Subgroups — C₆₇×#33426 #30712677

Order 17152.

520 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2875	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2166	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2165	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2153	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2115	C₂	1	2	yes	maximal
6	C₂×#347×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#131×C₆₇	C₄	1	1	yes	characteristic
8	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
9	C₄×#41×C₆₇	C₄	1	1	yes	characteristic
10	C₂×M₃₂×C₆₇	C₂²	1	4	yes	—
11	#405×C₆₇	C₄	1	1	yes	characteristic
12	#405×C₆₇	C₄	1	2	yes	—
13	#405×C₆₇	C₂²	1	2	yes	—
14	#405×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₆₇	C₄	1	1	yes	characteristic
16	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂×#97×C₆₇	C₄	1	2	yes	—
18	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
19	#364×C₆₇	C₂²	1	8	yes	—
20	#349×C₆₇	C₂²	1	8	yes	—
21	#347×C₆₇	C₂²	1	4	yes	—
22	#347×C₆₇	C₂²	1	4	yes	—
23	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
24	#131×C₆₇	—	2	4	no	—
25	#131×C₆₇	—	2	4	no	—
26	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
27	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
34	C₄×Q₈×C₆₇	—	2	1	no	—
35	C₄×D₄×C₆₇	—	2	2	no	—
36	C₄×D₄×C₆₇	—	2	1	no	—
37	#117×C₆₇	C₂×C₄	1	2	yes	—
38	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
41	M₃₂×C₆₇	—	2	4	no	—
42	M₃₂×C₆₇	C₂³	1	8	yes	—
43	C₂×C₁₆×C₆₇	—	2	4	no	—
44	#97×C₆₇	C₂×C₄	1	1	yes	characteristic
45	#97×C₆₇	C₂×C₄	1	2	yes	—
46	#97×C₆₇	C₂×C₄	1	2	yes	—
47	#97×C₆₇	C₂×C₄	1	2	yes	—
48	#97×C₆₇	C₂×C₄	1	1	yes	characteristic
49	#97×C₆₇	C₂³	1	1	yes	characteristic
50	#97×C₆₇	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₆₇	C₂³	1	2	yes	—
54	#41×C₆₇	—	2	2	no	—
55	#41×C₆₇	—	2	2	no	—
56	C₂×Q₈×C₆₇	M₁₆	1	1	yes	characteristic
57	C₂×D₄×C₆₇	M₁₆	1	2	yes	—
58	C₂×D₄×C₆₇	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	—	2	2	no	—
60	C₂²×C₄×C₆₇	—	2	1	no	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	M₁₆×C₆₇	—	2	2	no	—
64	M₁₆×C₆₇	—	2	4	no	—
65	M₁₆×C₆₇	—	2	4	no	—
66	M₁₆×C₆₇	M₁₆	1	4	yes	—
67	C₂×C₈×C₆₇	—	2	2	no	—
68	C₂×C₈×C₆₇	—	2	4	no	—
69	C₂×C₈×C₆₇	—	2	4	no	—
70	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₆₇	#41	1	4	yes	—
76	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
77	C₄⋊C₄×C₆₇	—	2	2	no	—
78	C₄⋊C₄×C₆₇	—	2	1	no	—
79	C₂²⋊C₄×C₆₇	—	2	2	no	—
80	C₂²⋊C₄×C₆₇	—	2	1	no	—
81	C₄²×C₆₇	—	2	2	no	—
82	C₄²×C₆₇	—	2	1	no	—
83	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
86	C₁₆×C₆₇	—	2	8	no	—
87	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
88	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
89	Q₈×C₆₇	—	2	2	no	—
90	D₄×C₆₇	—	2	4	no	—
91	D₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	1	no	—
96	C₂×C₄×C₆₇	—	2	2	no	—
97	C₂×C₄×C₆₇	—	8	1	no	—
98	C₂×C₄×C₆₇	—	8	2	no	—
99	C₂×C₄×C₆₇	#117	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
101	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
104	C₈×C₆₇	—	2	2	no	—
105	C₈×C₆₇	—	2	1	no	—
106	C₈×C₆₇	—	2	1	no	—
107	C₈×C₆₇	#117	1	2	yes	—
108	C₈×C₆₇	C₂×M₁₆	1	4	yes	—
109	C₈×C₆₇	C₂×#41	1	2	yes	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	4	1	no	—
112	C₂²×C₆₇	—	4	2	no	—
113	C₂²×C₆₇	#432	1	2	yes	—
114	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
115	C₄×C₆₇	—	2	2	no	—
116	C₄×C₆₇	—	4	1	no	—
117	C₄×C₆₇	—	4	2	no	—
118	C₄×C₆₇	—	4	1	no	—
119	C₄×C₆₇	#432	1	2	yes	—
120	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
121	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
122	#33426	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂×C₆₇	—	2	1	no	—
124	C₂×C₆₇	—	2	2	no	—
125	C₂×C₆₇	—	4	2	no	—
126	C₂×C₆₇	C₂×#432	1	1	yes	characteristic
127	#2875	C₂×C₆₇	1	1	yes	characteristic
128	#2166	C₂×C₆₇	1	4	yes	complemented
129	#2165	C₂×C₆₇	1	4	yes	complemented
130	#2153	C₂×C₆₇	1	2	yes	complemented
131	#2115	C₂×C₆₇	1	2	yes	—
132	C₂×#347	C₂×C₆₇	1	2	yes	complemented
133	C₆₇	#33426	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×#131	C₄×C₆₇	1	1	yes	characteristic
135	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
136	C₄×#41	C₄×C₆₇	1	1	yes	characteristic
137	C₂×M₃₂	C₂²×C₆₇	1	4	yes	—
138	#405	C₄×C₆₇	1	2	yes	—
139	#405	C₄×C₆₇	1	1	yes	characteristic
140	#405	C₂²×C₆₇	1	2	yes	—
141	#405	C₂²×C₆₇	1	1	yes	characteristic
142	C₄×M₁₆	C₄×C₆₇	1	1	yes	characteristic
143	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
144	C₂×#97	C₄×C₆₇	1	2	yes	—
145	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
146	#364	C₂²×C₆₇	1	8	yes	—
147	#349	C₂²×C₆₇	1	8	yes	—
148	#347	C₂²×C₆₇	1	4	yes	—
149	#347	C₂²×C₆₇	1	4	yes	—
150	C₂×#41	C₂×C₄×C₆₇	1	1	yes	characteristic
151	#131	—	2	4	no	—
152	#131	—	2	4	no	—
153	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
154	C₂×M₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
155	C₂×M₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
156	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
157	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
158	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
159	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
160	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
161	C₄×Q₈	—	2	1	no	—
162	C₄×D₄	—	2	2	no	—
163	C₄×D₄	—	2	1	no	—
164	#117	C₂×C₄×C₆₇	1	2	yes	—
165	#117	C₂×C₄×C₆₇	1	1	yes	characteristic
166	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
167	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
168	M₃₂	—	2	4	no	—
169	M₃₂	C₂³×C₆₇	1	8	yes	—
170	C₂×C₁₆	—	2	4	no	—
171	#97	C₂×C₄×C₆₇	1	1	yes	characteristic
172	#97	C₂×C₄×C₆₇	1	2	yes	—
173	#97	C₂×C₄×C₆₇	1	2	yes	—
174	#97	C₂×C₄×C₆₇	1	2	yes	—
175	#97	C₂×C₄×C₆₇	1	1	yes	characteristic
176	#97	C₂³×C₆₇	1	1	yes	characteristic
177	#97	C₂³×C₆₇	1	1	yes	characteristic
178	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
179	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
180	C₄×C₈	C₂³×C₆₇	1	2	yes	—
181	#41	—	2	2	no	—
182	#41	—	2	2	no	—
183	C₂×Q₈	M₁₆×C₆₇	1	1	yes	characteristic
184	C₂×D₄	M₁₆×C₆₇	1	2	yes	—
185	C₂×D₄	M₁₆×C₆₇	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
189	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
190	M₁₆	—	2	2	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	—	2	4	no	—
193	M₁₆	M₁₆×C₆₇	1	4	yes	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
198	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
199	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
200	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
201	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
202	C₂×C₈	#41×C₆₇	1	4	yes	—
203	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
211	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
212	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
213	C₁₆	—	2	8	no	—
214	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
215	C₂³	C₂×M₁₆×C₆₇	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	D₄	—	2	4	no	—
218	D₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	#117×C₆₇	1	2	yes	—
227	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
228	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
229	C₂×C₄	C₂×M₁₆×C₆₇	1	1	yes	characteristic
230	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	#117×C₆₇	1	2	yes	—
235	C₈	C₂×M₁₆×C₆₇	1	4	yes	—
236	C₈	C₂×#41×C₆₇	1	2	yes	—
237	C₂²	—	2	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	2	no	—
240	C₂²	#432×C₆₇	1	2	yes	—
241	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	2	no	—
246	C₄	#432×C₆₇	1	2	yes	—
247	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
248	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	2	2	no	—
251	C₂	—	4	2	no	—
252	C₂	C₂×#432×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₇	C₂×#117
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂×C₆₇	C₂×#432
prime radical	C₂×#41×C₆₇	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂×#97×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2166	C₂	semidirect
3. C₆₇×#2165	C₂	semidirect
4. C₆₇×#2153	C₂	semidirect
6. C₂×#347×C₆₇	C₂	semidirect
122. #33426	C₆₇	direct (N × H)
128. #2166	C₂×C₆₇	semidirect
129. #2165	C₂×C₆₇	semidirect
130. #2153	C₂×C₆₇	semidirect
132. C₂×#347	C₂×C₆₇	semidirect
133. C₆₇	#33426	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2875	2
C₆₇×#2166	2
C₆₇×#2165	2
C₆₇×#2153	2
C₆₇×#2115	2
C₂×#347×C₆₇	2
#33426	67

Subgroups — C67×#33426 #30712677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33426 #30712677