Subgroups — C₆₇×#33708 #30712959

Order 17152.

590 conjugacy classes of subgroups.

The subgroup lattice has 266 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	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2245	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2244	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2242	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2139	C₂	1	4	yes	maximal, complemented
7	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₆₇	—	2	1	no	—
9	C₂×#135×C₆₇	C₂²	1	2	yes	—
10	#532×C₆₇	—	2	1	no	—
11	C₂×#125×C₆₇	—	2	1	no	—
12	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
13	#499×C₆₇	—	2	1	no	—
14	#499×C₆₇	—	2	1	no	—
15	#499×C₆₇	C₂²	1	2	yes	—
16	#499×C₆₇	C₂²	1	2	yes	—
17	#495×C₆₇	—	2	1	no	—
18	#495×C₆₇	C₂²	1	2	yes	—
19	#494×C₆₇	C₂²	1	1	yes	characteristic
20	#493×C₆₇	C₂²	1	1	yes	characteristic
21	#492×C₆₇	—	2	1	no	—
22	C₂×#107×C₆₇	C₂²	1	2	yes	—
23	C₂×#106×C₆₇	—	2	1	no	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#358×C₆₇	C₂²	1	8	yes	—
26	#357×C₆₇	C₂²	1	8	yes	—
27	#346×C₆₇	C₂²	1	2	yes	—
28	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
29	C₂×#41×C₆₇	—	2	2	no	—
30	#135×C₆₇	—	2	8	no	—
31	#135×C₆₇	—	4	4	no	—
32	C₂×Q₁₆×C₆₇	—	2	1	no	—
33	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
34	C₂×SD₁₆×C₆₇	—	2	2	no	—
35	C₂×SD₁₆×C₆₇	—	2	2	no	—
36	C₂×D₈×C₆₇	—	2	1	no	—
37	C₂×D₈×C₆₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₆₇	—	2	1	no	—
41	#127×C₆₇	—	2	1	no	—
42	#125×C₆₇	—	2	2	no	—
43	#125×C₆₇	—	2	2	no	—
44	#125×C₆₇	—	4	2	no	—
45	#124×C₆₇	—	2	2	no	—
46	#121×C₆₇	—	4	2	no	—
47	C₄×D₄×C₆₇	—	4	2	no	—
48	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
50	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
51	M₃₂×C₆₇	—	2	4	no	—
52	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
53	#107×C₆₇	—	2	2	no	—
54	#107×C₆₇	C₂³	1	4	yes	—
55	#106×C₆₇	—	2	2	no	—
56	#106×C₆₇	—	2	2	no	—
57	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
58	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
59	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	#41×C₆₇	—	4	4	no	—
62	#41×C₆₇	—	4	4	no	—
63	C₂×Q₈×C₆₇	—	2	2	no	—
64	C₂×D₄×C₆₇	—	2	2	no	—
65	C₂×D₄×C₆₇	—	4	2	no	—
66	C₂×D₄×C₆₇	—	4	2	no	—
67	C₂²×C₄×C₆₇	—	4	2	no	—
68	C₂²×C₄×C₆₇	—	4	2	no	—
69	C₂²×C₄×C₆₇	D₈	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	Q₁₆×C₆₇	—	2	4	no	—
72	Q₁₆×C₆₇	—	4	2	no	—
73	SD₁₆×C₆₇	—	4	4	no	—
74	SD₁₆×C₆₇	—	4	4	no	—
75	D₈×C₆₇	—	2	4	no	—
76	D₈×C₆₇	—	4	2	no	—
77	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
78	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	#41	1	4	yes	—
82	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄×C₆₇	—	2	4	no	—
84	C₄⋊C₄×C₆₇	—	4	2	no	—
85	C₄⋊C₄×C₆₇	—	4	4	no	—
86	C₄⋊C₄×C₆₇	—	4	4	no	—
87	C₄⋊C₄×C₆₇	—	4	2	no	—
88	C₂²⋊C₄×C₆₇	—	4	4	no	—
89	C₄²×C₆₇	D₈	1	2	yes	—
90	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₁₆×C₆₇	—	2	4	no	—
93	C₂³×C₆₇	—	4	2	no	—
94	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
95	Q₈×C₆₇	—	4	2	no	—
96	Q₈×C₆₇	—	4	2	no	—
97	D₄×C₆₇	—	4	2	no	—
98	D₄×C₆₇	—	4	2	no	—
99	D₄×C₆₇	—	8	4	no	—
100	D₄×C₆₇	—	8	4	no	—
101	C₂×C₄×C₆₇	—	4	2	no	—
102	C₂×C₄×C₆₇	—	4	2	no	—
103	C₂×C₄×C₆₇	—	4	4	no	—
104	C₂×C₄×C₆₇	—	8	4	no	—
105	C₂×C₄×C₆₇	—	8	4	no	—
106	C₂×C₄×C₆₇	#124	1	2	yes	—
107	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
111	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
112	C₈×C₆₇	#124	1	2	yes	—
113	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
114	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
115	C₈×C₆₇	C₂×#41	1	2	yes	—
116	C₂²×C₆₇	—	2	2	no	—
117	C₂²×C₆₇	—	8	2	no	—
118	C₂²×C₆₇	—	8	4	no	—
119	C₂²×C₆₇	#486	1	2	yes	—
120	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
121	C₄×C₆₇	—	2	2	no	—
122	C₄×C₆₇	—	2	2	no	—
123	C₄×C₆₇	—	8	2	no	—
124	C₄×C₆₇	—	8	4	no	—
125	C₄×C₆₇	#486	1	2	yes	—
126	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
127	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
128	#33708	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₆₇	—	2	2	no	—
130	C₂×C₆₇	—	16	2	no	—
131	C₂×C₆₇	C₂×#486	1	1	yes	characteristic
132	C₂×C₆₇	#3414	1	2	yes	—
133	C₂×#499	C₂×C₆₇	1	1	yes	characteristic, complemented
134	#3147	C₂×C₆₇	1	1	yes	characteristic
135	#2245	C₂×C₆₇	1	2	yes	complemented
136	#2244	C₂×C₆₇	1	2	yes	complemented
137	#2242	C₂×C₆₇	1	4	yes	complemented
138	#2139	C₂×C₆₇	1	4	yes	complemented
139	C₄×M₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
140	C₆₇	#33708	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×#135	—	2	1	no	—
142	C₂×#135	C₂²×C₆₇	1	2	yes	—
143	#532	—	2	1	no	—
144	C₂×#125	—	2	1	no	—
145	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
146	#499	—	2	1	no	—
147	#499	—	2	1	no	—
148	#499	C₂²×C₆₇	1	2	yes	—
149	#499	C₂²×C₆₇	1	2	yes	—
150	#495	—	2	1	no	—
151	#495	C₂²×C₆₇	1	2	yes	—
152	#494	C₂²×C₆₇	1	1	yes	characteristic
153	#493	C₂²×C₆₇	1	1	yes	characteristic
154	#492	—	2	1	no	—
155	C₂×#107	C₂²×C₆₇	1	2	yes	—
156	C₂×#106	—	2	1	no	—
157	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
158	#358	C₂²×C₆₇	1	8	yes	—
159	#357	C₂²×C₆₇	1	8	yes	—
160	#346	C₂²×C₆₇	1	2	yes	—
161	C₄×C₁₆	C₂²×C₆₇	1	2	yes	—
162	C₂×#41	—	2	2	no	—
163	#135	—	2	8	no	—
164	#135	—	4	4	no	—
165	C₂×Q₁₆	—	2	1	no	—
166	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
167	C₂×SD₁₆	—	2	2	no	—
168	C₂×SD₁₆	—	2	2	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	C₂³×C₆₇	1	2	yes	—
171	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
173	C₄⋊Q₈	—	2	1	no	—
174	#127	—	2	1	no	—
175	#125	—	2	2	no	—
176	#125	—	2	2	no	—
177	#125	—	4	2	no	—
178	#124	—	2	2	no	—
179	#121	—	4	2	no	—
180	C₄×D₄	—	4	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	4	2	no	—
183	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
184	M₃₂	—	2	4	no	—
185	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
186	#107	—	2	2	no	—
187	#107	C₂³×C₆₇	1	4	yes	—
188	#106	—	2	2	no	—
189	#106	—	2	2	no	—
190	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
191	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
192	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
193	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
194	#41	—	4	4	no	—
195	#41	—	4	4	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	4	2	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	D₈×C₆₇	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
204	Q₁₆	—	2	4	no	—
205	Q₁₆	—	4	2	no	—
206	SD₁₆	—	4	4	no	—
207	SD₁₆	—	4	4	no	—
208	D₈	—	2	4	no	—
209	D₈	—	4	2	no	—
210	C₂×C₈	SD₁₆×C₆₇	1	4	yes	—
211	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
212	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
213	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
214	C₂×C₈	#41×C₆₇	1	4	yes	—
215	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	—	4	4	no	—
222	C₄²	D₈×C₆₇	1	2	yes	—
223	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
224	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
225	C₁₆	—	2	4	no	—
226	C₂³	—	4	2	no	—
227	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
228	Q₈	—	4	2	no	—
229	Q₈	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	8	4	no	—
233	D₄	—	8	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	—	8	4	no	—
239	C₂×C₄	#124×C₆₇	1	2	yes	—
240	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
241	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
242	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
243	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
244	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
245	C₈	#124×C₆₇	1	2	yes	—
246	C₈	C₂×SD₁₆×C₆₇	1	2	yes	—
247	C₈	C₂×SD₁₆×C₆₇	1	2	yes	—
248	C₈	C₂×#41×C₆₇	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	8	2	no	—
251	C₂²	—	8	4	no	—
252	C₂²	#486×C₆₇	1	2	yes	—
253	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	8	2	no	—
257	C₄	—	8	4	no	—
258	C₄	#486×C₆₇	1	2	yes	—
259	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
260	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	16	2	no	—
263	C₂	C₂×#486×C₆₇	1	1	yes	characteristic
264	C₂	C₆₇×#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₆₇×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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×#499×C₆₇	C₂	semidirect
3. C₆₇×#2245	C₂	semidirect
4. C₆₇×#2244	C₂	semidirect
5. C₆₇×#2242	C₂	semidirect
6. C₆₇×#2139	C₂	semidirect
7. C₄×M₃₂×C₆₇	C₂	semidirect
128. #33708	C₆₇	direct (N × H)
133. C₂×#499	C₂×C₆₇	semidirect
135. #2245	C₂×C₆₇	semidirect
136. #2244	C₂×C₆₇	semidirect
137. #2242	C₂×C₆₇	semidirect
138. #2139	C₂×C₆₇	semidirect
139. C₄×M₃₂	C₂×C₆₇	semidirect
140. C₆₇	#33708	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
C₆₇×#3147	2
C₆₇×#2245	2
C₆₇×#2244	2
C₆₇×#2242	2
C₆₇×#2139	2
C₄×M₃₂×C₆₇	2
#33708	67

Subgroups — C67×#33708 #30712959

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33708 #30712959