Subgroups — #30748957

Order 17152.

394 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#22558957	C₂	1	1	yes	maximal, characteristic
3	#22558637	C₂	1	2	yes	maximal
4	#22558104	C₂	1	2	yes	maximal
5	C₆₇×#1848	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₃₄	—	2	1	no	—
8	C₈×Dic₁₃₄	—	2	1	no	—
9	C₈×Dic₁₃₄	C₄	1	2	yes	—
10	C₈×Dic₁₃₄	C₄	1	2	yes	—
11	#16863449	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₆₇⋊C₈	—	2	1	no	—
14	C₂×C₈×Dic₆₇	—	2	1	no	—
15	#16863149	—	2	1	no	—
16	#16863147	—	2	1	no	—
17	#16863126	—	2	4	no	—
18	C₂×#108×C₆₇	—	2	1	no	—
19	C₂×#108×C₆₇	—	2	1	no	—
20	#423×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₃₄	Q₈	1	1	yes	characteristic
23	C₂²×C₆₇⋊C₈	—	2	1	no	—
24	C₄×Dic₁₃₄	—	2	1	no	—
25	C₄×Dic₁₃₄	—	2	1	no	—
26	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
27	C₄×Dic₁₃₄	D₄	1	2	yes	—
28	C₂×C₄⋊Dic₆₇	Q₈	1	1	yes	characteristic
29	C₂×Dic₆₇⋊C₄	—	2	1	no	—
30	C₂×C₄×Dic₆₇	—	2	1	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
34	Dic₆₇⋊C₈	—	2	2	no	—
35	Dic₆₇⋊C₈	—	2	2	no	—
36	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
37	C₄×C₆₇⋊C₈	—	2	2	no	—
38	C₈×Dic₆₇	—	2	2	no	—
39	C₈×Dic₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	#108×C₆₇	—	4	2	no	—
45	#108×C₆₇	—	4	2	no	—
46	C₄⋊C₈×C₆₇	—	2	2	no	—
47	C₄⋊C₈×C₆₇	—	2	2	no	—
48	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₂×Dic₁₃₄	—	2	1	no	—
53	C₂×Dic₁₃₄	—	2	4	no	—
54	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
55	C₂²×Dic₆₇	—	2	1	no	—
56	C₂²×Dic₆₇	—	2	1	no	—
57	C₂×C₆₇⋊C₈	—	2	2	no	—
58	C₂×C₆₇⋊C₈	—	4	1	no	—
59	C₂×C₆₇⋊C₈	—	4	1	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₄⋊Dic₆₇	—	2	1	no	—
62	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
63	Dic₆₇⋊C₄	—	2	2	no	—
64	Dic₆₇⋊C₄	—	2	2	no	—
65	C₄×Dic₆₇	—	2	2	no	—
66	C₄×Dic₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
70	M₁₆×C₆₇	—	4	2	no	—
71	M₁₆×C₆₇	—	4	2	no	—
72	C₂×C₈×C₆₇	—	2	1	no	—
73	C₂×C₈×C₆₇	—	2	1	no	—
74	C₂×C₈×C₆₇	—	4	2	no	—
75	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
80	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
81	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₇	#41	1	1	yes	characteristic
85	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	4	1	no	—
88	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
98	C₂×Dic₆₇	—	2	4	no	—
99	C₂×Dic₆₇	—	2	2	no	—
100	C₂×Dic₆₇	—	2	2	no	—
101	C₂×Dic₆₇	—	4	2	no	—
102	Dic₁₃₄	—	2	4	no	—
103	Dic₁₃₄	—	2	2	no	—
104	Dic₁₃₄	#108	1	4	yes	complemented
105	C₈×C₆₇	—	2	1	no	—
106	C₈×C₆₇	—	2	1	no	—
107	C₈×C₆₇	—	8	2	no	—
108	C₈×C₆₇	#103	1	2	yes	—
109	C₈×C₆₇	#103	1	2	yes	—
110	C₆₇⋊C₈	—	4	2	no	—
111	C₆₇⋊C₈	—	8	2	no	—
112	C₂²×C₆₇	—	2	1	no	—
113	C₂²×C₆₇	—	2	1	no	—
114	C₂²×C₆₇	#380	1	1	yes	characteristic
115	C₂²×C₆₇	#475	1	1	yes	characteristic
116	C₂²×C₆₇	#477	1	1	yes	characteristic
117	C₄×C₆₇	—	2	1	no	—
118	C₄×C₆₇	—	2	1	no	—
119	C₄×C₆₇	—	2	1	no	—
120	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
121	C₄×C₆₇	#417	1	1	yes	characteristic
122	C₄×C₆₇	C₂×#108	1	2	yes	—
123	C₄×C₆₇	#460	1	1	yes	characteristic
124	C₄×C₆₇	#462	1	1	yes	characteristic
125	Dic₆₇	—	2	4	no	—
126	Dic₆₇	—	4	2	no	—
127	#8659	—	67	1	no	maximal, Sylow
128	C₂×C₆₇	—	2	1	no	—
129	C₂×C₆₇	—	2	1	no	—
130	C₂×C₆₇	#1803	1	1	yes	characteristic
131	C₂×C₆₇	#1810	1	1	yes	characteristic
132	C₂×C₆₇	#1814	1	1	yes	characteristic
133	C₂×C₈×Q₈	—	67	1	no	—
134	#1848	D₆₇	1	1	yes	characteristic
135	#1564	—	67	1	no	—
136	#1487	—	67	2	no	—
137	#1301	—	67	2	no	—
138	C₆₇	#8659	1	1	yes	characteristic, complemented, Sylow
139	C₂×C₄×Q₈	—	67	1	no	—
140	C₈×Q₈	—	67	2	no	—
141	C₈×Q₈	—	67	2	no	—
142	C₈×Q₈	—	134	1	no	—
143	C₈×Q₈	—	134	1	no	—
144	#431	—	67	2	no	—
145	C₂×#108	—	2	1	no	—
146	C₂×#108	—	2	1	no	—
147	#423	C₂×D₆₇	1	2	yes	—
148	C₂×C₄⋊C₈	—	67	1	no	—
149	C₂×C₄⋊C₈	—	134	1	no	—
150	C₂×C₄×C₈	—	134	1	no	—
151	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
152	#335	—	134	1	no	—
153	#334	—	134	1	no	—
154	Q₈⋊C₈	—	134	4	no	—
155	C₂²×Q₈	—	67	1	no	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂²×C₈	—	134	1	no	—
158	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
160	C₄×Q₈	—	67	2	no	—
161	C₄×Q₈	—	67	2	no	—
162	C₄×Q₈	—	134	1	no	—
163	C₄×Q₈	—	134	1	no	—
164	C₂×C₄⋊C₄	—	67	1	no	—
165	C₂×C₄⋊C₄	—	134	1	no	—
166	C₂×C₄²	—	134	1	no	—
167	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
168	#108	—	4	2	no	—
169	#108	—	4	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	67	2	no	—
173	C₄⋊C₈	—	134	2	no	—
174	C₄⋊C₈	—	134	2	no	—
175	C₄⋊C₈	—	134	2	no	—
176	C₄⋊C₈	—	134	1	no	—
177	C₂²⋊C₈	—	134	2	no	—
178	C₄×C₈	—	134	2	no	—
179	C₄×C₈	—	134	2	no	—
180	C₄×C₈	—	134	2	no	—
181	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
182	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
183	C₄×C₈	D₂₆₈	1	1	yes	characteristic
184	C₄×C₈	D₂₆₈	1	1	yes	characteristic
185	C₂×Q₈	—	67	2	no	—
186	C₂×Q₈	—	134	4	no	—
187	C₂×Q₈	—	134	1	no	—
188	C₂²×C₄	—	134	1	no	—
189	C₂²×C₄	—	134	1	no	—
190	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
191	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
192	C₂²×C₄	#11761629	1	1	yes	characteristic
193	M₁₆	—	4	2	no	—
194	M₁₆	—	4	2	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	268	1	no	—
199	C₂×C₈	—	268	1	no	—
200	C₂×C₈	—	268	2	no	—
201	C₂×C₈	—	134	2	no	—
202	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
203	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
204	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
205	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
206	C₂×C₈	#11761616	1	2	yes	—
207	C₂×C₈	#11761618	1	2	yes	—
208	C₄⋊C₄	—	67	2	no	—
209	C₄⋊C₄	—	134	2	no	—
210	C₄⋊C₄	—	134	2	no	—
211	C₄⋊C₄	—	134	1	no	—
212	C₄²	—	134	2	no	—
213	C₄²	—	134	2	no	—
214	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
215	C₄²	D₄×D₆₇	1	1	yes	characteristic
216	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
217	C₄²	#11761627	1	1	yes	characteristic
218	C₂³	#14524019	1	1	yes	characteristic
219	Q₈	—	67	4	no	—
220	Q₈	—	134	4	no	—
221	Q₈	—	134	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	268	2	no	—
225	C₂×C₄	—	134	2	no	—
226	C₂×C₄	—	134	4	no	—
227	C₂×C₄	—	134	2	no	—
228	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
229	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
230	C₂×C₄	#14524001	1	1	yes	characteristic
231	C₂×C₄	#14524014	1	1	yes	characteristic
232	C₂×C₄	#14524018	1	1	yes	characteristic
233	C₂×C₄	#14524023	1	1	yes	characteristic
234	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
235	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
236	C₂×C₄	#14524069	1	1	yes	characteristic
237	C₂×C₄	#14524074	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	8	2	no	—
241	C₈	—	268	2	no	—
242	C₈	—	536	2	no	—
243	C₈	#14523966	1	2	yes	—
244	C₈	#14523966	1	2	yes	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	#16863253	1	1	yes	characteristic
248	C₂²	#16863678	1	1	yes	characteristic
249	C₂²	#16863688	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	268	2	no	—
254	C₄	—	134	4	no	—
255	C₄	C₂×#14523966	1	1	yes	characteristic
256	C₄	#16863395	1	1	yes	characteristic
257	C₄	#108×D₆₇	1	2	yes	—
258	C₄	#16863593	1	1	yes	characteristic
259	C₄	#16863603	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	#22560211	1	1	yes	characteristic
263	C₂	#22560244	1	1	yes	characteristic
264	C₂	#22560264	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524014
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1848	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1848	D₆₇
hypocenter	C₆₇	#8659
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
104. Dic₁₃₄	#108	semidirect
138. C₆₇	#8659	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₃₄	2
#22558957	2
#22558637	2
#22558104	2
C₆₇×#1848	2
#8659	67