Subgroups — C₆₇×#33287 #30712538

Order 17152.

710 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2862	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2859	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1990	C₂	1	2	yes	maximal
4	C₆₇×#1956	C₂	1	2	yes	maximal
5	C₆₇×#1935	C₂	1	8	yes	maximal
6	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#125×C₆₇	—	2	1	no	—
10	C₂×#103×C₆₇	C₂²	1	8	yes	—
11	C₂×#97×C₆₇	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
13	#391×C₆₇	—	2	1	no	—
14	#391×C₆₇	—	2	1	no	—
15	#391×C₆₇	C₂²	1	4	yes	—
16	#389×C₆₇	—	2	2	no	—
17	#389×C₆₇	C₂²	1	4	yes	—
18	#384×C₆₇	C₂²	1	2	yes	—
19	#382×C₆₇	—	2	1	no	—
20	#382×C₆₇	—	2	2	no	—
21	C₄³×C₆₇	C₂²	1	1	yes	characteristic
22	#340×C₆₇	C₂²	1	8	yes	—
23	C₂²×Q₈×C₆₇	—	2	1	no	—
24	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
25	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
26	C₄⋊Q₈×C₆₇	—	2	4	no	—
27	C₄⋊Q₈×C₆₇	—	2	4	no	—
28	#125×C₆₇	—	4	4	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
36	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
37	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	D₄	1	2	yes	—
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	#103×C₆₇	—	2	16	no	—
43	#97×C₆₇	—	2	8	no	—
44	C₄×C₈×C₆₇	—	2	4	no	—
45	#95×C₆₇	—	2	4	no	—
46	#95×C₆₇	—	2	4	no	—
47	#95×C₆₇	—	2	4	no	—
48	C₂×Q₈×C₆₇	—	2	8	no	—
49	C₂×Q₈×C₆₇	—	2	4	no	—
50	C₂×Q₈×C₆₇	—	4	2	no	—
51	C₂×Q₈×C₆₇	—	4	2	no	—
52	C₂×Q₈×C₆₇	—	4	2	no	—
53	C₂²×C₄×C₆₇	—	2	4	no	—
54	C₂²×C₄×C₆₇	—	2	2	no	—
55	C₂²×C₄×C₆₇	—	2	2	no	—
56	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
61	C₂×C₈×C₆₇	—	2	4	no	—
62	C₂×C₈×C₆₇	—	2	4	no	—
63	C₂×C₈×C₆₇	#41	1	8	yes	—
64	C₄⋊C₄×C₆₇	—	2	8	no	—
65	C₄⋊C₄×C₆₇	—	2	4	no	—
66	C₄⋊C₄×C₆₇	—	4	8	no	—
67	C₄⋊C₄×C₆₇	—	4	2	no	—
68	C₄⋊C₄×C₆₇	—	4	2	no	—
69	C₄⋊C₄×C₆₇	—	4	2	no	—
70	C₄⋊C₄×C₆₇	—	4	4	no	—
71	C₄⋊C₄×C₆₇	—	4	4	no	—
72	C₄²×C₆₇	—	2	2	no	—
73	C₄²×C₆₇	—	2	2	no	—
74	C₄²×C₆₇	—	2	4	no	—
75	C₄²×C₆₇	—	2	4	no	—
76	C₄²×C₆₇	#41	1	4	yes	—
77	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
78	Q₈×C₆₇	—	4	8	no	—
79	Q₈×C₆₇	—	8	4	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	4	2	no	—
87	C₂×C₄×C₆₇	—	4	2	no	—
88	C₂×C₄×C₆₇	—	4	4	no	—
89	C₂×C₄×C₆₇	—	4	4	no	—
90	C₂×C₄×C₆₇	—	4	4	no	—
91	C₂×C₄×C₆₇	—	4	2	no	—
92	C₂×C₄×C₆₇	—	4	4	no	—
93	C₂×C₄×C₆₇	—	4	2	no	—
94	C₂×C₄×C₆₇	#124	1	2	yes	—
95	C₂×C₄×C₆₇	#124	1	2	yes	—
96	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×C₄×C₆₇	#137	1	2	yes	—
98	C₂×C₄×C₆₇	#137	1	4	yes	—
99	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
101	C₈×C₆₇	—	2	8	no	—
102	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
103	C₂²×C₆₇	#532	1	2	yes	—
104	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
106	C₂²×C₆₇	C₂×#137	1	2	yes	—
107	C₄×C₆₇	—	4	2	no	—
108	C₄×C₆₇	—	4	2	no	—
109	C₄×C₆₇	—	8	2	no	—
110	C₄×C₆₇	—	8	2	no	—
111	C₄×C₆₇	—	8	4	no	—
112	C₄×C₆₇	#489	1	4	yes	—
113	C₄×C₆₇	#491	1	4	yes	—
114	C₄×C₆₇	#529	1	4	yes	—
115	#33287	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂×C₆₇	#2861	1	1	yes	characteristic
117	C₂×C₆₇	C₂×#489	1	1	yes	characteristic
118	C₂×C₆₇	C₂×#491	1	1	yes	characteristic
119	C₂×C₆₇	#3182	1	2	yes	—
120	C₂×C₆₇	#3185	1	2	yes	—
121	#2862	C₂×C₆₇	1	1	yes	characteristic
122	#2859	C₂×C₆₇	1	1	yes	characteristic
123	#1990	C₂×C₆₇	1	2	yes	—
124	#1956	C₂×C₆₇	1	2	yes	—
125	#1935	C₂×C₆₇	1	8	yes	—
126	C₄×#97	C₂×C₆₇	1	1	yes	characteristic
127	C₆₇	#33287	1	1	yes	characteristic, direct product factor, Sylow
128	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
129	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
130	C₂×#125	—	2	1	no	—
131	C₂×#103	C₂²×C₆₇	1	8	yes	—
132	C₂×#97	C₂²×C₆₇	1	4	yes	—
133	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
134	#391	—	2	1	no	—
135	#391	—	2	1	no	—
136	#391	C₂²×C₆₇	1	4	yes	—
137	#389	—	2	2	no	—
138	#389	C₂²×C₆₇	1	4	yes	—
139	#384	C₂²×C₆₇	1	2	yes	—
140	#382	—	2	1	no	—
141	#382	—	2	2	no	—
142	C₄³	C₂²×C₆₇	1	1	yes	characteristic
143	#340	C₂²×C₆₇	1	8	yes	—
144	C₂²×Q₈	—	2	1	no	—
145	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
146	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
147	C₄⋊Q₈	—	2	4	no	—
148	C₄⋊Q₈	—	2	4	no	—
149	#125	—	4	4	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	4	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
157	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
158	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
159	C₂×C₄²	D₄×C₆₇	1	2	yes	—
160	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
161	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
162	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
163	#103	—	2	16	no	—
164	#97	—	2	8	no	—
165	C₄×C₈	—	2	4	no	—
166	#95	—	2	4	no	—
167	#95	—	2	4	no	—
168	#95	—	2	4	no	—
169	C₂×Q₈	—	2	8	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
178	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
179	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
181	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	#41×C₆₇	1	8	yes	—
185	C₄⋊C₄	—	2	8	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	8	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	4	4	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	4	no	—
196	C₄²	—	2	4	no	—
197	C₄²	#41×C₆₇	1	4	yes	—
198	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
199	Q₈	—	4	8	no	—
200	Q₈	—	8	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	#124×C₆₇	1	2	yes	—
216	C₂×C₄	#124×C₆₇	1	2	yes	—
217	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
218	C₂×C₄	#137×C₆₇	1	2	yes	—
219	C₂×C₄	#137×C₆₇	1	4	yes	—
220	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
221	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
222	C₈	—	2	8	no	—
223	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
224	C₂²	#532×C₆₇	1	2	yes	—
225	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
226	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
227	C₂²	C₂×#137×C₆₇	1	2	yes	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	8	2	no	—
231	C₄	—	8	2	no	—
232	C₄	—	8	4	no	—
233	C₄	#489×C₆₇	1	4	yes	—
234	C₄	#491×C₆₇	1	4	yes	—
235	C₄	#529×C₆₇	1	4	yes	—
236	C₂	C₆₇×#2861	1	1	yes	characteristic
237	C₂	C₂×#489×C₆₇	1	1	yes	characteristic
238	C₂	C₂×#491×C₆₇	1	1	yes	characteristic
239	C₂	C₆₇×#3182	1	2	yes	—
240	C₂	C₆₇×#3185	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
115. #33287	C₆₇	direct (N × H)
127. C₆₇	#33287	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2862	2
C₆₇×#2859	2
C₆₇×#1990	2
C₆₇×#1956	2
C₆₇×#1935	2
C₄×#97×C₆₇	2
#33287	67

Subgroups — C67×#33287 #30712538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33287 #30712538