	C₂×Q₈⋊(C₆₇⋊C₈)	1	1	yes	characteristic
285	C₂	#22558704	1	1	yes	characteristic
286	C₂	#22558708	1	1	yes	characteristic
287	C₂	#22558804	1	1	yes	characteristic
288	C₂	C₂×#16863142	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
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	#7570
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
28. C₄⋊Q₈×C₆₇	C₈	semidirect
149. C₆₇	#7570	semidirect
169. C₄⋊Q₈	C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₆₇⋊C₈)	2
#22558032	2
C₄×C₄⋊Q₈×C₆₇	2
#7570	67