Subgroups — #30748970

Order 17152.

378 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560347	C₂	1	1	yes	maximal, characteristic
2	#22558957	C₂	1	1	yes	maximal, characteristic
3	#22558638	C₂	1	2	yes	maximal
4	#22558106	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	#16863420	C₂²	1	2	yes	—
8	#16863161	—	2	1	no	—
9	#16863161	—	2	1	no	—
10	#16863149	—	2	1	no	—
11	#16863147	—	2	1	no	—
12	Q₈⋊(C₆₇⋊C₈)	—	2	4	no	—
13	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₆₇	—	2	1	no	—
15	C₈×Q₈×C₆₇	—	2	1	no	—
16	C₈×Q₈×C₆₇	C₄	1	2	yes	—
17	C₈×Q₈×C₆₇	C₄	1	2	yes	—
18	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₇	—	2	1	no	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#11761612	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₄×C₆₇⋊C₈	—	2	2	no	—
28	C₂²×Q₈×C₆₇	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	—	2	1	no	—
30	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₆₇	—	2	1	no	—
33	C₄×Q₈×C₆₇	—	2	1	no	—
34	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₄×Q₈×C₆₇	D₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₆₇	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	—	2	1	no	—
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₆₇	—	2	2	no	—
41	C₄⋊C₈×C₆₇	—	2	2	no	—
42	C₄⋊C₈×C₆₇	—	2	1	no	—
43	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₆₇	—	2	2	no	—
45	C₄×C₈×C₆₇	—	2	2	no	—
46	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
50	#11761612	—	4	2	no	—
51	#11761612	—	4	2	no	—
52	C₂×C₆₇⋊C₈	—	4	2	no	—
53	C₂×C₆₇⋊C₈	—	4	2	no	—
54	C₂×Q₈×C₆₇	—	2	4	no	—
55	C₂×Q₈×C₆₇	—	2	1	no	—
56	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₆₇	—	2	1	no	—
58	C₂²×C₄×C₆₇	—	2	1	no	—
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	—	2	2	no	—
64	C₂×C₈×C₆₇	—	2	2	no	—
65	C₂×C₈×C₆₇	—	4	2	no	—
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₄⋊C₄×C₆₇	—	2	2	no	—
71	C₄⋊C₄×C₆₇	—	2	2	no	—
72	C₄⋊C₄×C₆₇	—	2	1	no	—
73	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
74	C₄²×C₆₇	—	2	2	no	—
75	C₄²×C₆₇	—	2	2	no	—
76	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₆₇	#41	1	1	yes	characteristic
80	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
81	Q₈×C₆₇	—	2	4	no	—
82	Q₈×C₆₇	—	2	2	no	—
83	Q₈×C₆₇	#108	1	4	yes	—
84	C₂×C₄×C₆₇	—	2	4	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	4	1	no	—
89	C₂×C₄×C₆₇	—	4	2	no	—
90	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#135	1	2	yes	—
97	C₂×C₄×C₆₇	#137	1	2	yes	—
98	C₈×C₆₇	—	2	1	no	—
99	C₈×C₆₇	—	2	1	no	—
100	C₈×C₆₇	—	2	1	no	—
101	C₈×C₆₇	—	2	1	no	—
102	C₈×C₆₇	—	4	2	no	—
103	C₆₇⋊C₈	—	8	2	no	—
104	C₆₇⋊C₈	—	8	2	no	—
105	C₂²×C₆₇	—	2	1	no	—
106	C₂²×C₆₇	—	2	1	no	—
107	C₂²×C₆₇	#380	1	1	yes	characteristic
108	C₂²×C₆₇	#479	1	2	yes	—
109	C₄×C₆₇	—	2	4	no	—
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	—	2	1	no	—
112	C₄×C₆₇	—	2	1	no	—
113	C₄×C₆₇	—	4	2	no	—
114	C₄×C₆₇	#416	1	1	yes	characteristic
115	C₄×C₆₇	#419	1	1	yes	characteristic
116	C₄×C₆₇	C₂×#108	1	2	yes	—
117	C₄×C₆₇	#464	1	2	yes	—
118	#8661	—	67	1	no	maximal, Sylow
119	C₂×C₆₇	—	2	2	no	—
120	C₂×C₆₇	#1805	1	1	yes	characteristic
121	C₂×C₆₇	#1810	1	1	yes	characteristic
122	C₂×C₆₇	#1814	1	1	yes	characteristic
123	C₂×C₈×Q₈	D₆₇	1	1	yes	characteristic
124	#1837	—	67	1	no	—
125	#1564	—	67	1	no	—
126	#1487	—	67	2	no	—
127	#1301	—	67	2	no	—
128	C₆₇	#8661	1	1	yes	characteristic, complemented, Sylow
129	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
130	C₈×Q₈	—	2	1	no	—
131	C₈×Q₈	—	2	1	no	—
132	C₈×Q₈	Dic₆₇	1	2	yes	—
133	C₈×Q₈	Dic₆₇	1	2	yes	—
134	#431	—	67	2	no	—
135	#423	—	67	2	no	—
136	C₂×C₄⋊C₈	—	2	1	no	—
137	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
138	C₂×C₄×C₈	—	2	1	no	—
139	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
140	#341	—	134	1	no	—
141	#341	—	134	1	no	—
142	#335	—	134	1	no	—
143	#334	—	134	1	no	—
144	Q₈⋊C₈	—	134	4	no	—
145	C₂²×Q₈	Dic₁₃₄	1	1	yes	characteristic
146	C₂×M₁₆	—	134	2	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
149	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
150	C₄×Q₈	—	2	1	no	—
151	C₄×Q₈	—	2	1	no	—
152	C₄×Q₈	D₂₆₈	1	2	yes	—
153	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	Dic₁₃₄	1	1	yes	characteristic
156	C₂×C₄²	—	2	1	no	—
157	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₄⋊C₈	—	134	2	no	—
162	C₄⋊C₈	—	134	2	no	—
163	C₄⋊C₈	—	134	2	no	—
164	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
165	C₂²⋊C₈	—	134	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	—	134	2	no	—
169	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
170	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
171	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
172	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	2	1	no	—
175	C₂×Q₈	C₄⋊Dic₆₇	1	2	yes	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
180	C₂²×C₄	#11761628	1	1	yes	characteristic
181	M₁₆	—	268	2	no	—
182	M₁₆	—	268	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	268	2	no	—
188	C₂×C₈	—	268	2	no	—
189	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
191	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
192	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
200	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
201	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
202	C₄²	#11761628	1	1	yes	characteristic
203	C₂³	#14524022	1	1	yes	characteristic
204	Q₈	—	2	4	no	—
205	Q₈	—	2	2	no	—
206	Q₈	#14523979	1	4	yes	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
215	C₂×C₄	#14524002	1	1	yes	characteristic
216	C₂×C₄	#14524017	1	1	yes	characteristic
217	C₂×C₄	#14524017	1	1	yes	characteristic
218	C₂×C₄	#14524022	1	1	yes	characteristic
219	C₂×C₄	#14524063	1	2	yes	—
220	C₂×C₄	#14524074	1	2	yes	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	4	2	no	—
226	C₈	—	536	2	no	—
227	C₈	—	536	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	#16863256	1	1	yes	characteristic
231	C₂²	#16863703	1	2	yes	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	4	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	2	no	—
237	C₄	#16863392	1	1	yes	characteristic
238	C₄	#16863405	1	1	yes	characteristic
239	C₄	C₂×#14523979	1	2	yes	—
240	C₄	#16863618	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	#22560224	1	1	yes	characteristic
243	C₂	#22560247	1	1	yes	characteristic
244	C₂	#22560263	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524017
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₈×Q₈×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₈×Q₈	D₆₇
hypocenter	C₆₇	#8661
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
128. C₆₇	#8661	semidirect

Maximal subgroups

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

subgroup	index
#22560347	2
#22558957	2
#22558638	2
#22558106	2
C₂×C₈×Q₈×C₆₇	2
#8661	67