Subgroups — #30754724

Order 17152.

482 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561146	C₂	1	2	yes	maximal
2	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal
4	C₂×#16863118	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	#16863151	—	2	4	no	—
10	#16863149	—	2	2	no	—
11	Q₈⋊(C₆₇⋊C₈)	—	2	4	no	—
12	#16863118	—	2	2	no	—
13	C₂²×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
15	C₄×Q₁₆×C₆₇	—	2	2	no	—
16	C₄×Q₁₆×C₆₇	—	2	2	no	—
17	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
18	C₂×#103×C₆₇	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	C₂²×C₆₇⋊C₈	—	2	1	no	—
22	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄×C₆₇⋊C₈	—	2	1	no	—
26	C₄×C₆₇⋊C₈	—	2	1	no	—
27	C₄×C₆₇⋊C₈	—	2	1	no	—
28	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
29	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×Q₁₆×C₆₇	—	2	4	no	—
31	C₂×Q₁₆×C₆₇	—	2	2	no	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₆₇	—	2	2	no	—
35	C₄×Q₈×C₆₇	—	2	2	no	—
36	C₄×Q₈×C₆₇	—	2	2	no	—
37	C₄×Q₈×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₇	—	2	2	no	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	#107×C₆₇	—	2	2	no	—
43	#103×C₆₇	—	2	4	no	—
44	C₄×C₈×C₆₇	D₄	1	2	yes	—
45	C₄×C₈×C₆₇	D₄	1	2	yes	—
46	C₂×C₆₇⋊C₈	—	2	2	no	—
47	C₂×C₆₇⋊C₈	—	2	2	no	—
48	C₂×C₆₇⋊C₈	—	2	2	no	—
49	C₂×C₆₇⋊C₈	—	4	2	no	—
50	C₂×C₆₇⋊C₈	—	4	2	no	—
51	C₂×C₆₇⋊C₈	—	4	4	no	—
52	C₂×C₆₇⋊C₈	Q₁₆	1	4	yes	complemented
53	C₂×Q₈×C₆₇	—	2	2	no	—
54	C₂×Q₈×C₆₇	—	2	2	no	—
55	C₂×Q₈×C₆₇	—	2	2	no	—
56	C₂×Q₈×C₆₇	—	2	2	no	—
57	C₂×Q₈×C₆₇	—	2	2	no	—
58	C₂×Q₈×C₆₇	M₁₆	1	4	yes	—
59	C₂²×C₄×C₆₇	—	2	2	no	—
60	C₂²×C₄×C₆₇	—	2	2	no	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
64	Q₁₆×C₆₇	—	2	8	no	—
65	C₂×C₈×C₆₇	—	2	2	no	—
66	C₂×C₈×C₆₇	—	2	2	no	—
67	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
69	C₄⋊C₄×C₆₇	—	2	2	no	—
70	C₄⋊C₄×C₆₇	—	2	2	no	—
71	C₄⋊C₄×C₆₇	—	4	2	no	—
72	C₄⋊C₄×C₆₇	—	4	2	no	—
73	C₄²×C₆₇	—	4	2	no	—
74	C₄²×C₆₇	—	4	2	no	—
75	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₆₇	#41	1	1	yes	characteristic
79	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
80	Q₈×C₆₇	—	2	4	no	—
81	Q₈×C₆₇	—	2	4	no	—
82	Q₈×C₆₇	—	4	2	no	—
83	Q₈×C₆₇	—	4	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	4	no	—
86	C₂×C₄×C₆₇	—	4	2	no	—
87	C₂×C₄×C₆₇	—	4	2	no	—
88	C₂×C₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	—	4	4	no	—
90	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₆₇	#131	1	2	yes	—
98	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
99	C₂×C₄×C₆₇	#135	1	2	yes	—
100	C₈×C₆₇	—	2	2	no	—
101	C₈×C₆₇	—	2	2	no	—
102	C₆₇⋊C₈	—	2	4	no	—
103	C₆₇⋊C₈	—	4	2	no	—
104	C₆₇⋊C₈	—	8	2	no	—
105	C₂²×C₆₇	#390	1	1	yes	characteristic
106	C₂²×C₆₇	#435	1	2	yes	—
107	C₂²×C₆₇	C₄×Q₁₆	1	2	yes	—
108	C₂²×C₆₇	#444	1	2	yes	—
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	—	4	4	no	—
112	C₄×C₆₇	—	8	2	no	—
113	C₄×C₆₇	#407	1	1	yes	characteristic
114	C₄×C₆₇	#408	1	1	yes	characteristic
115	C₄×C₆₇	#435	1	2	yes	—
116	C₄×C₆₇	#467	1	2	yes	—
117	C₄×C₆₇	#494	1	1	yes	characteristic
118	C₄×C₆₇	#495	1	1	yes	characteristic
119	#9662	—	67	1	no	maximal, Sylow
120	C₂×C₆₇	#1584	1	4	yes	—
121	C₂×C₆₇	#1955	1	1	yes	characteristic
122	C₂×C₆₇	#1969	1	1	yes	characteristic
123	C₂×C₆₇	#1975	1	1	yes	characteristic
124	C₂×C₄×Q₁₆	D₆₇	1	1	yes	characteristic
125	#1984	—	67	2	no	—
126	C₂×#335	—	67	1	no	—
127	C₂×Q₈⋊C₈	—	67	2	no	—
128	C₂×#322	—	67	1	no	—
129	C₆₇	#9662	1	1	yes	characteristic, complemented, Sylow
130	C₂²×Q₁₆	Dic₆₇	1	1	yes	characteristic
131	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	—
132	C₄×Q₁₆	—	2	2	no	—
133	C₄×Q₁₆	—	2	2	no	—
134	C₂×#107	Dic₆₇	1	1	yes	characteristic
135	C₂×C₄⋊C₈	—	67	2	no	—
136	C₂×#103	Dic₆₇	1	2	yes	—
137	C₂×C₄×C₈	—	67	1	no	—
138	C₂×C₄×C₈	—	67	1	no	—
139	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
140	#336	—	134	4	no	—
141	#335	—	134	2	no	—
142	Q₈⋊C₈	—	134	4	no	—
143	#322	—	134	2	no	—
144	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
145	C₂×Q₁₆	—	2	4	no	—
146	C₂×Q₁₆	—	2	2	no	—
147	C₂²×C₈	—	67	2	no	—
148	C₂²×C₈	—	134	2	no	—
149	C₂²×C₈	—	134	1	no	—
150	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
151	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
152	C₄×Q₈	—	2	2	no	—
153	C₄×Q₈	—	2	2	no	—
154	C₄×Q₈	—	2	2	no	—
155	C₄×Q₈	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
158	C₂×C₄²	—	2	2	no	—
159	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
160	#107	—	2	2	no	—
161	C₄⋊C₈	—	134	2	no	—
162	C₄⋊C₈	—	134	2	no	—
163	#103	—	2	4	no	—
164	C₄×C₈	—	67	2	no	—
165	C₄×C₈	—	134	1	no	—
166	C₄×C₈	—	134	1	no	—
167	C₄×C₈	—	134	1	no	—
168	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
169	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	#11761612	1	4	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
179	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
180	C₂²×C₄	#11761627	1	1	yes	characteristic
181	Q₁₆	—	2	8	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	268	2	no	—
185	C₂×C₈	—	268	2	no	—
186	C₂×C₈	—	268	4	no	—
187	C₂×C₈	—	67	4	no	—
188	C₂×C₈	—	134	2	no	—
189	C₂×C₈	—	134	2	no	—
190	C₂×C₈	—	134	2	no	—
191	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
192	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	—	4	2	no	—
199	C₄²	D₄×D₆₇	1	1	yes	characteristic
200	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
202	C₄²	#11761627	1	1	yes	characteristic
203	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
204	Q₈	—	2	4	no	—
205	Q₈	—	2	4	no	—
206	Q₈	—	4	2	no	—
207	Q₈	—	4	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
215	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
216	C₂×C₄	#14524014	1	1	yes	characteristic
217	C₂×C₄	#14524014	1	1	yes	characteristic
218	C₂×C₄	#14524030	1	1	yes	characteristic
219	C₂×C₄	#14524039	1	1	yes	characteristic
220	C₂×C₄	C₂×#11761612	1	2	yes	—
221	C₂×C₄	#14524050	1	2	yes	—
222	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
223	C₂×C₄	#14524066	1	2	yes	—
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	268	2	no	—
227	C₈	—	536	2	no	—
228	C₈	—	134	4	no	—
229	C₂²	#16863299	1	1	yes	characteristic
230	C₂²	#16863468	1	2	yes	—
231	C₂²	Q₁₆×Dic₆₇	1	2	yes	—
232	C₂²	#16863518	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	8	2	no	—
237	C₄	#16863360	1	1	yes	characteristic
238	C₄	#16863363	1	1	yes	characteristic
239	C₄	#16863468	1	2	yes	—
240	C₄	#16863634	1	2	yes	—
241	C₄	#16863774	1	1	yes	characteristic
242	C₄	#16863781	1	1	yes	characteristic
243	C₂	#22559060	1	4	yes	—
244	C₂	#22561008	1	1	yes	characteristic
245	C₂	#22561071	1	1	yes	characteristic
246	C₂	#22561103	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₄×Q₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₄×Q₁₆	D₆₇
hypocenter	C₆₇	#9662
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
52. C₂×C₆₇⋊C₈	Q₁₆	semidirect
129. C₆₇	#9662	semidirect

Maximal subgroups

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

subgroup	index
#22561146	2
C₂×#16863149	2
C₂×Q₈⋊(C₆₇⋊C₈)	2
C₂×#16863118	2
C₂×C₄×Q₁₆×C₆₇	2
#9662	67