Subgroups — C₆₇×#33713 #30712964

Order 17152.

834 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×#508×C₆₇	C₂	1	4	yes	maximal, complemented
2	C₆₇×#3147	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2249	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2248	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2247	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2246	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₆₇	—	2	2	no	—
9	C₂×#135×C₆₇	C₂²	1	4	yes	—
10	#532×C₆₇	—	2	2	no	—
11	#508×C₆₇	—	2	16	no	—
12	C₂×Q₃₂×C₆₇	C₂²	1	4	yes	—
13	C₂×SD₃₂×C₆₇	C₂²	1	8	yes	—
14	C₂×D₁₆×C₆₇	C₂²	1	4	yes	—
15	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
16	#495×C₆₇	—	2	2	no	—
17	#495×C₆₇	C₂²	1	4	yes	—
18	#494×C₆₇	C₂²	1	2	yes	—
19	#493×C₆₇	C₂²	1	2	yes	—
20	#492×C₆₇	—	2	2	no	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
25	C₂×#41×C₆₇	—	2	4	no	—
26	#135×C₆₇	—	2	16	no	—
27	#135×C₆₇	—	4	8	no	—
28	C₂×Q₁₆×C₆₇	—	2	2	no	—
29	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
30	C₂×SD₁₆×C₆₇	—	2	4	no	—
31	C₂×SD₁₆×C₆₇	—	2	4	no	—
32	C₂×D₈×C₆₇	—	2	2	no	—
33	C₂×D₈×C₆₇	C₂³	1	4	yes	—
34	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	C₄⋊Q₈×C₆₇	—	2	2	no	—
37	#127×C₆₇	—	2	2	no	—
38	#124×C₆₇	—	2	4	no	—
39	#121×C₆₇	—	4	4	no	—
40	C₄×D₄×C₆₇	—	4	4	no	—
41	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
42	Q₃₂×C₆₇	—	2	8	no	—
43	SD₃₂×C₆₇	—	2	16	no	—
44	D₁₆×C₆₇	—	2	8	no	—
45	C₂×C₁₆×C₆₇	D₄	1	8	yes	complemented
46	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
47	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
48	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	#41×C₆₇	—	4	8	no	—
53	#41×C₆₇	—	4	8	no	—
54	C₂×Q₈×C₆₇	—	2	4	no	—
55	C₂×D₄×C₆₇	—	2	4	no	—
56	C₂×D₄×C₆₇	—	4	4	no	—
57	C₂×D₄×C₆₇	—	4	4	no	—
58	C₂²×C₄×C₆₇	—	4	4	no	—
59	C₂²×C₄×C₆₇	D₈	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	Q₁₆×C₆₇	—	2	8	no	—
62	Q₁₆×C₆₇	—	4	4	no	—
63	SD₁₆×C₆₇	—	4	8	no	—
64	SD₁₆×C₆₇	—	4	8	no	—
65	D₈×C₆₇	—	2	8	no	—
66	D₈×C₆₇	—	4	4	no	—
67	C₂×C₈×C₆₇	D₈	1	4	yes	—
68	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
70	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₆₇	—	4	4	no	—
74	C₂²⋊C₄×C₆₇	—	4	8	no	—
75	C₄²×C₆₇	D₈	1	2	yes	—
76	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₆₇	C₂×D₄	1	4	yes	—
79	C₁₆×C₆₇	C₂×D₄	1	4	yes	—
80	C₂³×C₆₇	—	4	4	no	—
81	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
82	Q₈×C₆₇	—	4	4	no	—
83	Q₈×C₆₇	—	4	4	no	—
84	D₄×C₆₇	—	4	4	no	—
85	D₄×C₆₇	—	4	4	no	—
86	D₄×C₆₇	—	8	8	no	—
87	D₄×C₆₇	—	8	8	no	—
88	C₂×C₄×C₆₇	—	2	4	no	—
89	C₂×C₄×C₆₇	—	4	4	no	—
90	C₂×C₄×C₆₇	—	8	8	no	—
91	C₂×C₄×C₆₇	#127	1	2	yes	—
92	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
93	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
97	C₈×C₆₇	#127	1	2	yes	—
98	C₈×C₆₇	C₂×D₈	1	2	yes	—
99	C₈×C₆₇	C₂×D₈	1	2	yes	—
100	C₈×C₆₇	C₂²×D₄	1	2	yes	—
101	C₂²×C₆₇	—	2	2	no	—
102	C₂²×C₆₇	—	8	4	no	—
103	C₂²×C₆₇	—	8	8	no	—
104	C₂²×C₆₇	#493	1	2	yes	—
105	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
106	C₄×C₆₇	—	2	2	no	—
107	C₄×C₆₇	—	8	4	no	—
108	C₄×C₆₇	#493	1	2	yes	—
109	C₄×C₆₇	#508	1	4	yes	complemented
110	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
111	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
112	#33713	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₆₇	—	2	2	no	—
114	C₂×C₆₇	—	16	4	no	—
115	C₂×C₆₇	C₂×#493	1	1	yes	characteristic
116	C₂×C₆₇	C₂×#508	1	2	yes	—
117	C₂×#508	C₂×C₆₇	1	4	yes	complemented
118	#3147	C₂×C₆₇	1	2	yes	complemented
119	#2249	C₂×C₆₇	1	4	yes	complemented
120	#2248	C₂×C₆₇	1	2	yes	complemented
121	#2247	C₂×C₆₇	1	1	yes	characteristic, complemented
122	#2246	C₂×C₆₇	1	1	yes	characteristic, complemented
123	C₂×C₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
124	C₆₇	#33713	1	1	yes	characteristic, direct product factor, Sylow
125	C₂×#135	—	2	2	no	—
126	C₂×#135	C₂²×C₆₇	1	4	yes	—
127	#532	—	2	2	no	—
128	#508	—	2	16	no	—
129	C₂×Q₃₂	C₂²×C₆₇	1	4	yes	—
130	C₂×SD₃₂	C₂²×C₆₇	1	8	yes	—
131	C₂×D₁₆	C₂²×C₆₇	1	4	yes	—
132	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
133	#495	—	2	2	no	—
134	#495	C₂²×C₆₇	1	4	yes	—
135	#494	C₂²×C₆₇	1	2	yes	—
136	#493	C₂²×C₆₇	1	2	yes	—
137	#492	—	2	2	no	—
138	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
139	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
140	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
141	C₄×C₁₆	C₂²×C₆₇	1	2	yes	—
142	C₂×#41	—	2	4	no	—
143	#135	—	2	16	no	—
144	#135	—	4	8	no	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
147	C₂×SD₁₆	—	2	4	no	—
148	C₂×SD₁₆	—	2	4	no	—
149	C₂×D₈	—	2	2	no	—
150	C₂×D₈	C₂³×C₆₇	1	4	yes	—
151	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
152	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
153	C₄⋊Q₈	—	2	2	no	—
154	#127	—	2	2	no	—
155	#124	—	2	4	no	—
156	#121	—	4	4	no	—
157	C₄×D₄	—	4	4	no	—
158	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
159	Q₃₂	—	2	8	no	—
160	SD₃₂	—	2	16	no	—
161	D₁₆	—	2	8	no	—
162	C₂×C₁₆	D₄×C₆₇	1	8	yes	complemented
163	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
164	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
165	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
166	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
167	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
168	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
169	#41	—	4	8	no	—
170	#41	—	4	8	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×D₄	—	2	4	no	—
173	C₂×D₄	—	4	4	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂²×C₄	—	4	4	no	—
176	C₂²×C₄	D₈×C₆₇	1	2	yes	—
177	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
178	Q₁₆	—	2	8	no	—
179	Q₁₆	—	4	4	no	—
180	SD₁₆	—	4	8	no	—
181	SD₁₆	—	4	8	no	—
182	D₈	—	2	8	no	—
183	D₈	—	4	4	no	—
184	C₂×C₈	D₈×C₆₇	1	4	yes	—
185	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
186	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
187	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
188	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
189	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
190	C₄⋊C₄	—	4	4	no	—
191	C₂²⋊C₄	—	4	8	no	—
192	C₄²	D₈×C₆₇	1	2	yes	—
193	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
194	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
195	C₁₆	C₂×D₄×C₆₇	1	4	yes	—
196	C₁₆	C₂×D₄×C₆₇	1	4	yes	—
197	C₂³	—	4	4	no	—
198	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
199	Q₈	—	4	4	no	—
200	Q₈	—	4	4	no	—
201	D₄	—	4	4	no	—
202	D₄	—	4	4	no	—
203	D₄	—	8	8	no	—
204	D₄	—	8	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	8	8	no	—
208	C₂×C₄	#127×C₆₇	1	2	yes	—
209	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
210	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
211	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
212	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
213	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
214	C₈	#127×C₆₇	1	2	yes	—
215	C₈	C₂×D₈×C₆₇	1	2	yes	—
216	C₈	C₂×D₈×C₆₇	1	2	yes	—
217	C₈	C₂²×D₄×C₆₇	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	8	4	no	—
220	C₂²	—	8	8	no	—
221	C₂²	#493×C₆₇	1	2	yes	—
222	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	#493×C₆₇	1	2	yes	—
226	C₄	#508×C₆₇	1	4	yes	complemented
227	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
228	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	—	16	4	no	—
231	C₂	C₂×#493×C₆₇	1	1	yes	characteristic
232	C₂	C₂×#508×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#508×C₆₇	C₂	semidirect
2. C₆₇×#3147	C₂	semidirect
3. C₆₇×#2249	C₂	semidirect
4. C₆₇×#2248	C₂	semidirect
5. C₆₇×#2247	C₂	semidirect
6. C₆₇×#2246	C₂	semidirect
7. C₂×C₄×C₁₆×C₆₇	C₂	semidirect
45. C₂×C₁₆×C₆₇	D₄	semidirect
109. C₄×C₆₇	#508	semidirect
112. #33713	C₆₇	direct (N × H)
117. C₂×#508	C₂×C₆₇	semidirect
118. #3147	C₂×C₆₇	semidirect
119. #2249	C₂×C₆₇	semidirect
120. #2248	C₂×C₆₇	semidirect
121. #2247	C₂×C₆₇	semidirect
122. #2246	C₂×C₆₇	semidirect
123. C₂×C₄×C₁₆	C₂×C₆₇	semidirect
124. C₆₇	#33713	direct (N × H)
162. C₂×C₁₆	D₄×C₆₇	semidirect
226. C₄	#508×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508×C₆₇	2
C₆₇×#3147	2
C₆₇×#2249	2
C₆₇×#2248	2
C₆₇×#2247	2
C₆₇×#2246	2
C₂×C₄×C₁₆×C₆₇	2
#33713	67

Subgroups — C67×#33713 #30712964

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33713 #30712964