Subgroups — C₆₇×#25910 #30705161

Order 17152.

1066 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#491×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2024	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1992	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1873	C₂	1	4	yes	maximal
7	C₆₇×#1749	C₂	1	1	yes	maximal, characteristic
8	#533×C₆₇	—	2	8	no	—
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
11	C₂×#125×C₆₇	C₂²	1	2	yes	—
12	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
13	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
15	C₂×#117×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	#491×C₆₇	—	2	8	no	—
18	C₂×#103×C₆₇	C₂²	1	8	yes	—
19	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×#97×C₆₇	C₂²	1	2	yes	—
21	#391×C₆₇	C₂²	1	2	yes	—
22	#389×C₆₇	—	2	2	no	—
23	#384×C₆₇	C₂²	1	1	yes	characteristic
24	#380×C₆₇	—	2	2	no	—
25	#340×C₆₇	C₂²	1	8	yes	—
26	#336×C₆₇	C₂²	1	2	yes	—
27	C₂²×Q₈×C₆₇	—	2	2	no	—
28	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
29	C₂³×C₄×C₆₇	D₄	1	2	yes	—
30	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
32	C₄⋊Q₈×C₆₇	—	2	8	no	—
33	C₄⋊Q₈×C₆₇	—	2	4	no	—
34	#125×C₆₇	—	2	4	no	—
35	#125×C₆₇	D₄	1	8	yes	—
36	C₂²⋊Q₈×C₆₇	—	2	16	no	—
37	C₂²⋊Q₈×C₆₇	—	2	8	no	—
38	C₄×Q₈×C₆₇	—	2	16	no	—
39	#117×C₆₇	—	2	8	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
47	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₄²×C₆₇	—	2	2	no	—
50	C₂×C₄²×C₆₇	—	2	4	no	—
51	C₂×C₄²×C₆₇	D₄	1	2	yes	—
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	#103×C₆₇	—	2	16	no	—
55	C₂²⋊C₈×C₆₇	—	2	4	no	—
56	#97×C₆₇	—	2	4	no	—
57	#95×C₆₇	—	2	4	no	—
58	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂×Q₈×C₆₇	—	2	8	no	—
60	C₂×Q₈×C₆₇	—	2	4	no	—
61	C₂×Q₈×C₆₇	—	2	4	no	—
62	C₂×Q₈×C₆₇	—	2	4	no	—
63	C₂×Q₈×C₆₇	—	4	4	no	—
64	C₂×Q₈×C₆₇	—	4	4	no	—
65	C₂²×C₄×C₆₇	—	2	2	no	—
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	4	2	no	—
73	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₆₇	#41	1	4	yes	—
77	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₆₇	—	2	4	no	—
79	C₂×C₈×C₆₇	—	2	4	no	—
80	C₂×C₈×C₆₇	—	2	4	no	—
81	C₄⋊C₄×C₆₇	—	2	8	no	—
82	C₄⋊C₄×C₆₇	—	2	16	no	—
83	C₄⋊C₄×C₆₇	—	2	4	no	—
84	C₄⋊C₄×C₆₇	—	2	4	no	—
85	C₄⋊C₄×C₆₇	—	4	4	no	—
86	C₄⋊C₄×C₆₇	—	4	4	no	—
87	C₄⋊C₄×C₆₇	—	4	4	no	—
88	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
89	C₂²⋊C₄×C₆₇	—	2	8	no	—
90	C₂²⋊C₄×C₆₇	—	4	4	no	—
91	C₄²×C₆₇	—	2	8	no	—
92	C₄²×C₆₇	—	2	4	no	—
93	C₄²×C₆₇	—	2	2	no	—
94	C₄²×C₆₇	C₂×D₄	1	4	yes	—
95	C₂³×C₆₇	—	2	1	no	—
96	C₂³×C₆₇	—	2	1	no	—
97	C₂³×C₆₇	—	2	2	no	—
98	C₂³×C₆₇	—	2	2	no	—
99	C₂³×C₆₇	#124	1	2	yes	—
100	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
101	Q₈×C₆₇	—	4	16	no	—
102	Q₈×C₆₇	—	4	8	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	2	1	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	8	no	—
110	C₂×C₄×C₆₇	—	2	8	no	—
111	C₂×C₄×C₆₇	—	4	8	no	—
112	C₂×C₄×C₆₇	—	4	4	no	—
113	C₂×C₄×C₆₇	—	4	4	no	—
114	C₂×C₄×C₆₇	—	4	2	no	—
115	C₂×C₄×C₆₇	—	4	2	no	—
116	C₂×C₄×C₆₇	—	4	2	no	—
117	C₂×C₄×C₆₇	—	4	2	no	—
118	C₂×C₄×C₆₇	—	4	4	no	—
119	C₂×C₄×C₆₇	#120	1	4	yes	—
120	C₂×C₄×C₆₇	#124	1	2	yes	—
121	C₂×C₄×C₆₇	#137	1	4	yes	—
122	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
123	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
124	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
125	C₈×C₆₇	—	4	4	no	—
126	C₂²×C₆₇	—	2	2	no	—
127	C₂²×C₆₇	—	2	2	no	—
128	C₂²×C₆₇	—	2	2	no	—
129	C₂²×C₆₇	—	4	2	no	—
130	C₂²×C₆₇	—	4	2	no	—
131	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
132	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
133	C₂²×C₆₇	#536	1	1	yes	characteristic
134	C₂²×C₆₇	C₂×#137	1	2	yes	—
135	C₂²×C₆₇	#575	1	2	yes	—
136	C₄×C₆₇	—	2	2	no	—
137	C₄×C₆₇	—	2	4	no	—
138	C₄×C₆₇	—	2	4	no	—
139	C₄×C₆₇	—	4	8	no	—
140	C₄×C₆₇	—	8	2	no	—
141	C₄×C₆₇	—	8	2	no	—
142	C₄×C₆₇	#548	1	4	yes	—
143	#25910	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂×C₆₇	—	4	2	no	—
145	C₂×C₆₇	#2436	1	1	yes	characteristic
146	C₂×C₆₇	#3007	1	2	yes	—
147	C₂×C₆₇	#3135	1	1	yes	characteristic
148	C₂×C₆₇	#3139	1	1	yes	characteristic
149	C₂×C₆₇	#3237	1	2	yes	—
150	C₂×#533	C₂×C₆₇	1	1	yes	characteristic
151	C₂×#491	C₂×C₆₇	1	2	yes	complemented
152	#2857	C₂×C₆₇	1	1	yes	characteristic
153	#2024	C₂×C₆₇	1	4	yes	complemented
154	#1992	C₂×C₆₇	1	2	yes	complemented
155	#1873	C₂×C₆₇	1	4	yes	—
156	#1749	C₂×C₆₇	1	1	yes	characteristic
157	C₆₇	#25910	1	1	yes	characteristic, direct product factor, Sylow
158	#533	—	2	8	no	—
159	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
160	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
161	C₂×#125	C₂²×C₆₇	1	2	yes	—
162	C₂×C₂²⋊Q₈	—	2	2	no	—
163	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
164	C₂×C₄×Q₈	—	2	2	no	—
165	C₂×#117	C₂²×C₆₇	1	2	yes	—
166	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
167	#491	—	2	8	no	—
168	C₂×#103	C₂²×C₆₇	1	8	yes	—
169	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
170	C₂×#97	C₂²×C₆₇	1	2	yes	—
171	#391	C₂²×C₆₇	1	2	yes	—
172	#389	—	2	2	no	—
173	#384	C₂²×C₆₇	1	1	yes	characteristic
174	#380	—	2	2	no	—
175	#340	C₂²×C₆₇	1	8	yes	—
176	#336	C₂²×C₆₇	1	2	yes	—
177	C₂²×Q₈	—	2	2	no	—
178	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
179	C₂³×C₄	D₄×C₆₇	1	2	yes	—
180	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
182	C₄⋊Q₈	—	2	8	no	—
183	C₄⋊Q₈	—	2	4	no	—
184	#125	—	2	4	no	—
185	#125	D₄×C₆₇	1	8	yes	—
186	C₂²⋊Q₈	—	2	16	no	—
187	C₂²⋊Q₈	—	2	8	no	—
188	C₄×Q₈	—	2	16	no	—
189	#117	—	2	8	no	—
190	C₂×C₄⋊C₄	—	2	4	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
196	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₂²⋊C₄	—	2	2	no	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	2	4	no	—
201	C₂×C₄²	D₄×C₆₇	1	2	yes	—
202	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
203	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
204	#103	—	2	16	no	—
205	C₂²⋊C₈	—	2	4	no	—
206	#97	—	2	4	no	—
207	#95	—	2	4	no	—
208	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
209	C₂×Q₈	—	2	8	no	—
210	C₂×Q₈	—	2	4	no	—
211	C₂×Q₈	—	2	4	no	—
212	C₂×Q₈	—	2	4	no	—
213	C₂×Q₈	—	4	4	no	—
214	C₂×Q₈	—	4	4	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
226	C₂²×C₄	#41×C₆₇	1	4	yes	—
227	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	2	16	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
239	C₂²⋊C₄	—	2	8	no	—
240	C₂²⋊C₄	—	4	4	no	—
241	C₄²	—	2	8	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	2	no	—
244	C₄²	C₂×D₄×C₆₇	1	4	yes	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	#124×C₆₇	1	2	yes	—
250	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
251	Q₈	—	4	16	no	—
252	Q₈	—	4	8	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	4	8	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	#120×C₆₇	1	4	yes	—
270	C₂×C₄	#124×C₆₇	1	2	yes	—
271	C₂×C₄	#137×C₆₇	1	4	yes	—
272	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
273	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
274	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
275	C₈	—	4	4	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
282	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
283	C₂²	#536×C₆₇	1	1	yes	characteristic
284	C₂²	C₂×#137×C₆₇	1	2	yes	—
285	C₂²	#575×C₆₇	1	2	yes	—
286	C₄	—	2	2	no	—
287	C₄	—	2	4	no	—
288	C₄	—	2	4	no	—
289	C₄	—	4	8	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	2	no	—
292	C₄	#548×C₆₇	1	4	yes	—
293	C₂	—	4	2	no	—
294	C₂	C₆₇×#2436	1	1	yes	characteristic
295	C₂	C₆₇×#3007	1	2	yes	—
296	C₂	C₆₇×#3135	1	1	yes	characteristic
297	C₂	C₆₇×#3139	1	1	yes	characteristic
298	C₂	C₆₇×#3237	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#491×C₆₇	C₂	semidirect
4. C₆₇×#2024	C₂	semidirect
5. C₆₇×#1992	C₂	semidirect
143. #25910	C₆₇	direct (N × H)
151. C₂×#491	C₂×C₆₇	semidirect
153. #2024	C₂×C₆₇	semidirect
154. #1992	C₂×C₆₇	semidirect
157. C₆₇	#25910	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×#533×C₆₇	2
C₂×#491×C₆₇	2
C₆₇×#2857	2
C₆₇×#2024	2
C₆₇×#1992	2
C₆₇×#1873	2
C₆₇×#1749	2
#25910	67

Subgroups — C67×#25910 #30705161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25910 #30705161