Subgroups — C₆₇×#9066 #30688317

Order 17152.

570 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×#494×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2369	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1987	C₂	1	2	yes	maximal
4	C₂×#333×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1279	C₂	1	2	yes	maximal
6	C₂²×Q₁₆×C₆₇	—	2	2	no	—
7	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
9	#494×C₆₇	—	2	2	no	—
10	#494×C₆₇	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	#391×C₆₇	—	2	2	no	—
14	#389×C₆₇	—	2	2	no	—
15	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
16	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	#340×C₆₇	—	2	2	no	—
18	#340×C₆₇	—	2	2	no	—
19	#333×C₆₇	—	2	2	no	—
20	C₂²×Q₈×C₆₇	—	2	1	no	—
21	C₂²×Q₈×C₆₇	D₄	1	2	yes	—
22	C₂×Q₁₆×C₆₇	—	2	8	no	—
23	C₂×Q₁₆×C₆₇	—	4	4	no	—
24	C₂×Q₁₆×C₆₇	—	4	4	no	—
25	C₂²×C₈×C₆₇	—	2	1	no	—
26	C₂²×C₈×C₆₇	—	2	2	no	—
27	C₄⋊Q₈×C₆₇	—	2	1	no	—
28	C₄⋊Q₈×C₆₇	—	2	2	no	—
29	C₄⋊Q₈×C₆₇	—	2	2	no	—
30	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
31	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
38	C₂×C₄²×C₆₇	—	2	2	no	—
39	C₂×C₄²×C₆₇	—	2	2	no	—
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	C₂×C₄	1	2	yes	—
45	#95×C₆₇	—	2	4	no	—
46	#95×C₆₇	—	2	2	no	—
47	C₂×Q₈×C₆₇	—	2	2	no	—
48	C₂×Q₈×C₆₇	—	2	2	no	—
49	C₂×Q₈×C₆₇	—	4	2	no	—
50	C₂×Q₈×C₆₇	—	4	2	no	—
51	C₂×Q₈×C₆₇	—	4	2	no	—
52	C₂×Q₈×C₆₇	—	4	2	no	—
53	C₂×Q₈×C₆₇	—	4	1	no	—
54	C₂×Q₈×C₆₇	C₂²⋊C₄	1	4	yes	—
55	C₂²×C₄×C₆₇	—	2	2	no	—
56	C₂²×C₄×C₆₇	—	2	2	no	—
57	C₂²×C₄×C₆₇	—	2	2	no	—
58	C₂²×C₄×C₆₇	—	4	1	no	—
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
61	Q₁₆×C₆₇	—	4	16	no	—
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	—	4	2	no	—
67	C₂×C₈×C₆₇	—	4	2	no	—
68	C₄⋊C₄×C₆₇	—	2	4	no	—
69	C₄⋊C₄×C₆₇	—	2	4	no	—
70	C₄⋊C₄×C₆₇	—	2	2	no	—
71	C₄⋊C₄×C₆₇	—	2	4	no	—
72	C₄⋊C₄×C₆₇	—	2	2	no	—
73	C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₄⋊C₄×C₆₇	—	4	2	no	—
75	C₄⋊C₄×C₆₇	—	8	2	no	—
76	C₄⋊C₄×C₆₇	Q₁₆	1	4	yes	complemented
77	C₄²×C₆₇	—	4	2	no	—
78	C₄²×C₆₇	—	4	2	no	—
79	C₄²×C₆₇	—	8	2	no	—
80	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₆₇	#41	1	2	yes	—
83	C₂³×C₆₇	#120	1	1	yes	characteristic
84	Q₈×C₆₇	—	4	4	no	—
85	Q₈×C₆₇	—	4	4	no	—
86	Q₈×C₆₇	—	8	2	no	—
87	Q₈×C₆₇	—	8	2	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	2	4	no	—
91	C₂×C₄×C₆₇	—	2	4	no	—
92	C₂×C₄×C₆₇	—	4	2	no	—
93	C₂×C₄×C₆₇	—	4	2	no	—
94	C₂×C₄×C₆₇	—	4	2	no	—
95	C₂×C₄×C₆₇	—	4	2	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	—	8	1	no	—
101	C₂×C₄×C₆₇	—	8	1	no	—
102	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₆₇	#121	1	2	yes	—
105	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
107	C₂×C₄×C₆₇	#135	1	2	yes	—
108	C₂×C₄×C₆₇	#137	1	2	yes	—
109	C₂×C₄×C₆₇	#137	1	2	yes	—
110	C₈×C₆₇	—	2	4	no	—
111	C₈×C₆₇	—	8	2	no	—
112	C₂²×C₆₇	#386	1	1	yes	characteristic
113	C₂²×C₆₇	#451	1	2	yes	—
114	C₂²×C₆₇	#452	1	2	yes	—
115	C₂²×C₆₇	#453	1	1	yes	characteristic
116	C₂²×C₆₇	#453	1	1	yes	characteristic
117	C₄×C₆₇	—	2	1	no	—
118	C₄×C₆₇	—	2	1	no	—
119	C₄×C₆₇	—	4	4	no	—
120	C₄×C₆₇	—	4	4	no	—
121	C₄×C₆₇	—	8	2	no	—
122	C₄×C₆₇	—	8	2	no	—
123	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
124	C₄×C₆₇	#441	1	2	yes	—
125	C₄×C₆₇	#462	1	2	yes	—
126	C₄×C₆₇	#464	1	2	yes	—
127	#9066	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂×C₆₇	#1632	1	2	yes	—
129	C₂×C₆₇	#1633	1	1	yes	characteristic
130	C₂×C₆₇	#1634	1	1	yes	characteristic
131	C₂×C₆₇	#1878	1	2	yes	—
132	C₂×C₆₇	#1887	1	1	yes	characteristic
133	C₂×#494	C₂×C₆₇	1	1	yes	characteristic
134	#2369	C₂×C₆₇	1	1	yes	characteristic
135	#1987	C₂×C₆₇	1	2	yes	—
136	C₂×#333	C₂×C₆₇	1	1	yes	characteristic
137	#1279	C₂×C₆₇	1	2	yes	—
138	C₆₇	#9066	1	1	yes	characteristic, direct product factor, Sylow
139	C₂²×Q₁₆	—	2	2	no	—
140	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
141	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
142	#494	—	2	2	no	—
143	#494	C₄×C₆₇	1	4	yes	complemented
144	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
145	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
146	#391	—	2	2	no	—
147	#389	—	2	2	no	—
148	C₄×C₄⋊C₄	—	2	2	no	—
149	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
150	#340	—	2	2	no	—
151	#340	—	2	2	no	—
152	#333	—	2	2	no	—
153	C₂²×Q₈	—	2	1	no	—
154	C₂²×Q₈	D₄×C₆₇	1	2	yes	—
155	C₂×Q₁₆	—	2	8	no	—
156	C₂×Q₁₆	—	4	4	no	—
157	C₂×Q₁₆	—	4	4	no	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₄⋊Q₈	—	2	1	no	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
164	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
165	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
170	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
171	C₂×C₄²	—	2	2	no	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄×C₈	—	2	1	no	—
177	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
178	#95	—	2	4	no	—
179	#95	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	1	no	—
187	C₂×Q₈	C₂²⋊C₄×C₆₇	1	4	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	4	1	no	—
192	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
194	Q₁₆	—	4	16	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	8	2	no	—
209	C₄⋊C₄	Q₁₆×C₆₇	1	4	yes	complemented
210	C₄²	—	4	2	no	—
211	C₄²	—	4	2	no	—
212	C₄²	—	8	2	no	—
213	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
214	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
215	C₄²	#41×C₆₇	1	2	yes	—
216	C₂³	#120×C₆₇	1	1	yes	characteristic
217	Q₈	—	4	4	no	—
218	Q₈	—	4	4	no	—
219	Q₈	—	8	2	no	—
220	Q₈	—	8	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	8	1	no	—
234	C₂×C₄	—	8	1	no	—
235	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
236	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
237	C₂×C₄	#121×C₆₇	1	2	yes	—
238	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
239	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
240	C₂×C₄	#135×C₆₇	1	2	yes	—
241	C₂×C₄	#137×C₆₇	1	2	yes	—
242	C₂×C₄	#137×C₆₇	1	2	yes	—
243	C₈	—	2	4	no	—
244	C₈	—	8	2	no	—
245	C₂²	#386×C₆₇	1	1	yes	characteristic
246	C₂²	#451×C₆₇	1	2	yes	—
247	C₂²	#452×C₆₇	1	2	yes	—
248	C₂²	#453×C₆₇	1	1	yes	characteristic
249	C₂²	#453×C₆₇	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	4	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	2	no	—
255	C₄	—	8	2	no	—
256	C₄	C₄×Q₁₆×C₆₇	1	2	yes	—
257	C₄	#441×C₆₇	1	2	yes	—
258	C₄	#462×C₆₇	1	2	yes	—
259	C₄	#464×C₆₇	1	2	yes	—
260	C₂	C₆₇×#1632	1	2	yes	—
261	C₂	C₆₇×#1633	1	1	yes	characteristic
262	C₂	C₆₇×#1634	1	1	yes	characteristic
263	C₂	C₆₇×#1878	1	2	yes	—
264	C₂	C₆₇×#1887	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	#120
commutator	C₄²	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
10. #494×C₆₇	C₄	semidirect
76. C₄⋊C₄×C₆₇	Q₁₆	semidirect
127. #9066	C₆₇	direct (N × H)
138. C₆₇	#9066	direct (N × H)
143. #494	C₄×C₆₇	semidirect
209. C₄⋊C₄	Q₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#494×C₆₇	2
C₆₇×#2369	2
C₆₇×#1987	2
C₂×#333×C₆₇	2
C₆₇×#1279	2
#9066	67

Subgroups — C67×#9066 #30688317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#9066 #30688317