Subgroups — C₆₇×#9842 #30689093

Order 17152.

626 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#492×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1990	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1957	C₂	1	2	yes	maximal
4	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×SD₁₆×C₆₇	—	2	2	no	—
7	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
9	#492×C₆₇	—	2	2	no	—
10	#492×C₆₇	C₄	1	4	yes	complemented
11	C₂×#107×C₆₇	—	2	2	no	—
12	C₂×#102×C₆₇	—	2	4	no	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#384×C₆₇	C₂²	1	2	yes	—
16	#340×C₆₇	—	2	4	no	—
17	#322×C₆₇	—	2	2	no	—
18	C₂²×Q₈×C₆₇	—	2	1	no	—
19	C₂²×D₄×C₆₇	—	2	1	no	—
20	C₂²×D₄×C₆₇	—	2	2	no	—
21	C₂×SD₁₆×C₆₇	—	2	8	no	—
22	C₂×SD₁₆×C₆₇	—	4	4	no	—
23	C₂×SD₁₆×C₆₇	—	4	4	no	—
24	C₂²×C₈×C₆₇	D₄	1	4	yes	—
25	C₂²×C₈×C₆₇	D₄	1	2	yes	—
26	C₄⋊Q₈×C₆₇	—	2	1	no	—
27	C₄⋊Q₈×C₆₇	—	2	2	no	—
28	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
29	#127×C₆₇	—	2	1	no	—
30	#127×C₆₇	—	2	2	no	—
31	#127×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
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×C₄²×C₆₇	C₂³	1	1	yes	characteristic
37	#107×C₆₇	—	4	4	no	—
38	#102×C₆₇	—	4	8	no	—
39	C₄×C₈×C₆₇	—	2	2	no	—
40	C₄×C₈×C₆₇	—	2	2	no	—
41	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₆₇	D₄	1	2	yes	—
43	C₂⁴×C₆₇	—	4	1	no	—
44	C₂×Q₈×C₆₇	—	2	2	no	—
45	C₂×Q₈×C₆₇	—	4	1	no	—
46	C₂×Q₈×C₆₇	—	4	2	no	—
47	C₂×Q₈×C₆₇	—	4	2	no	—
48	C₂×D₄×C₆₇	—	2	2	no	—
49	C₂×D₄×C₆₇	—	2	4	no	—
50	C₂×D₄×C₆₇	—	4	2	no	—
51	C₂×D₄×C₆₇	—	4	1	no	—
52	C₂×D₄×C₆₇	—	4	2	no	—
53	C₂×D₄×C₆₇	—	4	2	no	—
54	C₂×D₄×C₆₇	—	4	4	no	—
55	C₂×D₄×C₆₇	—	4	4	no	—
56	C₂²×C₄×C₆₇	—	4	1	no	—
57	C₂²×C₄×C₆₇	—	4	2	no	—
58	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
60	SD₁₆×C₆₇	—	4	16	no	—
61	C₂×C₈×C₆₇	—	2	4	no	—
62	C₂×C₈×C₆₇	—	2	4	no	—
63	C₂×C₈×C₆₇	—	2	4	no	—
64	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
65	C₂×C₈×C₆₇	D₈	1	4	yes	—
66	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
67	C₄⋊C₄×C₆₇	—	2	4	no	—
68	C₄⋊C₄×C₆₇	—	4	2	no	—
69	C₄⋊C₄×C₆₇	—	4	4	no	—
70	C₄⋊C₄×C₆₇	—	4	4	no	—
71	C₄⋊C₄×C₆₇	—	4	4	no	—
72	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₆₇	#41	1	2	yes	—
75	C₂³×C₆₇	—	4	2	no	—
76	C₂³×C₆₇	—	8	1	no	—
77	C₂³×C₆₇	—	8	1	no	—
78	C₂³×C₆₇	—	8	2	no	—
79	C₂³×C₆₇	—	8	2	no	—
80	C₂³×C₆₇	#127	1	1	yes	characteristic
81	Q₈×C₆₇	—	4	4	no	—
82	Q₈×C₆₇	—	8	2	no	—
83	D₄×C₆₇	—	4	4	no	—
84	D₄×C₆₇	—	4	8	no	—
85	D₄×C₆₇	—	8	4	no	—
86	D₄×C₆₇	—	8	2	no	—
87	C₂×C₄×C₆₇	—	4	2	no	—
88	C₂×C₄×C₆₇	—	8	1	no	—
89	C₂×C₄×C₆₇	—	8	1	no	—
90	C₂×C₄×C₆₇	—	8	2	no	—
91	C₂×C₄×C₆₇	—	8	2	no	—
92	C₂×C₄×C₆₇	—	8	2	no	—
93	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₆₇	#121	1	2	yes	—
96	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
99	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
100	C₂×C₄×C₆₇	#137	1	4	yes	—
101	C₈×C₆₇	—	4	4	no	—
102	C₈×C₆₇	#102	1	8	yes	complemented
103	C₂²×C₆₇	—	8	2	no	—
104	C₂²×C₆₇	—	8	4	no	—
105	C₂²×C₆₇	—	16	2	no	—
106	C₂²×C₆₇	—	16	1	no	—
107	C₂²×C₆₇	—	16	1	no	—
108	C₂²×C₆₇	#390	1	1	yes	characteristic
109	C₂²×C₆₇	#492	1	1	yes	characteristic
110	C₂²×C₆₇	#493	1	1	yes	characteristic
111	C₂²×C₆₇	#496	1	2	yes	—
112	C₂²×C₆₇	#497	1	2	yes	—
113	C₄×C₆₇	—	8	2	no	—
114	C₄×C₆₇	—	16	2	no	—
115	C₄×C₆₇	C₂×#102	1	2	yes	—
116	C₄×C₆₇	#440	1	4	yes	—
117	C₄×C₆₇	#468	1	4	yes	—
118	C₄×C₆₇	#486	1	2	yes	—
119	#9842	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂×C₆₇	—	16	2	no	—
121	C₂×C₆₇	#1714	1	2	yes	—
122	C₂×C₆₇	#1721	1	1	yes	characteristic
123	C₂×C₆₇	#1722	1	1	yes	characteristic
124	C₂×C₆₇	#1957	1	1	yes	characteristic
125	C₂×C₆₇	#1973	1	2	yes	—
126	C₂×#492	C₂×C₆₇	1	1	yes	characteristic
127	#1990	C₂×C₆₇	1	2	yes	complemented
128	#1957	C₂×C₆₇	1	2	yes	—
129	#1845	C₂×C₆₇	1	1	yes	characteristic, complemented
130	C₂×#322	C₂×C₆₇	1	1	yes	characteristic, complemented
131	C₆₇	#9842	1	1	yes	characteristic, direct product factor, Sylow
132	C₂²×SD₁₆	—	2	2	no	—
133	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
134	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
135	#492	—	2	2	no	—
136	#492	C₄×C₆₇	1	4	yes	complemented
137	C₂×#107	—	2	2	no	—
138	C₂×#102	—	2	4	no	—
139	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
140	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
141	#384	C₂²×C₆₇	1	2	yes	—
142	#340	—	2	4	no	—
143	#322	—	2	2	no	—
144	C₂²×Q₈	—	2	1	no	—
145	C₂²×D₄	—	2	1	no	—
146	C₂²×D₄	—	2	2	no	—
147	C₂×SD₁₆	—	2	8	no	—
148	C₂×SD₁₆	—	4	4	no	—
149	C₂×SD₁₆	—	4	4	no	—
150	C₂²×C₈	D₄×C₆₇	1	4	yes	—
151	C₂²×C₈	D₄×C₆₇	1	2	yes	—
152	C₄⋊Q₈	—	2	1	no	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
155	#127	—	2	1	no	—
156	#127	—	2	2	no	—
157	#127	C₂×C₄×C₆₇	1	2	yes	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
163	#107	—	4	4	no	—
164	#102	—	4	8	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
168	C₄×C₈	D₄×C₆₇	1	2	yes	—
169	C₂⁴	—	4	1	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	1	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	4	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂²×C₄	—	4	1	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
186	SD₁₆	—	4	16	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
191	C₂×C₈	D₈×C₆₇	1	4	yes	—
192	C₂×C₈	SD₁₆×C₆₇	1	4	yes	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	4	4	no	—
198	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
199	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
200	C₄²	#41×C₆₇	1	2	yes	—
201	C₂³	—	4	2	no	—
202	C₂³	—	8	1	no	—
203	C₂³	—	8	1	no	—
204	C₂³	—	8	2	no	—
205	C₂³	—	8	2	no	—
206	C₂³	#127×C₆₇	1	1	yes	characteristic
207	Q₈	—	4	4	no	—
208	Q₈	—	8	2	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	8	no	—
211	D₄	—	8	4	no	—
212	D₄	—	8	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
220	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
221	C₂×C₄	#121×C₆₇	1	2	yes	—
222	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
223	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
224	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
225	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
226	C₂×C₄	#137×C₆₇	1	4	yes	—
227	C₈	—	4	4	no	—
228	C₈	#102×C₆₇	1	8	yes	complemented
229	C₂²	—	8	2	no	—
230	C₂²	—	8	4	no	—
231	C₂²	—	16	2	no	—
232	C₂²	—	16	1	no	—
233	C₂²	—	16	1	no	—
234	C₂²	#390×C₆₇	1	1	yes	characteristic
235	C₂²	#492×C₆₇	1	1	yes	characteristic
236	C₂²	#493×C₆₇	1	1	yes	characteristic
237	C₂²	#496×C₆₇	1	2	yes	—
238	C₂²	#497×C₆₇	1	2	yes	—
239	C₄	—	8	2	no	—
240	C₄	—	16	2	no	—
241	C₄	C₂×#102×C₆₇	1	2	yes	—
242	C₄	#440×C₆₇	1	4	yes	—
243	C₄	#468×C₆₇	1	4	yes	—
244	C₄	#486×C₆₇	1	2	yes	—
245	C₂	—	16	2	no	—
246	C₂	C₆₇×#1714	1	2	yes	—
247	C₂	C₆₇×#1721	1	1	yes	characteristic
248	C₂	C₆₇×#1722	1	1	yes	characteristic
249	C₂	C₆₇×#1957	1	1	yes	characteristic
250	C₂	C₆₇×#1973	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	#127
commutator	C₄²	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
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
2. C₆₇×#1990	C₂	semidirect
4. C₆₇×#1845	C₂	semidirect
5. C₂×#322×C₆₇	C₂	semidirect
10. #492×C₆₇	C₄	semidirect
102. C₈×C₆₇	#102	semidirect
119. #9842	C₆₇	direct (N × H)
127. #1990	C₂×C₆₇	semidirect
129. #1845	C₂×C₆₇	semidirect
130. C₂×#322	C₂×C₆₇	semidirect
131. C₆₇	#9842	direct (N × H)
136. #492	C₄×C₆₇	semidirect
228. C₈	#102×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#492×C₆₇	2
C₆₇×#1990	2
C₆₇×#1957	2
C₆₇×#1845	2
C₂×#322×C₆₇	2
#9842	67

Subgroups — C67×#9842 #30689093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#9842 #30689093