Subgroups — #30748870

Order 17152.

336 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	#22558968	C₂	1	1	yes	maximal, characteristic
2	#22558965	C₂	1	1	yes	maximal, characteristic
3	#22558947	C₂	1	2	yes	maximal
4	C₆₇⋊#1309	C₂	1	2	yes	maximal
5	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic
6	#16863420	C₂²	1	2	yes	—
7	#16863420	C₂²	1	2	yes	—
8	#16863149	—	2	1	no	—
9	#16863149	—	2	1	no	—
10	#16863147	—	2	1	no	—
11	#16863147	—	2	1	no	—
12	#16863142	—	2	2	no	—
13	#16863142	—	2	2	no	—
14	#533×C₆₇	C₂²	1	1	yes	characteristic
15	#446×C₆₇	—	2	1	no	—
16	#446×C₆₇	—	2	1	no	—
17	C₈×Q₈×C₆₇	—	2	1	no	—
18	C₈×Q₈×C₆₇	—	2	1	no	—
19	#424×C₆₇	C₄	1	2	yes	—
20	#423×C₆₇	C₂²	1	1	yes	characteristic
21	#405×C₆₇	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
34	#125×C₆₇	Q₈	1	2	yes	—
35	C₂²⋊Q₈×C₆₇	—	2	1	no	—
36	C₂²⋊Q₈×C₆₇	—	2	1	no	—
37	C₄×Q₈×C₆₇	—	2	1	no	—
38	C₄×Q₈×C₆₇	—	2	1	no	—
39	#117×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₆₇	—	2	2	no	—
42	C₄⋊C₈×C₆₇	—	2	2	no	—
43	C₄⋊C₈×C₆₇	—	2	1	no	—
44	C₄⋊C₈×C₆₇	—	2	1	no	—
45	#97×C₆₇	—	2	2	no	—
46	C₄×C₈×C₆₇	—	2	2	no	—
47	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
51	#11761612	—	8	2	no	—
52	#11761612	—	8	2	no	—
53	C₂×C₆₇⋊C₈	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	4	2	no	—
55	C₂×Q₈×C₆₇	—	2	1	no	—
56	C₂×Q₈×C₆₇	—	2	1	no	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
60	M₁₆×C₆₇	—	4	2	no	—
61	C₂×C₈×C₆₇	—	2	2	no	—
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	—	4	1	no	—
64	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₄⋊C₄×C₆₇	—	2	1	no	—
69	C₄⋊C₄×C₆₇	—	2	1	no	—
70	C₄⋊C₄×C₆₇	—	2	2	no	—
71	C₄⋊C₄×C₆₇	—	2	2	no	—
72	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
73	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
74	C₂²⋊C₄×C₆₇	—	2	2	no	—
75	C₄²×C₆₇	—	2	2	no	—
76	C₄²×C₆₇	C₂×D₄	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	C₂×Q₈	1	1	yes	characteristic
80	C₂³×C₆₇	#117	1	1	yes	characteristic
81	Q₈×C₆₇	—	4	2	no	—
82	Q₈×C₆₇	—	4	1	no	—
83	Q₈×C₆₇	—	4	1	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂×C₄⋊C₄	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	C₂²⋊Q₈	1	1	yes	characteristic
97	C₈×C₆₇	—	2	1	no	—
98	C₈×C₆₇	—	2	1	no	—
99	C₈×C₆₇	—	2	1	no	—
100	C₈×C₆₇	—	4	2	no	—
101	C₈×C₆₇	#101	1	2	yes	—
102	C₆₇⋊C₈	—	8	2	no	—
103	C₆₇⋊C₈	—	8	2	no	—
104	C₂²×C₆₇	—	2	1	no	—
105	C₂²×C₆₇	—	2	1	no	—
106	C₂²×C₆₇	—	2	1	no	—
107	C₂²×C₆₇	#380	1	1	yes	characteristic
108	C₄×C₆₇	—	2	1	no	—
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	—	4	2	no	—
112	C₄×C₆₇	—	4	2	no	—
113	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
114	C₄×C₆₇	#413	1	1	yes	characteristic
115	C₄×C₆₇	#471	1	2	yes	—
116	C₄×C₆₇	#472	1	1	yes	characteristic
117	C₄×C₆₇	#473	1	1	yes	characteristic
118	#8641	—	67	1	no	maximal, Sylow
119	C₂×C₆₇	—	4	1	no	—
120	C₂×C₆₇	#1801	1	1	yes	characteristic
121	C₂×C₆₇	#1815	1	1	yes	characteristic
122	C₂×C₆₇	#1816	1	1	yes	characteristic
123	#2960	D₆₇	1	1	yes	characteristic
124	#1567	—	67	1	no	—
125	#1566	—	67	1	no	—
126	#1561	—	67	2	no	—
127	#1309	—	67	2	no	—
128	C₆₇	#8641	1	1	yes	characteristic, complemented, Sylow
129	#533	C₂×D₆₇	1	1	yes	characteristic
130	#446	—	2	1	no	—
131	#446	—	2	1	no	—
132	C₈×Q₈	—	2	1	no	—
133	C₈×Q₈	—	2	1	no	—
134	#424	Dic₆₇	1	2	yes	—
135	#423	—	67	2	no	—
136	#423	—	67	2	no	—
137	#423	C₂×D₆₇	1	1	yes	characteristic
138	#405	Dic₆₇	1	2	yes	—
139	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
140	#335	—	134	1	no	—
141	#335	—	134	1	no	—
142	#334	—	134	1	no	—
143	#334	—	134	1	no	—
144	#332	—	134	2	no	—
145	#332	—	134	2	no	—
146	C₂×M₁₆	—	134	2	no	—
147	C₂×M₁₆	—	134	2	no	—
148	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
149	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
150	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
151	C₄⋊Q₈	D₂₆₈	1	1	yes	characteristic
152	C₄⋊Q₈	D₂₆₈	1	1	yes	characteristic
153	#125	Dic₁₃₄	1	2	yes	—
154	C₂²⋊Q₈	—	2	1	no	—
155	C₂²⋊Q₈	—	2	1	no	—
156	C₄×Q₈	—	2	1	no	—
157	C₄×Q₈	—	2	1	no	—
158	#117	C₂×Dic₆₇	1	2	yes	—
159	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	2	1	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	—	134	2	no	—
165	C₄⋊C₈	—	134	2	no	—
166	C₄⋊C₈	—	134	2	no	—
167	C₄⋊C₈	—	134	2	no	—
168	#97	—	2	2	no	—
169	C₄×C₈	—	2	2	no	—
170	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
171	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
172	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
173	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
174	C₂×Q₈	—	2	1	no	—
175	C₂×Q₈	—	2	1	no	—
176	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
177	C₂²×C₄	#11761628	1	1	yes	characteristic
178	C₂²×C₄	#11761628	1	1	yes	characteristic
179	M₁₆	—	4	2	no	—
180	M₁₆	—	536	2	no	—
181	M₁₆	—	536	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	—	268	2	no	—
186	C₂×C₈	—	268	2	no	—
187	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
188	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
196	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
197	C₂²⋊C₄	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
200	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
201	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
202	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
203	C₂³	#14524002	1	1	yes	characteristic
204	Q₈	—	4	2	no	—
205	Q₈	—	4	1	no	—
206	Q₈	—	4	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄⋊Dic₆₇	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	#14524022	1	1	yes	characteristic
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	4	2	no	—
224	C₈	—	536	2	no	—
225	C₈	—	536	2	no	—
226	C₈	#14523962	1	2	yes	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	#16863256	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	2	no	—
236	C₄	C₂×#14523962	1	1	yes	characteristic
237	C₄	#16863380	1	1	yes	characteristic
238	C₄	#16863659	1	2	yes	—
239	C₄	#16863666	1	1	yes	characteristic
240	C₄	#16863671	1	1	yes	characteristic
241	C₂	—	4	1	no	—
242	C₂	#22560205	1	1	yes	characteristic
243	C₂	#22560268	1	1	yes	characteristic
244	C₂	#22560273	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863256
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2960	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2960	D₆₇
hypocenter	C₆₇	#8641
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
128. C₆₇	#8641	semidirect

Maximal subgroups

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

subgroup	index
#22558968	2
#22558965	2
#22558947	2
C₆₇⋊#1309	2
C₆₇×#2960	2
#8641	67