Subgroups — C₆₇×#33277 #30712528

Order 17152.

858 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2859	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2858	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1990	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1957	C₂	1	2	yes	maximal
5	C₆₇×#1934	C₂	1	8	yes	maximal, complemented
6	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#125×C₆₇	—	2	1	no	—
10	C₂×#124×C₆₇	—	2	1	no	—
11	C₂×#124×C₆₇	—	2	1	no	—
12	C₂×#102×C₆₇	C₂²	1	8	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
15	#390×C₆₇	—	2	2	no	—
16	#390×C₆₇	C₂²	1	4	yes	—
17	#389×C₆₇	—	2	2	no	—
18	#389×C₆₇	C₂²	1	4	yes	—
19	#384×C₆₇	C₂²	1	2	yes	—
20	#382×C₆₇	—	2	1	no	—
21	C₄³×C₆₇	C₂²	1	1	yes	characteristic
22	#340×C₆₇	C₂²	1	8	yes	—
23	C₂²×Q₈×C₆₇	—	2	1	no	—
24	C₂²×D₄×C₆₇	—	2	1	no	—
25	C₂²×D₄×C₆₇	—	2	1	no	—
26	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
27	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
28	C₄⋊Q₈×C₆₇	—	2	4	no	—
29	#127×C₆₇	—	2	4	no	—
30	#125×C₆₇	—	4	4	no	—
31	#124×C₆₇	—	4	4	no	—
32	#124×C₆₇	—	4	4	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
38	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
39	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
40	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
41	C₂×C₄²×C₆₇	D₄	1	2	yes	—
42	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	#102×C₆₇	—	2	16	no	—
47	C₄×C₈×C₆₇	—	2	8	no	—
48	C₄×C₈×C₆₇	—	2	4	no	—
49	#95×C₆₇	—	2	4	no	—
50	C₂⁴×C₆₇	—	2	2	no	—
51	C₂×Q₈×C₆₇	—	4	2	no	—
52	C₂×Q₈×C₆₇	—	4	2	no	—
53	C₂×Q₈×C₆₇	—	4	2	no	—
54	C₂×D₄×C₆₇	—	2	8	no	—
55	C₂×D₄×C₆₇	—	2	4	no	—
56	C₂×D₄×C₆₇	—	4	2	no	—
57	C₂×D₄×C₆₇	—	4	2	no	—
58	C₂×D₄×C₆₇	—	4	2	no	—
59	C₂×D₄×C₆₇	—	4	2	no	—
60	C₂×D₄×C₆₇	—	4	4	no	—
61	C₂²×C₄×C₆₇	—	2	2	no	—
62	C₂²×C₄×C₆₇	—	2	4	no	—
63	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₆₇	—	2	4	no	—
69	C₂×C₈×C₆₇	—	2	4	no	—
70	C₂×C₈×C₆₇	#41	1	8	yes	—
71	C₄⋊C₄×C₆₇	—	2	8	no	—
72	C₄⋊C₄×C₆₇	—	2	4	no	—
73	C₄⋊C₄×C₆₇	—	4	2	no	—
74	C₄⋊C₄×C₆₇	—	4	8	no	—
75	C₄⋊C₄×C₆₇	—	4	4	no	—
76	C₄⋊C₄×C₆₇	—	4	4	no	—
77	C₂²⋊C₄×C₆₇	—	4	8	no	—
78	C₂²⋊C₄×C₆₇	—	4	8	no	—
79	C₄²×C₆₇	—	2	4	no	—
80	C₄²×C₆₇	—	2	4	no	—
81	C₄²×C₆₇	D₈	1	4	yes	—
82	C₄²×C₆₇	SD₁₆	1	4	yes	—
83	C₄²×C₆₇	#41	1	4	yes	—
84	C₂³×C₆₇	—	4	2	no	—
85	C₂³×C₆₇	—	4	4	no	—
86	C₂³×C₆₇	—	4	2	no	—
87	C₂³×C₆₇	—	4	2	no	—
88	C₂³×C₆₇	—	4	4	no	—
89	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
90	Q₈×C₆₇	—	8	4	no	—
91	D₄×C₆₇	—	4	8	no	—
92	D₄×C₆₇	—	8	4	no	—
93	D₄×C₆₇	—	8	4	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	2	no	—
97	C₂×C₄×C₆₇	—	2	2	no	—
98	C₂×C₄×C₆₇	—	4	2	no	—
99	C₂×C₄×C₆₇	—	4	4	no	—
100	C₂×C₄×C₆₇	—	4	2	no	—
101	C₂×C₄×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	—	4	2	no	—
103	C₂×C₄×C₆₇	—	4	4	no	—
104	C₂×C₄×C₆₇	#124	1	2	yes	—
105	C₂×C₄×C₆₇	#124	1	2	yes	—
106	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
107	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
109	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
110	C₂×C₄×C₆₇	#135	1	4	yes	—
111	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
112	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
113	C₈×C₆₇	—	2	8	no	—
114	C₂²×C₆₇	—	8	2	no	—
115	C₂²×C₆₇	—	8	2	no	—
116	C₂²×C₆₇	—	8	2	no	—
117	C₂²×C₆₇	—	8	4	no	—
118	C₂²×C₆₇	—	8	4	no	—
119	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
120	C₂²×C₆₇	#532	1	2	yes	—
121	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
122	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
123	C₂²×C₆₇	C₂×#135	1	2	yes	—
124	C₄×C₆₇	—	2	4	no	—
125	C₄×C₆₇	—	2	4	no	—
126	C₄×C₆₇	—	8	2	no	—
127	C₄×C₆₇	—	8	4	no	—
128	C₄×C₆₇	#486	1	4	yes	—
129	C₄×C₆₇	#488	1	4	yes	—
130	C₄×C₆₇	#529	1	4	yes	—
131	#33277	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂×C₆₇	—	16	2	no	—
133	C₂×C₆₇	#2861	1	1	yes	characteristic
134	C₂×C₆₇	C₂×#486	1	1	yes	characteristic
135	C₂×C₆₇	C₂×#488	1	1	yes	characteristic
136	C₂×C₆₇	#3167	1	2	yes	—
137	C₂×C₆₇	#3169	1	2	yes	—
138	#2859	C₂×C₆₇	1	1	yes	characteristic, complemented
139	#2858	C₂×C₆₇	1	1	yes	characteristic
140	#1990	C₂×C₆₇	1	2	yes	complemented
141	#1957	C₂×C₆₇	1	2	yes	—
142	#1934	C₂×C₆₇	1	8	yes	complemented
143	C₄²×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
144	C₆₇	#33277	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
146	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
147	C₂×#125	—	2	1	no	—
148	C₂×#124	—	2	1	no	—
149	C₂×#124	—	2	1	no	—
150	C₂×#102	C₂²×C₆₇	1	8	yes	—
151	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
152	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
153	#390	—	2	2	no	—
154	#390	C₂²×C₆₇	1	4	yes	—
155	#389	—	2	2	no	—
156	#389	C₂²×C₆₇	1	4	yes	—
157	#384	C₂²×C₆₇	1	2	yes	—
158	#382	—	2	1	no	—
159	C₄³	C₂²×C₆₇	1	1	yes	characteristic
160	#340	C₂²×C₆₇	1	8	yes	—
161	C₂²×Q₈	—	2	1	no	—
162	C₂²×D₄	—	2	1	no	—
163	C₂²×D₄	—	2	1	no	—
164	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
165	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
166	C₄⋊Q₈	—	2	4	no	—
167	#127	—	2	4	no	—
168	#125	—	4	4	no	—
169	#124	—	4	4	no	—
170	#124	—	4	4	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	4	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
176	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
177	C₂×C₂²⋊C₄	—	2	4	no	—
178	C₂×C₂²⋊C₄	—	2	4	no	—
179	C₂×C₄²	D₄×C₆₇	1	2	yes	—
180	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
181	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
182	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
183	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
184	#102	—	2	16	no	—
185	C₄×C₈	—	2	8	no	—
186	C₄×C₈	—	2	4	no	—
187	#95	—	2	4	no	—
188	C₂⁴	—	2	2	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	—	4	2	no	—
192	C₂×D₄	—	2	8	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂×D₄	—	4	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
202	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
205	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	#41×C₆₇	1	8	yes	—
209	C₄⋊C₄	—	2	8	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	8	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₂²⋊C₄	—	4	8	no	—
216	C₂²⋊C₄	—	4	8	no	—
217	C₄²	—	2	4	no	—
218	C₄²	—	2	4	no	—
219	C₄²	D₈×C₆₇	1	4	yes	—
220	C₄²	SD₁₆×C₆₇	1	4	yes	—
221	C₄²	#41×C₆₇	1	4	yes	—
222	C₂³	—	4	2	no	—
223	C₂³	—	4	4	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	—	4	4	no	—
227	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
228	Q₈	—	8	4	no	—
229	D₄	—	4	8	no	—
230	D₄	—	8	4	no	—
231	D₄	—	8	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	#124×C₆₇	1	2	yes	—
243	C₂×C₄	#124×C₆₇	1	2	yes	—
244	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
245	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
246	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
247	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
248	C₂×C₄	#135×C₆₇	1	4	yes	—
249	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
250	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
251	C₈	—	2	8	no	—
252	C₂²	—	8	2	no	—
253	C₂²	—	8	2	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	8	4	no	—
256	C₂²	—	8	4	no	—
257	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
258	C₂²	#532×C₆₇	1	2	yes	—
259	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
260	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
261	C₂²	C₂×#135×C₆₇	1	2	yes	—
262	C₄	—	2	4	no	—
263	C₄	—	2	4	no	—
264	C₄	—	8	2	no	—
265	C₄	—	8	4	no	—
266	C₄	#486×C₆₇	1	4	yes	—
267	C₄	#488×C₆₇	1	4	yes	—
268	C₄	#529×C₆₇	1	4	yes	—
269	C₂	—	16	2	no	—
270	C₂	C₆₇×#2861	1	1	yes	characteristic
271	C₂	C₂×#486×C₆₇	1	1	yes	characteristic
272	C₂	C₂×#488×C₆₇	1	1	yes	characteristic
273	C₂	C₆₇×#3167	1	2	yes	—
274	C₂	C₆₇×#3169	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2859	C₂	semidirect
3. C₆₇×#1990	C₂	semidirect
5. C₆₇×#1934	C₂	semidirect
6. C₄²×C₈×C₆₇	C₂	semidirect
131. #33277	C₆₇	direct (N × H)
138. #2859	C₂×C₆₇	semidirect
140. #1990	C₂×C₆₇	semidirect
142. #1934	C₂×C₆₇	semidirect
143. C₄²×C₈	C₂×C₆₇	semidirect
144. C₆₇	#33277	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×#2859	2
C₆₇×#2858	2
C₆₇×#1990	2
C₆₇×#1957	2
C₆₇×#1934	2
C₄²×C₈×C₆₇	2
#33277	67

Subgroups — C67×#33277 #30712528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33277 #30712528