Subgroups — C₆₇×#33224 #30712475

Order 17152.

1318 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	C₂×#486×C₆₇	C₂	1	4	yes	maximal, complemented
2	C₆₇×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2867	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1957	C₂	1	8	yes	maximal, complemented
5	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
7	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
8	C₂×#127×C₆₇	—	2	2	no	—
9	C₂×#127×C₆₇	—	2	2	no	—
10	C₂×#127×C₆₇	C₂²	1	2	yes	—
11	C₂×#127×C₆₇	C₂²	1	4	yes	—
12	#486×C₆₇	—	2	16	no	—
13	C₂×#102×C₆₇	C₂²	1	16	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
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×D₄×C₆₇	—	2	8	no	—
21	C₂²×D₄×C₆₇	—	2	2	no	—
22	C₂²×D₄×C₆₇	—	2	2	no	—
23	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
24	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
25	C₄⋊Q₈×C₆₇	—	2	8	no	—
26	C₄⋊Q₈×C₆₇	—	4	8	no	—
27	#127×C₆₇	—	2	8	no	—
28	#127×C₆₇	—	2	16	no	—
29	#127×C₆₇	—	4	8	no	—
30	#127×C₆₇	—	4	8	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
34	C₂×C₄²×C₆₇	D₄	1	2	yes	—
35	C₂×C₄²×C₆₇	D₄	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
38	#102×C₆₇	—	2	32	no	—
39	C₄×C₈×C₆₇	—	2	8	no	—
40	C₄×C₈×C₆₇	D₄	1	8	yes	complemented
41	C₂⁴×C₆₇	—	2	4	no	—
42	C₂×Q₈×C₆₇	—	4	4	no	—
43	C₂×Q₈×C₆₇	—	4	8	no	—
44	C₂×D₄×C₆₇	—	2	8	no	—
45	C₂×D₄×C₆₇	—	2	16	no	—
46	C₂×D₄×C₆₇	—	4	16	no	—
47	C₂×D₄×C₆₇	—	4	4	no	—
48	C₂×D₄×C₆₇	—	4	4	no	—
49	C₂×D₄×C₆₇	—	4	16	no	—
50	C₂×D₄×C₆₇	—	4	8	no	—
51	C₂×D₄×C₆₇	—	4	16	no	—
52	C₂×D₄×C₆₇	—	4	8	no	—
53	C₂²×C₄×C₆₇	—	2	4	no	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₆₇	—	2	8	no	—
59	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
60	C₄⋊C₄×C₆₇	—	2	8	no	—
61	C₄⋊C₄×C₆₇	—	4	4	no	—
62	C₄⋊C₄×C₆₇	—	4	16	no	—
63	C₄²×C₆₇	D₈	1	8	yes	—
64	C₄²×C₆₇	SD₁₆	1	8	yes	—
65	C₄²×C₆₇	C₂×D₄	1	4	yes	—
66	C₄²×C₆₇	#41	1	8	yes	—
67	C₂³×C₆₇	—	4	8	no	—
68	C₂³×C₆₇	—	4	4	no	—
69	C₂³×C₆₇	—	4	8	no	—
70	C₂³×C₆₇	—	4	8	no	—
71	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
72	Q₈×C₆₇	—	8	8	no	—
73	D₄×C₆₇	—	4	16	no	—
74	D₄×C₆₇	—	8	32	no	—
75	D₄×C₆₇	—	8	8	no	—
76	D₄×C₆₇	—	8	8	no	—
77	C₂×C₄×C₆₇	—	4	4	no	—
78	C₂×C₄×C₆₇	—	4	8	no	—
79	C₂×C₄×C₆₇	#124	1	8	yes	—
80	C₂×C₄×C₆₇	#127	1	2	yes	—
81	C₂×C₄×C₆₇	#127	1	2	yes	—
82	C₂×C₄×C₆₇	C₂×D₈	1	8	yes	—
83	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
84	C₂×C₄×C₆₇	C₂×SD₁₆	1	8	yes	—
85	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
86	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
87	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
88	C₈×C₆₇	—	2	8	no	—
89	C₂²×C₆₇	—	8	4	no	—
90	C₂²×C₆₇	—	8	16	no	—
91	C₂²×C₆₇	—	8	4	no	—
92	C₂²×C₆₇	—	8	4	no	—
93	C₂²×C₆₇	C₂×#124	1	2	yes	—
94	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
95	C₂²×C₆₇	C₂²×D₈	1	2	yes	—
96	C₂²×C₆₇	C₂²×SD₁₆	1	2	yes	—
97	C₄×C₆₇	—	8	4	no	—
98	C₄×C₆₇	#486	1	16	yes	complemented
99	C₄×C₆₇	#492	1	4	yes	—
100	C₄×C₆₇	#493	1	4	yes	—
101	C₄×C₆₇	#532	1	4	yes	—
102	#33224	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂×C₆₇	—	16	4	no	—
104	C₂×C₆₇	#2858	1	1	yes	characteristic
105	C₂×C₆₇	C₂×#486	1	4	yes	—
106	C₂×C₆₇	C₂×#492	1	1	yes	characteristic
107	C₂×C₆₇	C₂×#493	1	1	yes	characteristic
108	C₂×#486	C₂×C₆₇	1	4	yes	complemented
109	#2868	C₂×C₆₇	1	1	yes	characteristic, complemented
110	#2867	C₂×C₆₇	1	1	yes	characteristic
111	#1957	C₂×C₆₇	1	8	yes	complemented
112	C₄²×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
113	C₆₇	#33224	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×C₄⋊Q₈	—	2	2	no	—
115	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
116	C₂×#127	—	2	2	no	—
117	C₂×#127	—	2	2	no	—
118	C₂×#127	C₂²×C₆₇	1	2	yes	—
119	C₂×#127	C₂²×C₆₇	1	4	yes	—
120	#486	—	2	16	no	—
121	C₂×#102	C₂²×C₆₇	1	16	yes	—
122	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
123	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
124	#384	—	2	2	no	—
125	#384	C₂²×C₆₇	1	4	yes	—
126	C₄³	C₂²×C₆₇	1	1	yes	characteristic
127	C₂²×Q₈	—	2	2	no	—
128	C₂²×D₄	—	2	8	no	—
129	C₂²×D₄	—	2	2	no	—
130	C₂²×D₄	—	2	2	no	—
131	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
132	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
133	C₄⋊Q₈	—	2	8	no	—
134	C₄⋊Q₈	—	4	8	no	—
135	#127	—	2	8	no	—
136	#127	—	2	16	no	—
137	#127	—	4	8	no	—
138	#127	—	4	8	no	—
139	C₂×C₄⋊C₄	—	2	8	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
142	C₂×C₄²	D₄×C₆₇	1	2	yes	—
143	C₂×C₄²	D₄×C₆₇	1	2	yes	—
144	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
145	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
146	#102	—	2	32	no	—
147	C₄×C₈	—	2	8	no	—
148	C₄×C₈	D₄×C₆₇	1	8	yes	complemented
149	C₂⁴	—	2	4	no	—
150	C₂×Q₈	—	4	4	no	—
151	C₂×Q₈	—	4	8	no	—
152	C₂×D₄	—	2	8	no	—
153	C₂×D₄	—	2	16	no	—
154	C₂×D₄	—	4	16	no	—
155	C₂×D₄	—	4	4	no	—
156	C₂×D₄	—	4	4	no	—
157	C₂×D₄	—	4	16	no	—
158	C₂×D₄	—	4	8	no	—
159	C₂×D₄	—	4	16	no	—
160	C₂×D₄	—	4	8	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
163	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
165	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
166	C₂×C₈	—	2	8	no	—
167	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
168	C₄⋊C₄	—	2	8	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	4	16	no	—
171	C₄²	D₈×C₆₇	1	8	yes	—
172	C₄²	SD₁₆×C₆₇	1	8	yes	—
173	C₄²	C₂×D₄×C₆₇	1	4	yes	—
174	C₄²	#41×C₆₇	1	8	yes	—
175	C₂³	—	4	8	no	—
176	C₂³	—	4	4	no	—
177	C₂³	—	4	8	no	—
178	C₂³	—	4	8	no	—
179	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
180	Q₈	—	8	8	no	—
181	D₄	—	4	16	no	—
182	D₄	—	8	32	no	—
183	D₄	—	8	8	no	—
184	D₄	—	8	8	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	8	no	—
187	C₂×C₄	#124×C₆₇	1	8	yes	—
188	C₂×C₄	#127×C₆₇	1	2	yes	—
189	C₂×C₄	#127×C₆₇	1	2	yes	—
190	C₂×C₄	C₂×D₈×C₆₇	1	8	yes	—
191	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
192	C₂×C₄	C₂×SD₁₆×C₆₇	1	8	yes	—
193	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
194	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
195	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
196	C₈	—	2	8	no	—
197	C₂²	—	8	4	no	—
198	C₂²	—	8	16	no	—
199	C₂²	—	8	4	no	—
200	C₂²	—	8	4	no	—
201	C₂²	C₂×#124×C₆₇	1	2	yes	—
202	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
203	C₂²	C₂²×D₈×C₆₇	1	2	yes	—
204	C₂²	C₂²×SD₁₆×C₆₇	1	2	yes	—
205	C₄	—	8	4	no	—
206	C₄	#486×C₆₇	1	16	yes	complemented
207	C₄	#492×C₆₇	1	4	yes	—
208	C₄	#493×C₆₇	1	4	yes	—
209	C₄	#532×C₆₇	1	4	yes	—
210	C₂	—	16	4	no	—
211	C₂	C₆₇×#2858	1	1	yes	characteristic
212	C₂	C₂×#486×C₆₇	1	4	yes	—
213	C₂	C₂×#492×C₆₇	1	1	yes	characteristic
214	C₂	C₂×#493×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2867	C₂
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
1. C₂×#486×C₆₇	C₂	semidirect
2. C₆₇×#2868	C₂	semidirect
4. C₆₇×#1957	C₂	semidirect
5. C₄²×C₈×C₆₇	C₂	semidirect
40. C₄×C₈×C₆₇	D₄	semidirect
98. C₄×C₆₇	#486	semidirect
102. #33224	C₆₇	direct (N × H)
108. C₂×#486	C₂×C₆₇	semidirect
109. #2868	C₂×C₆₇	semidirect
111. #1957	C₂×C₆₇	semidirect
112. C₄²×C₈	C₂×C₆₇	semidirect
113. C₆₇	#33224	direct (N × H)
148. C₄×C₈	D₄×C₆₇	semidirect
206. C₄	#486×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#486×C₆₇	2
C₆₇×#2868	2
C₆₇×#2867	2
C₆₇×#1957	2
C₄²×C₈×C₆₇	2
#33224	67

Subgroups — C67×#33224 #30712475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33224 #30712475