Subgroups — C₆₇×#33337 #30712588

Order 17152.

1128 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3407	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2115	C₂	1	6	yes	maximal, complemented
4	C₂×#349×C₆₇	C₂	1	6	yes	maximal, complemented
5	C₂²×#41×C₆₇	C₄	1	1	yes	characteristic
6	#568×C₆₇	—	2	8	no	—
7	C₂×#131×C₆₇	C₄	1	3	yes	—
8	C₂×#131×C₆₇	C₄	1	1	yes	characteristic
9	C₂×#131×C₆₇	C₂²	1	3	yes	—
10	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₆₇	C₄	1	3	yes	—
12	C₂²×M₁₆×C₆₇	C₂²	1	3	yes	—
13	#504×C₆₇	—	2	8	no	—
14	C₂×M₃₂×C₆₇	C₂²	1	12	yes	complemented
15	#349×C₆₇	C₂²	1	4	yes	—
16	#349×C₆₇	C₂²	1	12	yes	complemented
17	C₂×#41×C₆₇	—	2	4	no	—
18	C₂×#41×C₆₇	—	2	4	no	—
19	C₂×#41×C₆₇	C₈	1	1	yes	characteristic
20	C₂×#41×C₆₇	C₈	1	3	yes	—
21	C₂×#41×C₆₇	C₂×C₄	1	3	yes	—
22	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×Q₈×C₆₇	C₈	1	1	yes	characteristic
24	C₂²×D₄×C₆₇	C₈	1	3	yes	—
25	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
26	#131×C₆₇	—	2	4	no	—
27	#131×C₆₇	—	2	12	no	—
28	#131×C₆₇	—	2	12	no	—
29	#131×C₆₇	D₄	1	8	yes	—
30	C₂×M₁₆×C₆₇	—	2	12	no	—
31	C₂×M₁₆×C₆₇	—	2	12	no	—
32	C₂×M₁₆×C₆₇	C₂×C₄	1	3	yes	—
33	C₂×M₁₆×C₆₇	C₂×C₄	1	3	yes	—
34	C₂×M₁₆×C₆₇	C₂×C₄	1	3	yes	—
35	C₂×M₁₆×C₆₇	C₂×C₄	1	3	yes	—
36	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₆₇	C₂³	1	3	yes	—
38	C₂²×C₈×C₆₇	C₂×C₄	1	6	yes	—
39	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
41	M₃₂×C₆₇	—	2	12	no	—
42	M₃₂×C₆₇	C₂³	1	8	yes	—
43	C₂×C₁₆×C₆₇	—	2	12	no	—
44	C₂⁴×C₆₇	C₂×C₈	1	3	yes	—
45	#41×C₆₇	—	2	4	no	—
46	#41×C₆₇	—	2	12	no	—
47	#41×C₆₇	—	2	6	no	—
48	#41×C₆₇	—	2	6	no	—
49	#41×C₆₇	C₂²⋊C₄	1	4	yes	—
50	#41×C₆₇	M₁₆	1	4	yes	—
51	C₂×Q₈×C₆₇	—	2	4	no	—
52	C₂×Q₈×C₆₇	C₂×C₈	1	3	yes	—
53	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
54	C₂×D₄×C₆₇	—	2	12	no	—
55	C₂×D₄×C₆₇	C₂×C₈	1	6	yes	—
56	C₂×D₄×C₆₇	C₂×C₈	1	3	yes	—
57	C₂×D₄×C₆₇	C₂×C₈	1	3	yes	—
58	C₂²×C₄×C₆₇	—	2	6	no	—
59	C₂²×C₄×C₆₇	—	2	6	no	—
60	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
61	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
62	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
66	M₁₆×C₆₇	—	2	6	no	—
67	M₁₆×C₆₇	—	2	6	no	—
68	M₁₆×C₆₇	—	2	6	no	—
69	M₁₆×C₆₇	—	2	2	no	—
70	M₁₆×C₆₇	—	2	6	no	—
71	M₁₆×C₆₇	C₂²⋊C₄	1	12	yes	—
72	C₂×C₈×C₆₇	—	2	6	no	—
73	C₂×C₈×C₆₇	—	2	12	no	—
74	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
75	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
77	C₂×C₈×C₆₇	C₂×D₄	1	12	yes	—
78	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
79	C₁₆×C₆₇	—	2	8	no	—
80	C₂³×C₆₇	—	2	6	no	—
81	C₂³×C₆₇	—	8	3	no	—
82	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
83	C₂³×C₆₇	C₂²×C₈	1	3	yes	—
84	C₂³×C₆₇	C₂²×C₈	1	3	yes	—
85	Q₈×C₆₇	—	2	6	no	—
86	Q₈×C₆₇	C₂²⋊C₈	1	4	yes	—
87	D₄×C₆₇	—	2	12	no	—
88	D₄×C₆₇	—	2	6	no	—
89	D₄×C₆₇	C₂²⋊C₈	1	12	yes	—
90	C₂×C₄×C₆₇	—	2	6	no	—
91	C₂×C₄×C₆₇	—	2	3	no	—
92	C₂×C₄×C₆₇	—	2	1	no	—
93	C₂×C₄×C₆₇	—	2	1	no	—
94	C₂×C₄×C₆₇	—	2	3	no	—
95	C₂×C₄×C₆₇	—	2	6	no	—
96	C₂×C₄×C₆₇	—	2	6	no	—
97	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
98	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂²×C₈	1	3	yes	—
100	C₂×C₄×C₆₇	C₂²×C₈	1	3	yes	—
101	C₂×C₄×C₆₇	C₂×M₁₆	1	6	yes	—
102	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
103	C₈×C₆₇	—	2	1	no	—
104	C₈×C₆₇	—	2	3	no	—
105	C₈×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
106	C₈×C₆₇	C₂²×D₄	1	2	yes	—
107	C₂²×C₆₇	—	2	1	no	—
108	C₂²×C₆₇	—	2	3	no	—
109	C₂²×C₆₇	—	4	3	no	—
110	C₂²×C₆₇	—	4	1	no	—
111	C₂²×C₆₇	—	4	3	no	—
112	C₂²×C₆₇	—	8	6	no	—
113	C₂²×C₆₇	C₂×C₂²⋊C₈	1	6	yes	—
114	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
115	C₄×C₆₇	—	2	1	no	—
116	C₄×C₆₇	—	2	3	no	—
117	C₄×C₆₇	C₂×C₂²⋊C₈	1	6	yes	—
118	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
120	#33337	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂×C₆₇	—	2	1	no	—
122	C₂×C₆₇	—	2	6	no	—
123	C₂×C₆₇	—	4	3	no	—
124	C₂×C₆₇	—	4	1	no	—
125	C₂×C₆₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
126	C₂×#568	C₂×C₆₇	1	1	yes	characteristic
127	#3407	C₂×C₆₇	1	2	yes	complemented
128	#2115	C₂×C₆₇	1	6	yes	complemented
129	C₂×#349	C₂×C₆₇	1	6	yes	complemented
130	C₆₇	#33337	1	1	yes	characteristic, direct product factor, Sylow
131	C₂²×#41	C₄×C₆₇	1	1	yes	characteristic
132	#568	—	2	8	no	—
133	C₂×#131	C₄×C₆₇	1	3	yes	—
134	C₂×#131	C₄×C₆₇	1	1	yes	characteristic
135	C₂×#131	C₂²×C₆₇	1	3	yes	—
136	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
137	C₂²×M₁₆	C₄×C₆₇	1	3	yes	—
138	C₂²×M₁₆	C₂²×C₆₇	1	3	yes	—
139	#504	—	2	8	no	—
140	C₂×M₃₂	C₂²×C₆₇	1	12	yes	complemented
141	#349	C₂²×C₆₇	1	4	yes	—
142	#349	C₂²×C₆₇	1	12	yes	complemented
143	C₂×#41	—	2	4	no	—
144	C₂×#41	—	2	4	no	—
145	C₂×#41	C₈×C₆₇	1	1	yes	characteristic
146	C₂×#41	C₈×C₆₇	1	3	yes	—
147	C₂×#41	C₂×C₄×C₆₇	1	3	yes	—
148	C₂×#41	C₂×C₄×C₆₇	1	1	yes	characteristic
149	C₂²×Q₈	C₈×C₆₇	1	1	yes	characteristic
150	C₂²×D₄	C₈×C₆₇	1	3	yes	—
151	C₂³×C₄	C₂×C₄×C₆₇	1	3	yes	—
152	#131	—	2	4	no	—
153	#131	—	2	12	no	—
154	#131	—	2	12	no	—
155	#131	D₄×C₆₇	1	8	yes	—
156	C₂×M₁₆	—	2	12	no	—
157	C₂×M₁₆	—	2	12	no	—
158	C₂×M₁₆	C₂×C₄×C₆₇	1	3	yes	—
159	C₂×M₁₆	C₂×C₄×C₆₇	1	3	yes	—
160	C₂×M₁₆	C₂×C₄×C₆₇	1	3	yes	—
161	C₂×M₁₆	C₂×C₄×C₆₇	1	3	yes	—
162	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
163	C₂×M₁₆	C₂³×C₆₇	1	3	yes	—
164	C₂²×C₈	C₂×C₄×C₆₇	1	6	yes	—
165	C₂²×C₈	C₂×C₄×C₆₇	1	3	yes	—
166	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
167	M₃₂	—	2	12	no	—
168	M₃₂	C₂³×C₆₇	1	8	yes	—
169	C₂×C₁₆	—	2	12	no	—
170	C₂⁴	C₂×C₈×C₆₇	1	3	yes	—
171	#41	—	2	4	no	—
172	#41	—	2	12	no	—
173	#41	—	2	6	no	—
174	#41	—	2	6	no	—
175	#41	C₂²⋊C₄×C₆₇	1	4	yes	—
176	#41	M₁₆×C₆₇	1	4	yes	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	C₂×C₈×C₆₇	1	3	yes	—
179	C₂×Q₈	C₂×C₈×C₆₇	1	1	yes	characteristic
180	C₂×D₄	—	2	12	no	—
181	C₂×D₄	C₂×C₈×C₆₇	1	6	yes	—
182	C₂×D₄	C₂×C₈×C₆₇	1	3	yes	—
183	C₂×D₄	C₂×C₈×C₆₇	1	3	yes	—
184	C₂²×C₄	—	2	6	no	—
185	C₂²×C₄	—	2	6	no	—
186	C₂²×C₄	C₂×C₈×C₆₇	1	3	yes	—
187	C₂²×C₄	C₂×C₈×C₆₇	1	3	yes	—
188	C₂²×C₄	C₂×C₈×C₆₇	1	3	yes	—
189	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
190	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
192	M₁₆	—	2	6	no	—
193	M₁₆	—	2	6	no	—
194	M₁₆	—	2	6	no	—
195	M₁₆	—	2	2	no	—
196	M₁₆	—	2	6	no	—
197	M₁₆	C₂²⋊C₄×C₆₇	1	12	yes	—
198	C₂×C₈	—	2	6	no	—
199	C₂×C₈	—	2	12	no	—
200	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
201	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
202	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
203	C₂×C₈	C₂×D₄×C₆₇	1	12	yes	—
204	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
205	C₁₆	—	2	8	no	—
206	C₂³	—	2	6	no	—
207	C₂³	—	8	3	no	—
208	C₂³	C₂²×C₈×C₆₇	1	1	yes	characteristic
209	C₂³	C₂²×C₈×C₆₇	1	3	yes	—
210	C₂³	C₂²×C₈×C₆₇	1	3	yes	—
211	Q₈	—	2	6	no	—
212	Q₈	C₂²⋊C₈×C₆₇	1	4	yes	—
213	D₄	—	2	12	no	—
214	D₄	—	2	6	no	—
215	D₄	C₂²⋊C₈×C₆₇	1	12	yes	—
216	C₂×C₄	—	2	6	no	—
217	C₂×C₄	—	2	3	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	3	no	—
221	C₂×C₄	—	2	6	no	—
222	C₂×C₄	—	2	6	no	—
223	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	6	yes	—
224	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₈×C₆₇	1	3	yes	—
226	C₂×C₄	C₂²×C₈×C₆₇	1	3	yes	—
227	C₂×C₄	C₂×M₁₆×C₆₇	1	6	yes	—
228	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	2	3	no	—
231	C₈	C₂×C₂²⋊C₄×C₆₇	1	6	yes	—
232	C₈	C₂²×D₄×C₆₇	1	2	yes	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	3	no	—
235	C₂²	—	4	3	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	3	no	—
238	C₂²	—	8	6	no	—
239	C₂²	C₂×C₂²⋊C₈×C₆₇	1	6	yes	—
240	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	3	no	—
243	C₄	C₂×C₂²⋊C₈×C₆₇	1	6	yes	—
244	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
245	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
246	C₂	—	2	1	no	—
247	C₂	—	2	6	no	—
248	C₂	—	4	1	no	—
249	C₂	—	4	3	no	—
250	C₂	C₂²×C₂²⋊C₈×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₇	C₂²×C₂²⋊C₄
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂×C₆₇	C₂²×C₂²⋊C₈
prime radical	C₂²×#41×C₆₇	C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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×#2115	C₂	semidirect
4. C₂×#349×C₆₇	C₂	semidirect
14. C₂×M₃₂×C₆₇	C₂²	semidirect
16. #349×C₆₇	C₂²	semidirect
120. #33337	C₆₇	direct (N × H)
127. #3407	C₂×C₆₇	semidirect
128. #2115	C₂×C₆₇	semidirect
129. C₂×#349	C₂×C₆₇	semidirect
130. C₆₇	#33337	direct (N × H)
140. C₂×M₃₂	C₂²×C₆₇	semidirect
142. #349	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×#568×C₆₇	2
C₆₇×#3407	2
C₆₇×#2115	2
C₂×#349×C₆₇	2
#33337	67

Subgroups — C67×#33337 #30712588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33337 #30712588