Subgroups — C₆₇×#33172 #30712423

Order 17152.

1142 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#467×C₆₇	C₂	1	4	yes	maximal
3	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2857	C₂	1	2	yes	maximal
5	C₆₇×#1969	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1956	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
9	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
11	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
12	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
13	#494×C₆₇	—	2	4	no	—
14	#467×C₆₇	—	2	16	no	—
15	C₂×#107×C₆₇	C₂²	1	4	yes	—
16	C₂×#103×C₆₇	C₂²	1	8	yes	—
17	C₂×C₄×C₈×C₆₇	—	2	4	no	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	#391×C₆₇	C₂²	1	4	yes	—
22	#389×C₆₇	—	2	4	no	—
23	#384×C₆₇	C₂²	1	2	yes	—
24	#380×C₆₇	—	2	4	no	—
25	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
26	C₂³×C₄×C₆₇	D₄	1	2	yes	—
27	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×Q₁₆×C₆₇	—	2	8	no	—
29	C₂×Q₁₆×C₆₇	—	2	8	no	—
30	C₂×Q₁₆×C₆₇	—	2	8	no	—
31	C₂²×C₈×C₆₇	—	2	4	no	—
32	C₂²×C₈×C₆₇	—	2	4	no	—
33	C₂²×C₈×C₆₇	D₄	1	8	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
36	C₄⋊Q₈×C₆₇	—	2	4	no	—
37	C₄⋊Q₈×C₆₇	—	2	4	no	—
38	C₂²⋊Q₈×C₆₇	—	2	16	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
43	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
44	C₂×C₄²×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	—	2	2	no	—
46	C₂×C₄²×C₆₇	D₄	1	2	yes	—
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	#107×C₆₇	—	2	8	no	—
50	#103×C₆₇	—	2	16	no	—
51	C₄×C₈×C₆₇	—	2	4	no	—
52	C₄×C₈×C₆₇	—	2	2	no	—
53	C₄×C₈×C₆₇	D₄	1	4	yes	—
54	#95×C₆₇	—	2	8	no	—
55	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×Q₈×C₆₇	—	2	8	no	—
57	C₂×Q₈×C₆₇	—	2	4	no	—
58	C₂×Q₈×C₆₇	—	2	4	no	—
59	C₂×Q₈×C₆₇	—	2	8	no	—
60	C₂²×C₄×C₆₇	—	2	4	no	—
61	C₂²×C₄×C₆₇	—	2	4	no	—
62	C₂²×C₄×C₆₇	—	2	1	no	—
63	C₂²×C₄×C₆₇	—	2	1	no	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	Q₁₆	1	8	yes	complemented
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
73	Q₁₆×C₆₇	—	4	16	no	—
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₂×C₈×C₆₇	—	2	2	no	—
76	C₂×C₈×C₆₇	—	2	16	no	—
77	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₆₇	#41	1	4	yes	—
81	C₄⋊C₄×C₆₇	—	2	8	no	—
82	C₄⋊C₄×C₆₇	—	4	8	no	—
83	C₄⋊C₄×C₆₇	—	4	8	no	—
84	C₄⋊C₄×C₆₇	—	4	8	no	—
85	C₂²⋊C₄×C₆₇	—	4	8	no	—
86	C₄²×C₆₇	—	2	4	no	—
87	C₄²×C₆₇	—	2	1	no	—
88	C₄²×C₆₇	—	2	1	no	—
89	C₄²×C₆₇	C₂×D₄	1	2	yes	—
90	C₄²×C₆₇	#41	1	2	yes	—
91	C₂³×C₆₇	—	2	1	no	—
92	C₂³×C₆₇	—	2	1	no	—
93	C₂³×C₆₇	—	2	2	no	—
94	C₂³×C₆₇	#127	1	2	yes	—
95	C₂³×C₆₇	C₂×Q₁₆	1	4	yes	—
96	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₆₇	—	4	8	no	—
98	Q₈×C₆₇	—	4	8	no	—
99	C₂×C₄×C₆₇	—	2	1	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	1	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	2	16	no	—
106	C₂×C₄×C₆₇	—	4	4	no	—
107	C₂×C₄×C₆₇	—	4	4	no	—
108	C₂×C₄×C₆₇	—	4	4	no	—
109	C₂×C₄×C₆₇	—	4	4	no	—
110	C₂×C₄×C₆₇	—	4	8	no	—
111	C₂×C₄×C₆₇	#121	1	4	yes	—
112	C₂×C₄×C₆₇	#127	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
114	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
115	C₂×C₄×C₆₇	#135	1	4	yes	—
116	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
118	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
119	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
120	C₈×C₆₇	—	2	4	no	—
121	C₈×C₆₇	#121	1	8	yes	—
122	C₂²×C₆₇	—	2	2	no	—
123	C₂²×C₆₇	—	2	2	no	—
124	C₂²×C₆₇	—	2	4	no	—
125	C₂²×C₆₇	—	2	4	no	—
126	C₂²×C₆₇	#494	1	4	yes	complemented
127	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
128	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
129	C₂²×C₆₇	#532	1	1	yes	characteristic
130	C₂²×C₆₇	C₂²×Q₁₆	1	2	yes	—
131	C₂²×C₆₇	C₂×#135	1	2	yes	—
132	C₄×C₆₇	—	2	2	no	—
133	C₄×C₆₇	—	2	4	no	—
134	C₄×C₆₇	—	8	4	no	—
135	C₄×C₆₇	—	8	4	no	—
136	C₄×C₆₇	#467	1	8	yes	complemented
137	C₄×C₆₇	C₂×#121	1	2	yes	—
138	C₄×C₆₇	#532	1	2	yes	—
139	#33172	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂×C₆₇	—	2	4	no	—
141	C₂×C₆₇	#2856	1	1	yes	characteristic
142	C₂×C₆₇	C₂×#467	1	2	yes	—
143	C₂×C₆₇	C₂×#494	1	1	yes	characteristic
144	C₂×C₆₇	#3147	1	1	yes	characteristic
145	C₂×C₆₇	#3170	1	2	yes	—
146	C₂×#494	C₂×C₆₇	1	1	yes	characteristic, complemented
147	C₂×#467	C₂×C₆₇	1	4	yes	—
148	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
149	#2857	C₂×C₆₇	1	2	yes	—
150	#1969	C₂×C₆₇	1	4	yes	complemented
151	#1956	C₂×C₆₇	1	2	yes	complemented
152	#1845	C₂×C₆₇	1	1	yes	characteristic, complemented
153	C₆₇	#33172	1	1	yes	characteristic, direct product factor, Sylow
154	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	—
155	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
156	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
157	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
158	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
159	#494	—	2	4	no	—
160	#467	—	2	16	no	—
161	C₂×#107	C₂²×C₆₇	1	4	yes	—
162	C₂×#103	C₂²×C₆₇	1	8	yes	—
163	C₂×C₄×C₈	—	2	4	no	—
164	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
166	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
167	#391	C₂²×C₆₇	1	4	yes	—
168	#389	—	2	4	no	—
169	#384	C₂²×C₆₇	1	2	yes	—
170	#380	—	2	4	no	—
171	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
172	C₂³×C₄	D₄×C₆₇	1	2	yes	—
173	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
174	C₂×Q₁₆	—	2	8	no	—
175	C₂×Q₁₆	—	2	8	no	—
176	C₂×Q₁₆	—	2	8	no	—
177	C₂²×C₈	—	2	4	no	—
178	C₂²×C₈	—	2	4	no	—
179	C₂²×C₈	D₄×C₆₇	1	8	yes	—
180	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
181	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
182	C₄⋊Q₈	—	2	4	no	—
183	C₄⋊Q₈	—	2	4	no	—
184	C₂²⋊Q₈	—	2	16	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	—	2	4	no	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
189	C₂×C₂²⋊C₄	—	2	4	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	D₄×C₆₇	1	2	yes	—
193	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
195	#107	—	2	8	no	—
196	#103	—	2	16	no	—
197	C₄×C₈	—	2	4	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	D₄×C₆₇	1	4	yes	—
200	#95	—	2	8	no	—
201	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
202	C₂×Q₈	—	2	8	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	2	4	no	—
205	C₂×Q₈	—	2	8	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	Q₁₆×C₆₇	1	8	yes	complemented
214	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
216	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
218	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
219	Q₁₆	—	4	16	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	16	no	—
223	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
224	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
225	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
226	C₂×C₈	#41×C₆₇	1	4	yes	—
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	4	8	no	—
229	C₄⋊C₄	—	4	8	no	—
230	C₄⋊C₄	—	4	8	no	—
231	C₂²⋊C₄	—	4	8	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	C₂×D₄×C₆₇	1	2	yes	—
236	C₄²	#41×C₆₇	1	2	yes	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	2	no	—
240	C₂³	#127×C₆₇	1	2	yes	—
241	C₂³	C₂×Q₁₆×C₆₇	1	4	yes	—
242	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
243	Q₈	—	4	8	no	—
244	Q₈	—	4	8	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	#121×C₆₇	1	4	yes	—
258	C₂×C₄	#127×C₆₇	1	2	yes	—
259	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	—
260	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	—
261	C₂×C₄	#135×C₆₇	1	4	yes	—
262	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
264	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
265	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₈	#121×C₆₇	1	8	yes	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	4	no	—
272	C₂²	#494×C₆₇	1	4	yes	complemented
273	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
274	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
275	C₂²	#532×C₆₇	1	1	yes	characteristic
276	C₂²	C₂²×Q₁₆×C₆₇	1	2	yes	—
277	C₂²	C₂×#135×C₆₇	1	2	yes	—
278	C₄	—	2	4	no	—
279	C₄	—	2	2	no	—
280	C₄	—	8	4	no	—
281	C₄	—	8	4	no	—
282	C₄	#467×C₆₇	1	8	yes	complemented
283	C₄	C₂×#121×C₆₇	1	2	yes	—
284	C₄	#532×C₆₇	1	2	yes	—
285	C₂	—	2	4	no	—
286	C₂	C₆₇×#2856	1	1	yes	characteristic
287	C₂	C₂×#467×C₆₇	1	2	yes	—
288	C₂	C₂×#494×C₆₇	1	1	yes	characteristic
289	C₂	C₆₇×#3147	1	1	yes	characteristic
290	C₂	C₆₇×#3170	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
1. C₂×#494×C₆₇	C₂	semidirect
5. C₆₇×#1969	C₂	semidirect
6. C₆₇×#1956	C₂	semidirect
7. C₆₇×#1845	C₂	semidirect
67. C₂²×C₄×C₆₇	Q₁₆	semidirect
126. C₂²×C₆₇	#494	semidirect
136. C₄×C₆₇	#467	semidirect
139. #33172	C₆₇	direct (N × H)
146. C₂×#494	C₂×C₆₇	semidirect
150. #1969	C₂×C₆₇	semidirect
151. #1956	C₂×C₆₇	semidirect
152. #1845	C₂×C₆₇	semidirect
153. C₆₇	#33172	direct (N × H)
213. C₂²×C₄	Q₁₆×C₆₇	semidirect
272. C₂²	#494×C₆₇	semidirect
282. C₄	#467×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#494×C₆₇	2
C₂×#467×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₆₇×#2857	2
C₆₇×#1969	2
C₆₇×#1956	2
C₆₇×#1845	2
#33172	67

Subgroups — C67×#33172 #30712423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33172 #30712423