Subgroups — C₆₇×#33392 #30712643

Order 17152.

1154 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#576×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3407	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2916	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2148	C₂	1	6	yes	maximal, complemented
5	C₂×#361×C₆₇	C₂	1	6	yes	maximal, complemented
6	C₂×H(5,𝔽₂)×C₆₇	—	2	1	no	—
7	#576×C₆₇	—	2	4	no	—
8	#576×C₆₇	C₄	1	8	yes	—
9	C₂×AGL(1,ℤ/8)×C₆₇	—	2	3	no	—
10	C₂×#135×C₆₇	C₂²	1	3	yes	—
11	C₂²×D₈×C₆₇	C₂²	1	3	yes	—
12	#568×C₆₇	—	2	1	no	—
13	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
14	#504×C₆₇	—	2	4	no	—
15	C₂×M₃₂×C₆₇	C₂²	1	6	yes	complemented
16	#430×C₆₇	—	2	3	no	—
17	C₂×#108×C₆₇	C₂²	1	6	yes	complemented
18	#362×C₆₇	C₂²	1	4	yes	complemented
19	#361×C₆₇	C₂²	1	12	yes	complemented
20	H(5,𝔽₂)×C₆₇	—	2	8	no	—
21	H(5,𝔽₂)×C₆₇	—	4	4	no	—
22	C₂×#41×C₆₇	—	2	3	no	—
23	C₂×#41×C₆₇	—	4	1	no	—
24	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
25	C₂²×D₄×C₆₇	—	2	3	no	—
26	C₂²×D₄×C₆₇	—	4	3	no	—
27	AGL(1,ℤ/8)×C₆₇	—	2	24	no	—
28	#135×C₆₇	—	2	6	no	—
29	#135×C₆₇	C₂×C₄	1	12	yes	—
30	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
31	C₂×SD₁₆×C₆₇	—	2	3	no	—
32	C₂×D₈×C₆₇	—	2	3	no	—
33	C₂×D₈×C₆₇	—	2	6	no	—
34	C₂×D₈×C₆₇	C₂×C₄	1	12	yes	—
35	C₂×D₈×C₆₇	C₂³	1	3	yes	—
36	#131×C₆₇	—	4	4	no	—
37	#131×C₆₇	C₂×C₄	1	4	yes	—
38	#131×C₆₇	D₄	1	4	yes	—
39	C₂×M₁₆×C₆₇	—	2	6	no	—
40	C₂×M₁₆×C₆₇	D₄	1	3	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
42	M₃₂×C₆₇	—	2	6	no	—
43	M₃₂×C₆₇	C₂³	1	4	yes	—
44	C₂×C₁₆×C₆₇	—	2	6	no	—
45	#108×C₆₇	—	2	6	no	—
46	#108×C₆₇	C₂³	1	4	yes	—
47	C₂⁴×C₆₇	—	4	3	no	—
48	#41×C₆₇	—	2	12	no	—
49	#41×C₆₇	—	4	8	no	—
50	#41×C₆₇	—	4	6	no	—
51	#41×C₆₇	C₂²⋊C₄	1	4	yes	—
52	#41×C₆₇	D₈	1	4	yes	—
53	C₂×Q₈×C₆₇	—	2	1	no	—
54	C₂×Q₈×C₆₇	SD₁₆	1	1	yes	characteristic
55	C₂×D₄×C₆₇	—	2	12	no	—
56	C₂×D₄×C₆₇	—	2	3	no	—
57	C₂×D₄×C₆₇	—	4	3	no	—
58	C₂×D₄×C₆₇	—	4	24	no	—
59	C₂×D₄×C₆₇	—	4	6	no	—
60	C₂×D₄×C₆₇	—	4	3	no	—
61	C₂×D₄×C₆₇	SD₁₆	1	3	yes	—
62	C₂²×C₄×C₆₇	—	4	3	no	—
63	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
64	Q₁₆×C₆₇	—	2	1	no	—
65	Q₁₆×C₆₇	C₂²×C₄	1	2	yes	—
66	SD₁₆×C₆₇	—	2	12	no	—
67	D₈×C₆₇	—	2	12	no	—
68	D₈×C₆₇	—	2	3	no	—
69	D₈×C₆₇	C₂²×C₄	1	6	yes	—
70	M₁₆×C₆₇	—	2	4	no	—
71	M₁₆×C₆₇	—	4	6	no	—
72	M₁₆×C₆₇	C₂²⋊C₄	1	12	yes	—
73	C₂×C₈×C₆₇	—	4	6	no	—
74	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
75	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
76	C₂×C₈×C₆₇	C₂×D₄	1	6	yes	—
77	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
78	C₁₆×C₆₇	—	2	4	no	—
79	C₂³×C₆₇	—	4	3	no	—
80	C₂³×C₆₇	—	4	12	no	—
81	C₂³×C₆₇	—	16	6	no	—
82	C₂³×C₆₇	C₂×SD₁₆	1	3	yes	—
83	Q₈×C₆₇	—	2	2	no	—
84	Q₈×C₆₇	—	4	1	no	—
85	Q₈×C₆₇	#102	1	4	yes	—
86	D₄×C₆₇	—	2	6	no	—
87	D₄×C₆₇	—	4	12	no	—
88	D₄×C₆₇	—	4	3	no	—
89	D₄×C₆₇	—	4	12	no	—
90	D₄×C₆₇	#102	1	12	yes	—
91	C₂×C₄×C₆₇	—	4	1	no	—
92	C₂×C₄×C₆₇	—	4	12	no	—
93	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
94	C₂×C₄×C₆₇	C₂×D₈	1	6	yes	—
95	C₂×C₄×C₆₇	C₂×SD₁₆	1	3	yes	—
96	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
97	C₈×C₆₇	—	4	4	no	—
98	C₈×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
99	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
100	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₆₇	—	4	3	no	—
102	C₂²×C₆₇	—	4	6	no	—
103	C₂²×C₆₇	—	8	6	no	—
104	C₂²×C₆₇	—	16	12	no	—
105	C₂²×C₆₇	C₂×#102	1	6	yes	—
106	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
107	C₄×C₆₇	—	4	2	no	—
108	C₄×C₆₇	C₂×#102	1	6	yes	—
109	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
110	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
111	#33392	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₆₇	—	2	1	no	—
113	C₂×C₆₇	—	2	6	no	—
114	C₂×C₆₇	—	8	6	no	—
115	C₂×C₆₇	C₂²×#102	1	1	yes	characteristic
116	C₂×#576	C₂×C₆₇	1	1	yes	characteristic
117	#3407	C₂×C₆₇	1	1	yes	characteristic, complemented
118	#2916	C₂×C₆₇	1	1	yes	characteristic, complemented
119	#2148	C₂×C₆₇	1	6	yes	complemented
120	C₂×#361	C₂×C₆₇	1	6	yes	complemented
121	C₆₇	#33392	1	1	yes	characteristic, direct product factor, Sylow
122	C₂×H(5,𝔽₂)	—	2	1	no	—
123	#576	—	2	4	no	—
124	#576	C₄×C₆₇	1	8	yes	—
125	C₂×AGL(1,ℤ/8)	—	2	3	no	—
126	C₂×#135	C₂²×C₆₇	1	3	yes	—
127	C₂²×D₈	C₂²×C₆₇	1	3	yes	—
128	#568	—	2	1	no	—
129	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
130	#504	—	2	4	no	—
131	C₂×M₃₂	C₂²×C₆₇	1	6	yes	complemented
132	#430	—	2	3	no	—
133	C₂×#108	C₂²×C₆₇	1	6	yes	complemented
134	#362	C₂²×C₆₇	1	4	yes	complemented
135	#361	C₂²×C₆₇	1	12	yes	complemented
136	H(5,𝔽₂)	—	2	8	no	—
137	H(5,𝔽₂)	—	4	4	no	—
138	C₂×#41	—	2	3	no	—
139	C₂×#41	—	4	1	no	—
140	C₂×#41	D₄×C₆₇	1	1	yes	characteristic
141	C₂²×D₄	—	2	3	no	—
142	C₂²×D₄	—	4	3	no	—
143	AGL(1,ℤ/8)	—	2	24	no	—
144	#135	—	2	6	no	—
145	#135	C₂×C₄×C₆₇	1	12	yes	—
146	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
147	C₂×SD₁₆	—	2	3	no	—
148	C₂×D₈	—	2	3	no	—
149	C₂×D₈	—	2	6	no	—
150	C₂×D₈	C₂×C₄×C₆₇	1	12	yes	—
151	C₂×D₈	C₂³×C₆₇	1	3	yes	—
152	#131	—	4	4	no	—
153	#131	C₂×C₄×C₆₇	1	4	yes	—
154	#131	D₄×C₆₇	1	4	yes	—
155	C₂×M₁₆	—	2	6	no	—
156	C₂×M₁₆	D₄×C₆₇	1	3	yes	—
157	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
158	M₃₂	—	2	6	no	—
159	M₃₂	C₂³×C₆₇	1	4	yes	—
160	C₂×C₁₆	—	2	6	no	—
161	#108	—	2	6	no	—
162	#108	C₂³×C₆₇	1	4	yes	—
163	C₂⁴	—	4	3	no	—
164	#41	—	2	12	no	—
165	#41	—	4	8	no	—
166	#41	—	4	6	no	—
167	#41	C₂²⋊C₄×C₆₇	1	4	yes	—
168	#41	D₈×C₆₇	1	4	yes	—
169	C₂×Q₈	—	2	1	no	—
170	C₂×Q₈	SD₁₆×C₆₇	1	1	yes	characteristic
171	C₂×D₄	—	2	12	no	—
172	C₂×D₄	—	2	3	no	—
173	C₂×D₄	—	4	3	no	—
174	C₂×D₄	—	4	24	no	—
175	C₂×D₄	—	4	6	no	—
176	C₂×D₄	—	4	3	no	—
177	C₂×D₄	SD₁₆×C₆₇	1	3	yes	—
178	C₂²×C₄	—	4	3	no	—
179	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
180	Q₁₆	—	2	1	no	—
181	Q₁₆	C₂²×C₄×C₆₇	1	2	yes	—
182	SD₁₆	—	2	12	no	—
183	D₈	—	2	12	no	—
184	D₈	—	2	3	no	—
185	D₈	C₂²×C₄×C₆₇	1	6	yes	—
186	M₁₆	—	2	4	no	—
187	M₁₆	—	4	6	no	—
188	M₁₆	C₂²⋊C₄×C₆₇	1	12	yes	—
189	C₂×C₈	—	4	6	no	—
190	C₂×C₈	C₂²×C₄×C₆₇	1	6	yes	—
191	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
192	C₂×C₈	C₂×D₄×C₆₇	1	6	yes	—
193	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
194	C₁₆	—	2	4	no	—
195	C₂³	—	4	3	no	—
196	C₂³	—	4	12	no	—
197	C₂³	—	16	6	no	—
198	C₂³	C₂×SD₁₆×C₆₇	1	3	yes	—
199	Q₈	—	2	2	no	—
200	Q₈	—	4	1	no	—
201	Q₈	#102×C₆₇	1	4	yes	—
202	D₄	—	2	6	no	—
203	D₄	—	4	12	no	—
204	D₄	—	4	3	no	—
205	D₄	—	4	12	no	—
206	D₄	#102×C₆₇	1	12	yes	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	12	no	—
209	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	6	yes	—
210	C₂×C₄	C₂×D₈×C₆₇	1	6	yes	—
211	C₂×C₄	C₂×SD₁₆×C₆₇	1	3	yes	—
212	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
213	C₈	—	4	4	no	—
214	C₈	C₂×C₂²⋊C₄×C₆₇	1	6	yes	—
215	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
216	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
217	C₂²	—	4	3	no	—
218	C₂²	—	4	6	no	—
219	C₂²	—	8	6	no	—
220	C₂²	—	16	12	no	—
221	C₂²	C₂×#102×C₆₇	1	6	yes	—
222	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
223	C₄	—	4	2	no	—
224	C₄	C₂×#102×C₆₇	1	6	yes	—
225	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
226	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
227	C₂	—	2	1	no	—
228	C₂	—	2	6	no	—
229	C₂	—	8	6	no	—
230	C₂	C₂²×#102×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	C₂²×#102
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂×#576×C₆₇	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂×#41×C₆₇ ◁ C₂×#131×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3407	C₂	semidirect
3. C₆₇×#2916	C₂	semidirect
4. C₆₇×#2148	C₂	semidirect
5. C₂×#361×C₆₇	C₂	semidirect
15. C₂×M₃₂×C₆₇	C₂²	semidirect
17. C₂×#108×C₆₇	C₂²	semidirect
18. #362×C₆₇	C₂²	semidirect
19. #361×C₆₇	C₂²	semidirect
111. #33392	C₆₇	direct (N × H)
117. #3407	C₂×C₆₇	semidirect
118. #2916	C₂×C₆₇	semidirect
119. #2148	C₂×C₆₇	semidirect
120. C₂×#361	C₂×C₆₇	semidirect
121. C₆₇	#33392	direct (N × H)
131. C₂×M₃₂	C₂²×C₆₇	semidirect
133. C₂×#108	C₂²×C₆₇	semidirect
134. #362	C₂²×C₆₇	semidirect
135. #361	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#576×C₆₇	2
C₆₇×#3407	2
C₆₇×#2916	2
C₆₇×#2148	2
C₂×#361×C₆₇	2
#33392	67

Subgroups — C67×#33392 #30712643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33392 #30712643