Subgroups — C₆₇×#33248 #30712499

Order 17152.

1006 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#493×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#490×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2641	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1972	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1970	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
9	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
10	C₂×#127×C₆₇	C₂²	1	2	yes	—
11	C₂×#125×C₆₇	C₂²	1	2	yes	—
12	#499×C₆₇	—	2	4	no	—
13	#493×C₆₇	—	2	4	no	—
14	#490×C₆₇	—	2	8	no	—
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	C₂×#106×C₆₇	C₂²	1	2	yes	—
17	C₂×#102×C₆₇	C₂²	1	8	yes	—
18	C₂×#97×C₆₇	C₂²	1	2	yes	—
19	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#394×C₆₇	—	2	8	no	—
23	#390×C₆₇	C₂²	1	8	yes	—
24	#376×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×D₄×C₆₇	—	2	4	no	—
26	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
27	C₂×D₈×C₆₇	—	2	4	no	—
28	C₂×D₈×C₆₇	—	2	4	no	—
29	C₂×D₈×C₆₇	—	2	8	no	—
30	C₂×D₈×C₆₇	—	2	8	no	—
31	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
34	#127×C₆₇	—	2	4	no	—
35	#127×C₆₇	—	2	4	no	—
36	#125×C₆₇	—	2	4	no	—
37	#125×C₆₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
40	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
41	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	#107×C₆₇	—	2	4	no	—
44	#106×C₆₇	—	2	4	no	—
45	#102×C₆₇	—	2	16	no	—
46	#97×C₆₇	—	2	4	no	—
47	#97×C₆₇	D₄	1	4	yes	—
48	C₄×C₈×C₆₇	—	2	4	no	—
49	C₄×C₈×C₆₇	D₄	1	4	yes	—
50	#95×C₆₇	D₄	1	4	yes	—
51	C₂⁴×C₆₇	—	2	4	no	—
52	C₂×D₄×C₆₇	—	2	4	no	—
53	C₂×D₄×C₆₇	—	2	4	no	—
54	C₂×D₄×C₆₇	—	2	8	no	—
55	C₂×D₄×C₆₇	—	2	8	no	—
56	C₂×D₄×C₆₇	—	4	8	no	—
57	C₂×D₄×C₆₇	—	8	8	no	—
58	C₂²×C₄×C₆₇	—	2	4	no	—
59	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
60	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
62	D₈×C₆₇	—	4	8	no	—
63	D₈×C₆₇	—	4	8	no	—
64	C₂×C₈×C₆₇	—	2	2	no	—
65	C₂×C₈×C₆₇	—	2	2	no	—
66	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
67	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₆₇	#41	1	4	yes	—
70	C₂×C₈×C₆₇	#41	1	4	yes	—
71	C₄⋊C₄×C₆₇	—	2	4	no	—
72	C₄⋊C₄×C₆₇	—	2	4	no	—
73	C₄⋊C₄×C₆₇	—	4	8	no	—
74	C₂²⋊C₄×C₆₇	—	8	16	no	—
75	C₄²×C₆₇	—	2	2	no	—
76	C₄²×C₆₇	—	2	2	no	—
77	C₄²×C₆₇	C₂×D₄	1	2	yes	—
78	C₄²×C₆₇	C₂×D₄	1	2	yes	—
79	C₂³×C₆₇	—	4	4	no	—
80	C₂³×C₆₇	—	4	8	no	—
81	C₂³×C₆₇	—	4	16	no	—
82	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
83	D₄×C₆₇	—	4	4	no	—
84	D₄×C₆₇	—	4	4	no	—
85	D₄×C₆₇	—	4	8	no	—
86	D₄×C₆₇	—	8	16	no	—
87	C₂×C₄×C₆₇	—	2	4	no	—
88	C₂×C₄×C₆₇	—	2	4	no	—
89	C₂×C₄×C₆₇	—	2	8	no	—
90	C₂×C₄×C₆₇	—	4	4	no	—
91	C₂×C₄×C₆₇	—	4	8	no	—
92	C₂×C₄×C₆₇	#127	1	4	yes	—
93	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
96	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
97	C₈×C₆₇	—	2	4	no	—
98	C₈×C₆₇	#124	1	4	yes	—
99	C₈×C₆₇	#124	1	4	yes	—
100	C₂²×C₆₇	—	8	4	no	—
101	C₂²×C₆₇	—	8	8	no	—
102	C₂²×C₆₇	—	8	16	no	—
103	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
104	C₂²×C₆₇	#532	1	2	yes	—
105	C₂²×C₆₇	#576	1	4	yes	—
106	C₄×C₆₇	—	2	4	no	—
107	C₄×C₆₇	—	4	4	no	—
108	C₄×C₆₇	—	8	4	no	—
109	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
110	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
111	C₄×C₆₇	#532	1	2	yes	—
112	#33248	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₆₇	—	16	4	no	—
114	C₂×C₆₇	#2858	1	1	yes	characteristic
115	C₂×C₆₇	#3155	1	2	yes	—
116	C₂×C₆₇	#3176	1	4	yes	—
117	C₂×#499	C₂×C₆₇	1	1	yes	characteristic, complemented
118	C₂×#493	C₂×C₆₇	1	1	yes	characteristic
119	C₂×#490	C₂×C₆₇	1	2	yes	complemented
120	#2641	C₂×C₆₇	1	2	yes	complemented
121	#1972	C₂×C₆₇	1	4	yes	complemented
122	#1970	C₂×C₆₇	1	4	yes	complemented
123	#1726	C₂×C₆₇	1	1	yes	characteristic, complemented
124	C₆₇	#33248	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
126	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
127	C₂×#127	C₂²×C₆₇	1	2	yes	—
128	C₂×#125	C₂²×C₆₇	1	2	yes	—
129	#499	—	2	4	no	—
130	#493	—	2	4	no	—
131	#490	—	2	8	no	—
132	C₂×#107	C₂²×C₆₇	1	2	yes	—
133	C₂×#106	C₂²×C₆₇	1	2	yes	—
134	C₂×#102	C₂²×C₆₇	1	8	yes	—
135	C₂×#97	C₂²×C₆₇	1	2	yes	—
136	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
139	#394	—	2	8	no	—
140	#390	C₂²×C₆₇	1	8	yes	—
141	#376	C₂²×C₆₇	1	1	yes	characteristic
142	C₂²×D₄	—	2	4	no	—
143	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
144	C₂×D₈	—	2	4	no	—
145	C₂×D₈	—	2	4	no	—
146	C₂×D₈	—	2	8	no	—
147	C₂×D₈	—	2	8	no	—
148	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
149	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
150	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
151	#127	—	2	4	no	—
152	#127	—	2	4	no	—
153	#125	—	2	4	no	—
154	#125	—	2	4	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
157	C₂×C₂²⋊C₄	—	2	16	no	—
158	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
159	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
160	#107	—	2	4	no	—
161	#106	—	2	4	no	—
162	#102	—	2	16	no	—
163	#97	—	2	4	no	—
164	#97	D₄×C₆₇	1	4	yes	—
165	C₄×C₈	—	2	4	no	—
166	C₄×C₈	D₄×C₆₇	1	4	yes	—
167	#95	D₄×C₆₇	1	4	yes	—
168	C₂⁴	—	2	4	no	—
169	C₂×D₄	—	2	4	no	—
170	C₂×D₄	—	2	4	no	—
171	C₂×D₄	—	2	8	no	—
172	C₂×D₄	—	2	8	no	—
173	C₂×D₄	—	4	8	no	—
174	C₂×D₄	—	8	8	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
177	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
178	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
179	D₈	—	4	8	no	—
180	D₈	—	4	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
184	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
185	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
186	C₂×C₈	#41×C₆₇	1	4	yes	—
187	C₂×C₈	#41×C₆₇	1	4	yes	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	4	8	no	—
191	C₂²⋊C₄	—	8	16	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
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	16	no	—
199	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
200	D₄	—	4	4	no	—
201	D₄	—	4	4	no	—
202	D₄	—	4	8	no	—
203	D₄	—	8	16	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	8	no	—
209	C₂×C₄	#127×C₆₇	1	4	yes	—
210	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
211	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
212	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
213	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
214	C₈	—	2	4	no	—
215	C₈	#124×C₆₇	1	4	yes	—
216	C₈	#124×C₆₇	1	4	yes	—
217	C₂²	—	8	4	no	—
218	C₂²	—	8	8	no	—
219	C₂²	—	8	16	no	—
220	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
221	C₂²	#532×C₆₇	1	2	yes	—
222	C₂²	#576×C₆₇	1	4	yes	—
223	C₄	—	2	4	no	—
224	C₄	—	4	4	no	—
225	C₄	—	8	4	no	—
226	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
227	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
228	C₄	#532×C₆₇	1	2	yes	—
229	C₂	—	16	4	no	—
230	C₂	C₆₇×#2858	1	1	yes	characteristic
231	C₂	C₆₇×#3155	1	2	yes	—
232	C₂	C₆₇×#3176	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 ◁ #376×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#499×C₆₇	C₂	semidirect
3. C₂×#490×C₆₇	C₂	semidirect
4. C₆₇×#2641	C₂	semidirect
5. C₆₇×#1972	C₂	semidirect
6. C₆₇×#1970	C₂	semidirect
7. C₆₇×#1726	C₂	semidirect
112. #33248	C₆₇	direct (N × H)
117. C₂×#499	C₂×C₆₇	semidirect
119. C₂×#490	C₂×C₆₇	semidirect
120. #2641	C₂×C₆₇	semidirect
121. #1972	C₂×C₆₇	semidirect
122. #1970	C₂×C₆₇	semidirect
123. #1726	C₂×C₆₇	semidirect
124. C₆₇	#33248	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#499×C₆₇	2
C₂×#493×C₆₇	2
C₂×#490×C₆₇	2
C₆₇×#2641	2
C₆₇×#1972	2
C₆₇×#1970	2
C₆₇×#1726	2
#33248	67

Subgroups — C67×#33248 #30712499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33248 #30712499