Subgroups — C₆₇×#9835 #30689086

Order 17152.

526 conjugacy classes of subgroups.

The subgroup lattice has 214 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×#1990	C₂	1	2	yes	maximal
3	C₆₇×#1990	C₂	1	2	yes	maximal
4	C₆₇×#1834	C₂	1	1	yes	maximal, characteristic
5	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₆₇	—	2	2	no	—
7	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
8	#494×C₆₇	—	2	2	no	—
9	#494×C₆₇	C₄	1	4	yes	complemented
10	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	#384×C₆₇	C₂²	1	2	yes	—
14	#340×C₆₇	—	2	4	no	—
15	#340×C₆₇	—	2	4	no	—
16	#340×C₆₇	—	2	2	no	—
17	#322×C₆₇	—	2	2	no	—
18	C₂²×Q₈×C₆₇	—	2	2	no	—
19	C₂×Q₁₆×C₆₇	—	2	8	no	—
20	C₂×Q₁₆×C₆₇	—	4	8	no	—
21	C₂²×C₈×C₆₇	D₄	1	2	yes	—
22	C₂²×C₈×C₆₇	D₄	1	2	yes	—
23	C₂²×C₈×C₆₇	D₄	1	2	yes	—
24	C₄⋊Q₈×C₆₇	—	2	2	no	—
25	C₄⋊Q₈×C₆₇	—	2	2	no	—
26	C₄⋊Q₈×C₆₇	—	2	2	no	—
27	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
28	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
33	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₆₇	—	2	1	no	—
35	C₄×C₈×C₆₇	—	2	1	no	—
36	C₄×C₈×C₆₇	—	2	1	no	—
37	C₄×C₈×C₆₇	—	2	1	no	—
38	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₆₇	D₄	1	2	yes	—
40	C₂×Q₈×C₆₇	—	2	4	no	—
41	C₂×Q₈×C₆₇	—	4	2	no	—
42	C₂×Q₈×C₆₇	—	4	8	no	—
43	C₂²×C₄×C₆₇	—	4	2	no	—
44	C₂²×C₄×C₆₇	—	4	2	no	—
45	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	Q₁₆×C₆₇	—	4	16	no	—
49	C₂×C₈×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	2	2	no	—
51	C₂×C₈×C₆₇	—	2	4	no	—
52	C₂×C₈×C₆₇	—	2	2	no	—
53	C₂×C₈×C₆₇	—	2	4	no	—
54	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
55	C₂×C₈×C₆₇	Q₁₆	1	4	yes	complemented
56	C₄⋊C₄×C₆₇	—	2	4	no	—
57	C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₄⋊C₄×C₆₇	—	4	2	no	—
59	C₄⋊C₄×C₆₇	—	4	2	no	—
60	C₄⋊C₄×C₆₇	—	4	4	no	—
61	C₄⋊C₄×C₆₇	—	4	4	no	—
62	C₄⋊C₄×C₆₇	—	4	4	no	—
63	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₆₇	#41	1	1	yes	characteristic
66	C₄²×C₆₇	#41	1	1	yes	characteristic
67	C₂³×C₆₇	#127	1	1	yes	characteristic
68	Q₈×C₆₇	—	4	8	no	—
69	Q₈×C₆₇	—	8	4	no	—
70	C₂×C₄×C₆₇	—	4	4	no	—
71	C₂×C₄×C₆₇	—	8	4	no	—
72	C₂×C₄×C₆₇	—	8	2	no	—
73	C₂×C₄×C₆₇	—	8	2	no	—
74	C₂×C₄×C₆₇	—	8	2	no	—
75	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₆₇	#135	1	2	yes	—
83	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
84	C₂×C₄×C₆₇	#137	1	4	yes	—
85	C₈×C₆₇	—	2	4	no	—
86	C₈×C₆₇	—	2	4	no	—
87	C₈×C₆₇	—	4	2	no	—
88	C₂²×C₆₇	#390	1	1	yes	characteristic
89	C₂²×C₆₇	#494	1	1	yes	characteristic
90	C₂²×C₆₇	#495	1	1	yes	characteristic
91	C₂²×C₆₇	#496	1	2	yes	—
92	C₂²×C₆₇	#497	1	2	yes	—
93	C₄×C₆₇	—	8	4	no	—
94	C₄×C₆₇	—	16	2	no	—
95	C₄×C₆₇	#418	1	2	yes	—
96	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
97	C₄×C₆₇	#441	1	2	yes	—
98	C₄×C₆₇	#467	1	2	yes	—
99	C₄×C₆₇	#470	1	2	yes	—
100	C₄×C₆₇	#490	1	2	yes	—
101	#9835	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂×C₆₇	#1720	1	2	yes	—
103	C₂×C₆₇	#1723	1	2	yes	—
104	C₂×C₆₇	#1961	1	1	yes	characteristic
105	C₂×C₆₇	#1969	1	1	yes	characteristic
106	C₂×C₆₇	#1971	1	1	yes	characteristic
107	C₂×#494	C₂×C₆₇	1	1	yes	characteristic
108	#1990	C₂×C₆₇	1	2	yes	—
109	#1990	C₂×C₆₇	1	2	yes	—
110	#1834	C₂×C₆₇	1	1	yes	characteristic
111	C₂×#322	C₂×C₆₇	1	1	yes	characteristic
112	C₆₇	#9835	1	1	yes	characteristic, direct product factor, Sylow
113	C₂²×Q₁₆	—	2	2	no	—
114	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
115	#494	—	2	2	no	—
116	#494	C₄×C₆₇	1	4	yes	complemented
117	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
120	#384	C₂²×C₆₇	1	2	yes	—
121	#340	—	2	2	no	—
122	#340	—	2	4	no	—
123	#340	—	2	4	no	—
124	#322	—	2	2	no	—
125	C₂²×Q₈	—	2	2	no	—
126	C₂×Q₁₆	—	2	8	no	—
127	C₂×Q₁₆	—	4	8	no	—
128	C₂²×C₈	D₄×C₆₇	1	2	yes	—
129	C₂²×C₈	D₄×C₆₇	1	2	yes	—
130	C₂²×C₈	D₄×C₆₇	1	2	yes	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
146	C₄×C₈	D₄×C₆₇	1	2	yes	—
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	4	8	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
155	Q₁₆	—	4	16	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
162	C₂×C₈	Q₁₆×C₆₇	1	4	yes	complemented
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₄⋊C₄	—	4	4	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
171	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
172	C₄²	#41×C₆₇	1	1	yes	characteristic
173	C₄²	#41×C₆₇	1	1	yes	characteristic
174	C₂³	#127×C₆₇	1	1	yes	characteristic
175	Q₈	—	4	8	no	—
176	Q₈	—	8	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	8	4	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
183	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
184	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
185	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
186	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
187	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
188	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
189	C₂×C₄	#135×C₆₇	1	2	yes	—
190	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
191	C₂×C₄	#137×C₆₇	1	4	yes	—
192	C₈	—	2	4	no	—
193	C₈	—	2	4	no	—
194	C₈	—	4	2	no	—
195	C₂²	#390×C₆₇	1	1	yes	characteristic
196	C₂²	#494×C₆₇	1	1	yes	characteristic
197	C₂²	#495×C₆₇	1	1	yes	characteristic
198	C₂²	#496×C₆₇	1	2	yes	—
199	C₂²	#497×C₆₇	1	2	yes	—
200	C₄	—	8	4	no	—
201	C₄	—	16	2	no	—
202	C₄	#418×C₆₇	1	2	yes	—
203	C₄	C₄×Q₁₆×C₆₇	1	2	yes	—
204	C₄	#441×C₆₇	1	2	yes	—
205	C₄	#467×C₆₇	1	2	yes	—
206	C₄	#470×C₆₇	1	2	yes	—
207	C₄	#490×C₆₇	1	2	yes	—
208	C₂	C₆₇×#1720	1	2	yes	—
209	C₂	C₆₇×#1723	1	2	yes	—
210	C₂	C₆₇×#1961	1	1	yes	characteristic
211	C₂	C₆₇×#1969	1	1	yes	characteristic
212	C₂	C₆₇×#1971	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	#127
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
9. #494×C₆₇	C₄	semidirect
55. C₂×C₈×C₆₇	Q₁₆	semidirect
101. #9835	C₆₇	direct (N × H)
112. C₆₇	#9835	direct (N × H)
116. #494	C₄×C₆₇	semidirect
162. C₂×C₈	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×#1990	2
C₆₇×#1834	2
C₂×#322×C₆₇	2
#9835	67

Subgroups — C67×#9835 #30689086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#9835 #30689086