Subgroups — C₆₇×#33294 #30712545

Order 17152.

750 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2868	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2859	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1992	C₂	1	4	yes	maximal
4	C₆₇×#1990	C₂	1	4	yes	maximal
5	C₆₇×#1836	C₂	1	4	yes	maximal
6	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
8	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
9	C₂×#125×C₆₇	C₂²	1	2	yes	—
10	C₂×#97×C₆₇	C₂²	1	2	yes	—
11	C₂×#97×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
13	#389×C₆₇	—	2	2	no	—
14	#389×C₆₇	—	2	2	no	—
15	#384×C₆₇	—	2	2	no	—
16	#384×C₆₇	C₂²	1	2	yes	—
17	#384×C₆₇	C₂²	1	2	yes	—
18	#384×C₆₇	C₂²	1	2	yes	—
19	#382×C₆₇	C₂²	1	2	yes	—
20	C₄³×C₆₇	C₂²	1	1	yes	characteristic
21	#340×C₆₇	C₂²	1	16	yes	—
22	C₂²×Q₈×C₆₇	—	2	2	no	—
23	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
24	C₄⋊Q₈×C₆₇	—	2	8	no	—
25	C₄⋊Q₈×C₆₇	—	4	8	no	—
26	#125×C₆₇	—	2	8	no	—
27	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
28	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	4	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×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
34	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
35	C₂×C₄²×C₆₇	D₄	1	2	yes	—
36	C₂×C₄²×C₆₇	D₄	1	2	yes	—
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	#97×C₆₇	—	2	4	no	—
41	#97×C₆₇	Q₈	1	8	yes	complemented
42	C₄×C₈×C₆₇	—	2	4	no	—
43	#95×C₆₇	—	2	4	no	—
44	C₂×Q₈×C₆₇	—	4	4	no	—
45	C₂×Q₈×C₆₇	—	4	8	no	—
46	C₂²×C₄×C₆₇	—	2	4	no	—
47	C₂²×C₄×C₆₇	—	2	4	no	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
53	C₂×C₈×C₆₇	—	2	4	no	—
54	C₂×C₈×C₆₇	—	2	4	no	—
55	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
56	C₄⋊C₄×C₆₇	—	2	8	no	—
57	C₄⋊C₄×C₆₇	—	2	8	no	—
58	C₄⋊C₄×C₆₇	—	4	8	no	—
59	C₄⋊C₄×C₆₇	—	4	8	no	—
60	C₄⋊C₄×C₆₇	—	4	8	no	—
61	C₄⋊C₄×C₆₇	—	4	4	no	—
62	C₄⋊C₄×C₆₇	—	4	4	no	—
63	C₄⋊C₄×C₆₇	—	4	4	no	—
64	C₄²×C₆₇	—	2	4	no	—
65	C₄²×C₆₇	—	2	4	no	—
66	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
67	C₄²×C₆₇	#41	1	4	yes	—
68	C₄²×C₆₇	#41	1	4	yes	—
69	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
70	Q₈×C₆₇	—	8	8	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	—	2	2	no	—
74	C₂×C₄×C₆₇	—	2	2	no	—
75	C₂×C₄×C₆₇	—	4	4	no	—
76	C₂×C₄×C₆₇	—	4	4	no	—
77	C₂×C₄×C₆₇	—	4	8	no	—
78	C₂×C₄×C₆₇	—	4	4	no	—
79	C₂×C₄×C₆₇	—	4	4	no	—
80	C₂×C₄×C₆₇	#124	1	4	yes	—
81	C₂×C₄×C₆₇	#124	1	2	yes	—
82	C₂×C₄×C₆₇	#124	1	2	yes	—
83	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
84	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
85	C₂×C₄×C₆₇	#137	1	4	yes	—
86	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
87	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
88	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
89	C₈×C₆₇	—	2	8	no	—
90	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
91	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
92	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
94	C₂²×C₆₇	C₂×#137	1	2	yes	—
95	C₄×C₆₇	—	2	4	no	—
96	C₄×C₆₇	—	2	4	no	—
97	C₄×C₆₇	—	8	4	no	—
98	C₄×C₆₇	—	8	4	no	—
99	C₄×C₆₇	#490	1	4	yes	—
100	C₄×C₆₇	#491	1	4	yes	—
101	C₄×C₆₇	#533	1	4	yes	—
102	#33294	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂×C₆₇	#2862	1	1	yes	characteristic
104	C₂×C₆₇	C₂×#490	1	1	yes	characteristic
105	C₂×C₆₇	C₂×#491	1	1	yes	characteristic
106	C₂×C₆₇	#3141	1	2	yes	—
107	C₂×C₆₇	#3165	1	2	yes	—
108	#2868	C₂×C₆₇	1	1	yes	characteristic
109	#2859	C₂×C₆₇	1	1	yes	characteristic
110	#1992	C₂×C₆₇	1	4	yes	—
111	#1990	C₂×C₆₇	1	4	yes	—
112	#1836	C₂×C₆₇	1	4	yes	—
113	C₄×#97	C₂×C₆₇	1	1	yes	characteristic
114	C₆₇	#33294	1	1	yes	characteristic, direct product factor, Sylow
115	C₂×C₄⋊Q₈	—	2	2	no	—
116	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
117	C₂×#125	C₂²×C₆₇	1	2	yes	—
118	C₂×#97	C₂²×C₆₇	1	2	yes	—
119	C₂×#97	C₂²×C₆₇	1	2	yes	—
120	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
121	#389	—	2	2	no	—
122	#389	—	2	2	no	—
123	#384	—	2	2	no	—
124	#384	C₂²×C₆₇	1	2	yes	—
125	#384	C₂²×C₆₇	1	2	yes	—
126	#384	C₂²×C₆₇	1	2	yes	—
127	#382	C₂²×C₆₇	1	2	yes	—
128	C₄³	C₂²×C₆₇	1	1	yes	characteristic
129	#340	C₂²×C₆₇	1	16	yes	—
130	C₂²×Q₈	—	2	2	no	—
131	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
132	C₄⋊Q₈	—	2	8	no	—
133	C₄⋊Q₈	—	4	8	no	—
134	#125	—	2	8	no	—
135	C₂×C₄⋊C₄	—	2	4	no	—
136	C₂×C₄⋊C₄	—	2	4	no	—
137	C₂×C₄⋊C₄	—	2	4	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
142	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
143	C₂×C₄²	D₄×C₆₇	1	2	yes	—
144	C₂×C₄²	D₄×C₆₇	1	2	yes	—
145	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
146	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
147	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
148	#97	—	2	4	no	—
149	#97	Q₈×C₆₇	1	8	yes	complemented
150	C₄×C₈	—	2	4	no	—
151	#95	—	2	4	no	—
152	C₂×Q₈	—	4	4	no	—
153	C₂×Q₈	—	4	8	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
157	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
158	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
159	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
160	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	C₂×Q₈×C₆₇	1	8	yes	—
164	C₄⋊C₄	—	2	8	no	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	—	4	8	no	—
167	C₄⋊C₄	—	4	8	no	—
168	C₄⋊C₄	—	4	8	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	4	4	no	—
171	C₄⋊C₄	—	4	4	no	—
172	C₄²	—	2	4	no	—
173	C₄²	—	2	4	no	—
174	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
175	C₄²	#41×C₆₇	1	4	yes	—
176	C₄²	#41×C₆₇	1	4	yes	—
177	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
178	Q₈	—	8	8	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	—	4	8	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	#124×C₆₇	1	4	yes	—
189	C₂×C₄	#124×C₆₇	1	2	yes	—
190	C₂×C₄	#124×C₆₇	1	2	yes	—
191	C₂×C₄	C₄⋊Q₈×C₆₇	1	4	yes	—
192	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
193	C₂×C₄	#137×C₆₇	1	4	yes	—
194	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
195	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
196	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
197	C₈	—	2	8	no	—
198	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
199	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
200	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
201	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
202	C₂²	C₂×#137×C₆₇	1	2	yes	—
203	C₄	—	2	4	no	—
204	C₄	—	2	4	no	—
205	C₄	—	8	4	no	—
206	C₄	—	8	4	no	—
207	C₄	#490×C₆₇	1	4	yes	—
208	C₄	#491×C₆₇	1	4	yes	—
209	C₄	#533×C₆₇	1	4	yes	—
210	C₂	C₆₇×#2862	1	1	yes	characteristic
211	C₂	C₂×#490×C₆₇	1	1	yes	characteristic
212	C₂	C₂×#491×C₆₇	1	1	yes	characteristic
213	C₂	C₆₇×#3141	1	2	yes	—
214	C₂	C₆₇×#3165	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
41. #97×C₆₇	Q₈	semidirect
102. #33294	C₆₇	direct (N × H)
114. C₆₇	#33294	direct (N × H)
149. #97	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2868	2
C₆₇×#2859	2
C₆₇×#1992	2
C₆₇×#1990	2
C₆₇×#1836	2
C₄×#97×C₆₇	2
#33294	67

Subgroups — C67×#33294 #30712545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33294 #30712545