Subgroups — C₆₇×#33241 #30712492

Order 17152.

1174 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#493×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#486×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2858	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1970	C₂	1	8	yes	maximal, complemented
6	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₆₇	C₂²	1	4	yes	—
8	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
9	C₂×#127×C₆₇	C₂²	1	2	yes	—
10	C₂×#124×C₆₇	—	2	4	no	—
11	#500×C₆₇	—	2	4	no	—
12	#493×C₆₇	—	2	4	no	—
13	#486×C₆₇	—	2	8	no	—
14	C₂×#107×C₆₇	C₂²	1	4	yes	—
15	C₂×#102×C₆₇	C₂²	1	8	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#390×C₆₇	—	2	4	no	—
20	#390×C₆₇	C₂²	1	8	yes	—
21	C₄³×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×Q₈×C₆₇	—	2	2	no	—
23	C₂²×D₄×C₆₇	—	2	2	no	—
24	C₂²×D₄×C₆₇	—	2	2	no	—
25	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
26	C₂×D₈×C₆₇	—	2	8	no	—
27	C₂×D₈×C₆₇	—	2	16	no	—
28	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
29	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
30	C₄⋊Q₈×C₆₇	—	2	4	no	—
31	C₄⋊Q₈×C₆₇	—	2	4	no	—
32	#127×C₆₇	—	2	4	no	—
33	#127×C₆₇	—	2	4	no	—
34	#124×C₆₇	—	4	16	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
37	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
38	C₂×C₄²×C₆₇	D₄	1	2	yes	—
39	C₂×C₄²×C₆₇	D₄	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	#107×C₆₇	—	2	8	no	—
43	#102×C₆₇	—	2	16	no	—
44	C₄×C₈×C₆₇	—	2	8	no	—
45	C₄×C₈×C₆₇	D₄	1	4	yes	—
46	C₄×C₈×C₆₇	D₄	1	4	yes	—
47	C₂⁴×C₆₇	—	2	4	no	—
48	C₂×Q₈×C₆₇	—	4	4	no	—
49	C₂×Q₈×C₆₇	—	4	8	no	—
50	C₂×D₄×C₆₇	—	2	4	no	—
51	C₂×D₄×C₆₇	—	2	4	no	—
52	C₂×D₄×C₆₇	—	2	16	no	—
53	C₂×D₄×C₆₇	—	4	4	no	—
54	C₂×D₄×C₆₇	—	4	4	no	—
55	C₂×D₄×C₆₇	—	4	8	no	—
56	C₂×D₄×C₆₇	—	4	8	no	—
57	C₂²×C₄×C₆₇	—	2	4	no	—
58	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
62	D₈×C₆₇	—	4	16	no	—
63	C₂×C₈×C₆₇	—	2	4	no	—
64	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
66	C₂×C₈×C₆₇	#41	1	8	yes	—
67	C₄⋊C₄×C₆₇	—	2	4	no	—
68	C₄⋊C₄×C₆₇	—	2	4	no	—
69	C₄⋊C₄×C₆₇	—	4	4	no	—
70	C₂²⋊C₄×C₆₇	—	4	32	no	—
71	C₄²×C₆₇	—	2	4	no	—
72	C₄²×C₆₇	—	2	4	no	—
73	C₄²×C₆₇	D₈	1	8	yes	complemented
74	C₄²×C₆₇	C₂×D₄	1	2	yes	—
75	C₄²×C₆₇	C₂×D₄	1	2	yes	—
76	C₂³×C₆₇	—	4	4	no	—
77	C₂³×C₆₇	—	4	8	no	—
78	C₂³×C₆₇	—	4	8	no	—
79	C₂³×C₆₇	—	4	8	no	—
80	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
81	Q₈×C₆₇	—	8	8	no	—
82	D₄×C₆₇	—	4	8	no	—
83	D₄×C₆₇	—	4	8	no	—
84	D₄×C₆₇	—	8	8	no	—
85	D₄×C₆₇	—	8	8	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	2	4	no	—
88	C₂×C₄×C₆₇	—	4	4	no	—
89	C₂×C₄×C₆₇	—	4	8	no	—
90	C₂×C₄×C₆₇	#127	1	2	yes	—
91	C₂×C₄×C₆₇	#127	1	2	yes	—
92	C₂×C₄×C₆₇	C₂×D₈	1	8	yes	—
93	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
94	C₂×C₄×C₆₇	#135	1	4	yes	—
95	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
98	C₈×C₆₇	—	2	4	no	—
99	C₈×C₆₇	#124	1	8	yes	complemented
100	C₂²×C₆₇	—	8	4	no	—
101	C₂²×C₆₇	—	8	4	no	—
102	C₂²×C₆₇	—	8	4	no	—
103	C₂²×C₆₇	—	8	16	no	—
104	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
105	C₂²×C₆₇	#532	1	2	yes	—
106	C₂²×C₆₇	C₂²×D₈	1	2	yes	—
107	C₂²×C₆₇	C₂×#135	1	2	yes	—
108	C₄×C₆₇	—	2	8	no	—
109	C₄×C₆₇	—	8	4	no	—
110	C₄×C₆₇	#493	1	4	yes	—
111	C₄×C₆₇	#495	1	4	yes	—
112	C₄×C₆₇	C₂×#124	1	2	yes	—
113	C₄×C₆₇	#532	1	2	yes	—
114	#33241	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×C₆₇	—	16	4	no	—
116	C₂×C₆₇	#2858	1	1	yes	characteristic
117	C₂×C₆₇	C₂×#493	1	1	yes	characteristic
118	C₂×C₆₇	C₂×#495	1	1	yes	characteristic
119	C₂×C₆₇	#3169	1	4	yes	—
120	C₂×#500	C₂×C₆₇	1	1	yes	characteristic, complemented
121	C₂×#493	C₂×C₆₇	1	1	yes	characteristic
122	C₂×#486	C₂×C₆₇	1	2	yes	complemented
123	#2858	C₂×C₆₇	1	2	yes	complemented
124	#1970	C₂×C₆₇	1	8	yes	complemented
125	C₄²×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
126	C₆₇	#33241	1	1	yes	characteristic, direct product factor, Sylow
127	C₂²×D₈	C₂²×C₆₇	1	4	yes	—
128	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
129	C₂×#127	C₂²×C₆₇	1	2	yes	—
130	C₂×#124	—	2	4	no	—
131	#500	—	2	4	no	—
132	#493	—	2	4	no	—
133	#486	—	2	8	no	—
134	C₂×#107	C₂²×C₆₇	1	4	yes	—
135	C₂×#102	C₂²×C₆₇	1	8	yes	—
136	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
137	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
139	#390	—	2	4	no	—
140	#390	C₂²×C₆₇	1	8	yes	—
141	C₄³	C₂²×C₆₇	1	1	yes	characteristic
142	C₂²×Q₈	—	2	2	no	—
143	C₂²×D₄	—	2	2	no	—
144	C₂²×D₄	—	2	2	no	—
145	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
146	C₂×D₈	—	2	8	no	—
147	C₂×D₈	—	2	16	no	—
148	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
149	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
150	C₄⋊Q₈	—	2	4	no	—
151	C₄⋊Q₈	—	2	4	no	—
152	#127	—	2	4	no	—
153	#127	—	2	4	no	—
154	#124	—	4	16	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
157	C₂×C₂²⋊C₄	—	2	16	no	—
158	C₂×C₄²	D₄×C₆₇	1	2	yes	—
159	C₂×C₄²	D₄×C₆₇	1	2	yes	—
160	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
161	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
162	#107	—	2	8	no	—
163	#102	—	2	16	no	—
164	C₄×C₈	—	2	8	no	—
165	C₄×C₈	D₄×C₆₇	1	4	yes	—
166	C₄×C₈	D₄×C₆₇	1	4	yes	—
167	C₂⁴	—	2	4	no	—
168	C₂×Q₈	—	4	4	no	—
169	C₂×Q₈	—	4	8	no	—
170	C₂×D₄	—	2	4	no	—
171	C₂×D₄	—	2	4	no	—
172	C₂×D₄	—	2	16	no	—
173	C₂×D₄	—	4	4	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂×D₄	—	4	8	no	—
176	C₂×D₄	—	4	8	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
179	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
181	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
182	D₈	—	4	16	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
185	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
186	C₂×C₈	#41×C₆₇	1	8	yes	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₂²⋊C₄	—	4	32	no	—
191	C₄²	—	2	4	no	—
192	C₄²	—	2	4	no	—
193	C₄²	D₈×C₆₇	1	8	yes	complemented
194	C₄²	C₂×D₄×C₆₇	1	2	yes	—
195	C₄²	C₂×D₄×C₆₇	1	2	yes	—
196	C₂³	—	4	4	no	—
197	C₂³	—	4	8	no	—
198	C₂³	—	4	8	no	—
199	C₂³	—	4	8	no	—
200	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
201	Q₈	—	8	8	no	—
202	D₄	—	4	8	no	—
203	D₄	—	4	8	no	—
204	D₄	—	8	8	no	—
205	D₄	—	8	8	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	8	no	—
210	C₂×C₄	#127×C₆₇	1	2	yes	—
211	C₂×C₄	#127×C₆₇	1	2	yes	—
212	C₂×C₄	C₂×D₈×C₆₇	1	8	yes	—
213	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
214	C₂×C₄	#135×C₆₇	1	4	yes	—
215	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
216	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
217	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
218	C₈	—	2	4	no	—
219	C₈	#124×C₆₇	1	8	yes	complemented
220	C₂²	—	8	4	no	—
221	C₂²	—	8	4	no	—
222	C₂²	—	8	4	no	—
223	C₂²	—	8	16	no	—
224	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
225	C₂²	#532×C₆₇	1	2	yes	—
226	C₂²	C₂²×D₈×C₆₇	1	2	yes	—
227	C₂²	C₂×#135×C₆₇	1	2	yes	—
228	C₄	—	2	8	no	—
229	C₄	—	8	4	no	—
230	C₄	#493×C₆₇	1	4	yes	—
231	C₄	#495×C₆₇	1	4	yes	—
232	C₄	C₂×#124×C₆₇	1	2	yes	—
233	C₄	#532×C₆₇	1	2	yes	—
234	C₂	—	16	4	no	—
235	C₂	C₆₇×#2858	1	1	yes	characteristic
236	C₂	C₂×#493×C₆₇	1	1	yes	characteristic
237	C₂	C₂×#495×C₆₇	1	1	yes	characteristic
238	C₂	C₆₇×#3169	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#493×C₆₇	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×#500×C₆₇	C₂	semidirect
3. C₂×#486×C₆₇	C₂	semidirect
4. C₆₇×#2858	C₂	semidirect
5. C₆₇×#1970	C₂	semidirect
6. C₄²×C₈×C₆₇	C₂	semidirect
73. C₄²×C₆₇	D₈	semidirect
99. C₈×C₆₇	#124	semidirect
114. #33241	C₆₇	direct (N × H)
120. C₂×#500	C₂×C₆₇	semidirect
122. C₂×#486	C₂×C₆₇	semidirect
123. #2858	C₂×C₆₇	semidirect
124. #1970	C₂×C₆₇	semidirect
125. C₄²×C₈	C₂×C₆₇	semidirect
126. C₆₇	#33241	direct (N × H)
193. C₄²	D₈×C₆₇	semidirect
219. C₈	#124×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₆₇	2
C₂×#493×C₆₇	2
C₂×#486×C₆₇	2
C₆₇×#2858	2
C₆₇×#1970	2
C₄²×C₈×C₆₇	2
#33241	67

Subgroups — C67×#33241 #30712492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33241 #30712492