Subgroups — C₆₇×#33225 #30712476

Order 17152.

1006 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#487×C₆₇	C₂	1	4	yes	maximal
2	C₆₇×#2868	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2868	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1956	C₂	1	8	yes	maximal
5	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
7	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
10	#487×C₆₇	—	2	16	no	—
11	C₂×#103×C₆₇	C₂²	1	16	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
14	#384×C₆₇	—	2	2	no	—
15	#384×C₆₇	—	2	2	no	—
16	#384×C₆₇	—	2	2	no	—
17	#384×C₆₇	C₂²	1	4	yes	—
18	C₄³×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₆₇	—	2	2	no	—
20	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
21	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
22	C₄⋊Q₈×C₆₇	—	2	8	no	—
23	C₄⋊Q₈×C₆₇	—	2	8	no	—
24	C₄⋊Q₈×C₆₇	—	2	16	no	—
25	C₄⋊Q₈×C₆₇	—	4	8	no	—
26	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
27	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
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	C₂³	1	4	yes	—
32	C₂×C₄²×C₆₇	D₄	1	2	yes	—
33	C₂×C₄²×C₆₇	D₄	1	2	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
36	#103×C₆₇	—	2	32	no	—
37	C₄×C₈×C₆₇	—	2	8	no	—
38	C₄×C₈×C₆₇	D₄	1	8	yes	—
39	C₂×Q₈×C₆₇	—	2	8	no	—
40	C₂×Q₈×C₆₇	—	2	16	no	—
41	C₂×Q₈×C₆₇	—	4	4	no	—
42	C₂×Q₈×C₆₇	—	4	8	no	—
43	C₂²×C₄×C₆₇	—	2	4	no	—
44	C₂²×C₄×C₆₇	—	2	4	no	—
45	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
49	C₂×C₈×C₆₇	—	2	8	no	—
50	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
51	C₄⋊C₄×C₆₇	—	2	8	no	—
52	C₄⋊C₄×C₆₇	—	4	16	no	—
53	C₄⋊C₄×C₆₇	—	4	4	no	—
54	C₄⋊C₄×C₆₇	—	4	4	no	—
55	C₄⋊C₄×C₆₇	—	4	4	no	—
56	C₄⋊C₄×C₆₇	—	4	16	no	—
57	C₄²×C₆₇	SD₁₆	1	8	yes	—
58	C₄²×C₆₇	Q₁₆	1	8	yes	—
59	C₄²×C₆₇	C₂×D₄	1	4	yes	—
60	C₄²×C₆₇	#41	1	8	yes	—
61	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
62	Q₈×C₆₇	—	4	16	no	—
63	Q₈×C₆₇	—	8	8	no	—
64	C₂×C₄×C₆₇	—	4	8	no	—
65	C₂×C₄×C₆₇	—	4	4	no	—
66	C₂×C₄×C₆₇	—	4	4	no	—
67	C₂×C₄×C₆₇	—	4	8	no	—
68	C₂×C₄×C₆₇	#124	1	8	yes	—
69	C₂×C₄×C₆₇	#127	1	2	yes	—
70	C₂×C₄×C₆₇	#127	1	2	yes	—
71	C₂×C₄×C₆₇	C₂×SD₁₆	1	8	yes	—
72	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
73	C₂×C₄×C₆₇	C₂×Q₁₆	1	8	yes	—
74	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
75	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
76	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
77	C₈×C₆₇	—	2	8	no	—
78	C₂²×C₆₇	C₂×#124	1	2	yes	—
79	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
80	C₂²×C₆₇	C₂²×SD₁₆	1	2	yes	—
81	C₂²×C₆₇	C₂²×Q₁₆	1	2	yes	—
82	C₄×C₆₇	—	8	4	no	—
83	C₄×C₆₇	—	8	4	no	—
84	C₄×C₆₇	#487	1	16	yes	complemented
85	C₄×C₆₇	#492	1	4	yes	—
86	C₄×C₆₇	#494	1	4	yes	—
87	C₄×C₆₇	#532	1	4	yes	—
88	#33225	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂×C₆₇	#2858	1	1	yes	characteristic
90	C₂×C₆₇	C₂×#487	1	4	yes	—
91	C₂×C₆₇	C₂×#492	1	1	yes	characteristic
92	C₂×C₆₇	C₂×#494	1	1	yes	characteristic
93	C₂×#487	C₂×C₆₇	1	4	yes	—
94	#2868	C₂×C₆₇	1	1	yes	characteristic
95	#2868	C₂×C₆₇	1	1	yes	characteristic
96	#1956	C₂×C₆₇	1	8	yes	—
97	C₄²×C₈	C₂×C₆₇	1	1	yes	characteristic
98	C₆₇	#33225	1	1	yes	characteristic, direct product factor, Sylow
99	C₂×C₄⋊Q₈	—	2	2	no	—
100	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
101	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
102	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
103	#487	—	2	16	no	—
104	C₂×#103	C₂²×C₆₇	1	16	yes	—
105	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
106	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
107	#384	—	2	2	no	—
108	#384	—	2	2	no	—
109	#384	—	2	2	no	—
110	#384	C₂²×C₆₇	1	4	yes	—
111	C₄³	C₂²×C₆₇	1	1	yes	characteristic
112	C₂²×Q₈	—	2	2	no	—
113	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
114	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
115	C₄⋊Q₈	—	2	8	no	—
116	C₄⋊Q₈	—	2	8	no	—
117	C₄⋊Q₈	—	2	16	no	—
118	C₄⋊Q₈	—	4	8	no	—
119	C₂×C₄⋊C₄	—	2	8	no	—
120	C₂×C₄⋊C₄	—	2	8	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
125	C₂×C₄²	D₄×C₆₇	1	2	yes	—
126	C₂×C₄²	D₄×C₆₇	1	2	yes	—
127	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
128	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
129	#103	—	2	32	no	—
130	C₄×C₈	—	2	8	no	—
131	C₄×C₈	D₄×C₆₇	1	8	yes	—
132	C₂×Q₈	—	2	8	no	—
133	C₂×Q₈	—	2	16	no	—
134	C₂×Q₈	—	4	4	no	—
135	C₂×Q₈	—	4	8	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
139	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
141	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
142	C₂×C₈	—	2	8	no	—
143	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
144	C₄⋊C₄	—	2	8	no	—
145	C₄⋊C₄	—	4	16	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	16	no	—
150	C₄²	SD₁₆×C₆₇	1	8	yes	—
151	C₄²	Q₁₆×C₆₇	1	8	yes	—
152	C₄²	C₂×D₄×C₆₇	1	4	yes	—
153	C₄²	#41×C₆₇	1	8	yes	—
154	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
155	Q₈	—	4	16	no	—
156	Q₈	—	8	8	no	—
157	C₂×C₄	—	4	8	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	4	8	no	—
161	C₂×C₄	#124×C₆₇	1	8	yes	—
162	C₂×C₄	#127×C₆₇	1	2	yes	—
163	C₂×C₄	#127×C₆₇	1	2	yes	—
164	C₂×C₄	C₂×SD₁₆×C₆₇	1	8	yes	—
165	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
166	C₂×C₄	C₂×Q₁₆×C₆₇	1	8	yes	—
167	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	—
168	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
169	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
170	C₈	—	2	8	no	—
171	C₂²	C₂×#124×C₆₇	1	2	yes	—
172	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
173	C₂²	C₂²×SD₁₆×C₆₇	1	2	yes	—
174	C₂²	C₂²×Q₁₆×C₆₇	1	2	yes	—
175	C₄	—	8	4	no	—
176	C₄	—	8	4	no	—
177	C₄	#487×C₆₇	1	16	yes	complemented
178	C₄	#492×C₆₇	1	4	yes	—
179	C₄	#494×C₆₇	1	4	yes	—
180	C₄	#532×C₆₇	1	4	yes	—
181	C₂	C₆₇×#2858	1	1	yes	characteristic
182	C₂	C₂×#487×C₆₇	1	4	yes	—
183	C₂	C₂×#492×C₆₇	1	1	yes	characteristic
184	C₂	C₂×#494×C₆₇	1	1	yes	characteristic

	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
84. C₄×C₆₇	#487	semidirect
88. #33225	C₆₇	direct (N × H)
98. C₆₇	#33225	direct (N × H)
177. C₄	#487×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#487×C₆₇	2
C₆₇×#2868	2
C₆₇×#1956	2
C₄²×C₈×C₆₇	2
#33225	67

Subgroups — C67×#33225 #30712476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33225 #30712476