Subgroups — C₆₇×#34638 #30713889

Order 17152.

1598 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2647	C₂	1	8	yes	maximal, complemented
2	C₆₇×#2646	C₂	1	8	yes	maximal
3	C₆₇×#2402	C₂	1	8	yes	maximal, complemented
4	C₆₇×#2296	C₂	1	4	yes	maximal
5	C₆₇×#2285	C₂	1	2	yes	maximal
6	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#117×C₆₇	C₂²	1	4	yes	—
8	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
9	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
10	#400×C₆₇	C₂²	1	32	yes	—
11	#398×C₆₇	C₂²	1	32	yes	—
12	#389×C₆₇	C₂²	1	16	yes	—
13	#385×C₆₇	C₂²	1	16	yes	—
14	#384×C₆₇	C₂²	1	4	yes	—
15	#382×C₆₇	C₂²	1	4	yes	—
16	#380×C₆₇	C₂²	1	8	yes	—
17	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	—
18	C₄×C₄⋊C₄×C₆₇	C₂²	1	16	yes	—
19	#376×C₆₇	C₂²	1	4	yes	—
20	#376×C₆₇	C₂²	1	4	yes	—
21	C₂×#95×C₆₇	C₂²	1	4	yes	—
22	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
23	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
24	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
25	#117×C₆₇	—	2	16	no	—
26	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
27	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
28	C₂×C₄⋊C₄×C₆₇	C₂³	1	32	yes	—
29	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
30	C₂×C₄²×C₆₇	—	2	8	no	—
31	C₂×C₄²×C₆₇	Q₈	1	8	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
33	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
35	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
37	#95×C₆₇	C₂³	1	16	yes	—
38	#95×C₆₇	C₂³	1	16	yes	—
39	#95×C₆₇	C₂³	1	32	yes	—
40	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	—	2	8	no	—
42	C₂²×C₄×C₆₇	—	4	4	no	—
43	C₂²×C₄×C₆₇	—	4	8	no	—
44	C₂²×C₄×C₆₇	C₂×Q₈	1	8	yes	—
45	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
46	C₂²×C₄×C₆₇	#41	1	8	yes	—
47	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
48	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
49	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
50	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
52	C₄⋊C₄×C₆₇	—	2	16	no	—
53	C₄⋊C₄×C₆₇	—	4	16	no	—
54	C₄⋊C₄×C₆₇	—	4	32	no	—
55	C₂²⋊C₄×C₆₇	—	2	16	no	—
56	C₄²×C₆₇	—	2	8	no	—
57	C₄²×C₆₇	—	2	8	no	—
58	C₄²×C₆₇	—	2	16	no	—
59	C₄²×C₆₇	C₂×Q₈	1	8	yes	—
60	C₄²×C₆₇	C₂×Q₈	1	8	yes	—
61	C₂³×C₆₇	—	2	4	no	—
62	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
63	C₂³×C₆₇	C₂×#41	1	4	yes	—
64	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
65	C₂×C₄×C₆₇	—	2	8	no	—
66	C₂×C₄×C₆₇	—	2	8	no	—
67	C₂×C₄×C₆₇	—	2	16	no	—
68	C₂×C₄×C₆₇	—	2	32	no	—
69	C₂×C₄×C₆₇	—	2	16	no	—
70	C₂×C₄×C₆₇	—	4	8	no	—
71	C₂×C₄×C₆₇	—	4	16	no	—
72	C₂×C₄×C₆₇	#125	1	16	yes	—
73	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
74	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
75	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
76	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
77	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
78	C₂²×C₆₇	—	2	8	no	—
79	C₂²×C₆₇	—	2	4	no	—
80	C₂²×C₆₇	C₂×#125	1	4	yes	—
81	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
82	C₂²×C₆₇	C₂²×#41	1	2	yes	—
83	C₂²×C₆₇	#585	1	4	yes	—
84	C₄×C₆₇	—	2	8	no	—
85	C₄×C₆₇	—	4	16	no	—
86	C₄×C₆₇	—	4	8	no	—
87	C₄×C₆₇	C₂×#125	1	4	yes	—
88	C₄×C₆₇	C₂×#125	1	4	yes	—
89	#34638	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂×C₆₇	—	2	4	no	—
91	C₂×C₆₇	C₂²×#125	1	1	yes	characteristic
92	C₂×C₆₇	#3455	1	4	yes	—
93	C₂×C₆₇	#3458	1	2	yes	—
94	#2647	C₂×C₆₇	1	8	yes	complemented
95	#2646	C₂×C₆₇	1	8	yes	—
96	#2402	C₂×C₆₇	1	8	yes	complemented
97	#2296	C₂×C₆₇	1	4	yes	—
98	#2285	C₂×C₆₇	1	2	yes	—
99	C₂×#376	C₂×C₆₇	1	1	yes	characteristic
100	C₆₇	#34638	1	1	yes	characteristic, direct product factor, Sylow
101	C₂×#117	C₂²×C₆₇	1	4	yes	—
102	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
103	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
104	#400	C₂²×C₆₇	1	32	yes	—
105	#398	C₂²×C₆₇	1	32	yes	—
106	#389	C₂²×C₆₇	1	16	yes	—
107	#385	C₂²×C₆₇	1	16	yes	—
108	#384	C₂²×C₆₇	1	4	yes	—
109	#382	C₂²×C₆₇	1	4	yes	—
110	#380	C₂²×C₆₇	1	8	yes	—
111	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
112	C₄×C₄⋊C₄	C₂²×C₆₇	1	16	yes	—
113	#376	C₂²×C₆₇	1	4	yes	—
114	#376	C₂²×C₆₇	1	4	yes	—
115	C₂×#95	C₂²×C₆₇	1	4	yes	—
116	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
117	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
118	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
119	#117	—	2	16	no	—
120	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
121	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
122	C₂×C₄⋊C₄	C₂³×C₆₇	1	32	yes	—
123	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
124	C₂×C₄²	—	2	8	no	—
125	C₂×C₄²	Q₈×C₆₇	1	8	yes	—
126	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
127	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
128	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
129	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
130	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
131	#95	C₂³×C₆₇	1	16	yes	—
132	#95	C₂³×C₆₇	1	16	yes	—
133	#95	C₂³×C₆₇	1	32	yes	—
134	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	—	4	4	no	—
137	C₂²×C₄	—	4	8	no	—
138	C₂²×C₄	C₂×Q₈×C₆₇	1	8	yes	—
139	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
140	C₂²×C₄	#41×C₆₇	1	8	yes	—
141	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
142	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
143	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
144	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
145	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
146	C₄⋊C₄	—	2	16	no	—
147	C₄⋊C₄	—	4	16	no	—
148	C₄⋊C₄	—	4	32	no	—
149	C₂²⋊C₄	—	2	16	no	—
150	C₄²	—	2	8	no	—
151	C₄²	—	2	8	no	—
152	C₄²	—	2	16	no	—
153	C₄²	C₂×Q₈×C₆₇	1	8	yes	—
154	C₄²	C₂×Q₈×C₆₇	1	8	yes	—
155	C₂³	—	2	4	no	—
156	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
157	C₂³	C₂×#41×C₆₇	1	4	yes	—
158	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	16	no	—
162	C₂×C₄	—	2	32	no	—
163	C₂×C₄	—	2	16	no	—
164	C₂×C₄	—	4	8	no	—
165	C₂×C₄	—	4	16	no	—
166	C₂×C₄	#125×C₆₇	1	16	yes	—
167	C₂×C₄	C₂²×Q₈×C₆₇	1	8	yes	—
168	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
169	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
170	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
171	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
172	C₂²	—	2	8	no	—
173	C₂²	—	2	4	no	—
174	C₂²	C₂×#125×C₆₇	1	4	yes	—
175	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
176	C₂²	C₂²×#41×C₆₇	1	2	yes	—
177	C₂²	#585×C₆₇	1	4	yes	—
178	C₄	—	2	8	no	—
179	C₄	—	4	8	no	—
180	C₄	—	4	16	no	—
181	C₄	C₂×#125×C₆₇	1	4	yes	—
182	C₄	C₂×#125×C₆₇	1	4	yes	—
183	C₂	—	2	4	no	—
184	C₂	C₂²×#125×C₆₇	1	1	yes	characteristic
185	C₂	C₆₇×#3455	1	4	yes	—
186	C₂	C₆₇×#3458	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂⁴×C₆₇	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2647	C₂	semidirect
3. C₆₇×#2402	C₂	semidirect
89. #34638	C₆₇	direct (N × H)
94. #2647	C₂×C₆₇	semidirect
96. #2402	C₂×C₆₇	semidirect
100. C₆₇	#34638	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×#2647	2
C₆₇×#2646	2
C₆₇×#2402	2
C₆₇×#2296	2
C₆₇×#2285	2
C₂×#376×C₆₇	2
#34638	67

Subgroups — C67×#34638 #30713889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34638 #30713889