Subgroups — C₆₇×#32932 #30712183

Order 17152.

1436 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#465×C₆₇	C₂	1	3	yes	maximal, complemented
3	C₆₇×#2616	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2615	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1895	C₂	1	3	yes	maximal, complemented
6	C₆₇×#1894	C₂	1	3	yes	maximal, complemented
7	C₆₇×#1807	C₂	1	3	yes	maximal, complemented
8	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#131×C₆₇	—	2	12	no	—
11	C₂²×M₁₆×C₆₇	C₂²	1	3	yes	—
12	C₂³×C₈×C₆₇	C₂²	1	3	yes	—
13	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	3	yes	—
14	C₂×#121×C₆₇	C₂²	1	3	yes	—
15	#465×C₆₇	—	2	12	no	—
16	C₂×#106×C₆₇	C₂²	1	3	yes	—
17	C₂×#103×C₆₇	C₂²	1	3	yes	—
18	C₂×#102×C₆₇	C₂²	1	3	yes	—
19	#395×C₆₇	—	2	1	no	—
20	#394×C₆₇	—	2	3	no	—
21	#393×C₆₇	—	2	1	no	—
22	#392×C₆₇	—	2	3	no	—
23	#380×C₆₇	C₂²	1	3	yes	—
24	#380×C₆₇	C₂²	1	3	yes	—
25	#340×C₆₇	C₂²	1	6	yes	—
26	C₂×#41×C₆₇	—	2	6	no	—
27	C₂×#41×C₆₇	—	2	6	no	—
28	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
29	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
30	C₂²×D₄×C₆₇	—	2	3	no	—
31	C₂²×D₄×C₆₇	D₄	1	3	yes	—
32	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
34	C₂×SD₁₆×C₆₇	—	2	6	no	—
35	#131×C₆₇	—	2	32	no	—
36	C₂×M₁₆×C₆₇	—	2	6	no	—
37	C₂×M₁₆×C₆₇	—	2	12	no	—
38	C₂²×C₈×C₆₇	—	2	12	no	—
39	C₂²×C₈×C₆₇	D₄	1	12	yes	complemented
40	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²⋊Q₈×C₆₇	—	2	6	no	—
43	C₂²⋊Q₈×C₆₇	—	2	6	no	—
44	#121×C₆₇	—	2	6	no	—
45	#121×C₆₇	—	2	6	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
47	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
48	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
49	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
50	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
51	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
52	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
53	#106×C₆₇	—	2	6	no	—
54	#103×C₆₇	—	2	6	no	—
55	#102×C₆₇	—	2	6	no	—
56	#95×C₆₇	—	2	3	no	—
57	#95×C₆₇	—	2	3	no	—
58	C₂⁴×C₆₇	—	2	1	no	—
59	C₂⁴×C₆₇	C₂×D₄	1	3	yes	—
60	#41×C₆₇	—	2	8	no	—
61	#41×C₆₇	—	2	24	no	—
62	C₂×Q₈×C₆₇	—	2	6	no	—
63	C₂×Q₈×C₆₇	—	2	3	no	—
64	C₂×Q₈×C₆₇	—	2	3	no	—
65	C₂×D₄×C₆₇	—	2	12	no	—
66	C₂×D₄×C₆₇	—	2	6	no	—
67	C₂×D₄×C₆₇	—	2	3	no	—
68	C₂×D₄×C₆₇	—	2	3	no	—
69	C₂×D₄×C₆₇	—	4	6	no	—
70	C₂×D₄×C₆₇	—	8	6	no	—
71	C₂²×C₄×C₆₇	—	2	6	no	—
72	C₂²×C₄×C₆₇	—	2	6	no	—
73	C₂²×C₄×C₆₇	—	2	1	no	—
74	C₂²×C₄×C₆₇	—	2	3	no	—
75	C₂²×C₄×C₆₇	—	2	3	no	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
78	C₂²×C₄×C₆₇	#41	1	6	yes	—
79	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
80	SD₁₆×C₆₇	—	4	4	no	—
81	M₁₆×C₆₇	—	2	24	no	—
82	C₂×C₈×C₆₇	—	2	6	no	—
83	C₂×C₈×C₆₇	—	2	24	no	—
84	C₂×C₈×C₆₇	C₂×D₄	1	6	yes	—
85	C₂×C₈×C₆₇	C₂×D₄	1	6	yes	—
86	C₄⋊C₄×C₆₇	—	2	3	no	—
87	C₄⋊C₄×C₆₇	—	2	3	no	—
88	C₄⋊C₄×C₆₇	—	2	3	no	—
89	C₄⋊C₄×C₆₇	—	2	3	no	—
90	C₄⋊C₄×C₆₇	—	4	6	no	—
91	C₂²⋊C₄×C₆₇	—	4	6	no	—
92	C₂²⋊C₄×C₆₇	—	4	6	no	—
93	C₂²⋊C₄×C₆₇	—	8	3	no	—
94	C₂²⋊C₄×C₆₇	—	8	3	no	—
95	C₂³×C₆₇	—	2	6	no	—
96	C₂³×C₆₇	—	2	3	no	—
97	C₂³×C₆₇	—	2	3	no	—
98	C₂³×C₆₇	—	2	6	no	—
99	C₂³×C₆₇	—	4	3	no	—
100	C₂³×C₆₇	—	4	3	no	—
101	C₂³×C₆₇	—	4	1	no	—
102	C₂³×C₆₇	#121	1	6	yes	—
103	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₆₇	—	2	8	no	—
105	Q₈×C₆₇	—	4	1	no	—
106	Q₈×C₆₇	—	4	3	no	—
107	D₄×C₆₇	—	2	24	no	—
108	D₄×C₆₇	—	4	1	no	—
109	D₄×C₆₇	—	4	3	no	—
110	D₄×C₆₇	—	8	12	no	—
111	C₂×C₄×C₆₇	—	2	6	no	—
112	C₂×C₄×C₆₇	—	2	6	no	—
113	C₂×C₄×C₆₇	—	2	6	no	—
114	C₂×C₄×C₆₇	—	2	12	no	—
115	C₂×C₄×C₆₇	—	4	3	no	—
116	C₂×C₄×C₆₇	—	4	3	no	—
117	C₂×C₄×C₆₇	—	4	3	no	—
118	C₂×C₄×C₆₇	—	4	3	no	—
119	C₂×C₄×C₆₇	—	4	6	no	—
120	C₂×C₄×C₆₇	—	4	3	no	—
121	C₂×C₄×C₆₇	#121	1	6	yes	—
122	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
123	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
124	C₈×C₆₇	—	2	6	no	—
125	C₈×C₆₇	#120	1	4	yes	—
126	C₂²×C₆₇	—	2	6	no	—
127	C₂²×C₆₇	—	2	6	no	—
128	C₂²×C₆₇	—	2	6	no	—
129	C₂²×C₆₇	—	4	6	no	—
130	C₂²×C₆₇	—	4	6	no	—
131	C₂²×C₆₇	—	8	1	no	—
132	C₂²×C₆₇	—	8	3	no	—
133	C₂²×C₆₇	—	8	3	no	—
134	C₂²×C₆₇	C₂×#121	1	3	yes	—
135	C₂²×C₆₇	#577	1	3	yes	—
136	C₂²×C₆₇	#577	1	1	yes	characteristic
137	C₄×C₆₇	—	2	6	no	—
138	C₄×C₆₇	—	8	1	no	—
139	C₄×C₆₇	—	8	3	no	—
140	C₄×C₆₇	—	8	3	no	—
141	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
142	C₄×C₆₇	#525	1	3	yes	—
143	#32932	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂×C₆₇	—	4	6	no	—
145	C₂×C₆₇	—	16	1	no	—
146	C₂×C₆₇	#2847	1	1	yes	characteristic
147	C₂×C₆₇	#3057	1	3	yes	—
148	C₂×C₆₇	#3060	1	3	yes	—
149	C₂²×#131	C₂×C₆₇	1	1	yes	characteristic, complemented
150	C₂×#465	C₂×C₆₇	1	3	yes	complemented
151	#2616	C₂×C₆₇	1	1	yes	characteristic, complemented
152	#2615	C₂×C₆₇	1	1	yes	characteristic
153	#1895	C₂×C₆₇	1	3	yes	complemented
154	#1894	C₂×C₆₇	1	3	yes	complemented
155	#1807	C₂×C₆₇	1	3	yes	complemented
156	C₆₇	#32932	1	1	yes	characteristic, direct product factor, Sylow
157	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
158	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
159	C₂×#131	—	2	12	no	—
160	C₂²×M₁₆	C₂²×C₆₇	1	3	yes	—
161	C₂³×C₈	C₂²×C₆₇	1	3	yes	—
162	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	3	yes	—
163	C₂×#121	C₂²×C₆₇	1	3	yes	—
164	#465	—	2	12	no	—
165	C₂×#106	C₂²×C₆₇	1	3	yes	—
166	C₂×#103	C₂²×C₆₇	1	3	yes	—
167	C₂×#102	C₂²×C₆₇	1	3	yes	—
168	#395	—	2	1	no	—
169	#394	—	2	3	no	—
170	#393	—	2	1	no	—
171	#392	—	2	3	no	—
172	#380	C₂²×C₆₇	1	3	yes	—
173	#380	C₂²×C₆₇	1	3	yes	—
174	#340	C₂²×C₆₇	1	6	yes	—
175	C₂×#41	—	2	6	no	—
176	C₂×#41	—	2	6	no	—
177	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
178	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
179	C₂²×D₄	—	2	3	no	—
180	C₂²×D₄	D₄×C₆₇	1	3	yes	—
181	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
182	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
183	C₂×SD₁₆	—	2	6	no	—
184	#131	—	2	32	no	—
185	C₂×M₁₆	—	2	6	no	—
186	C₂×M₁₆	—	2	12	no	—
187	C₂²×C₈	—	2	12	no	—
188	C₂²×C₈	D₄×C₆₇	1	12	yes	complemented
189	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
190	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
191	C₂²⋊Q₈	—	2	6	no	—
192	C₂²⋊Q₈	—	2	6	no	—
193	#121	—	2	6	no	—
194	#121	—	2	6	no	—
195	C₂×C₄⋊C₄	—	2	3	no	—
196	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
197	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
198	C₂×C₂²⋊C₄	—	2	3	no	—
199	C₂×C₂²⋊C₄	—	2	3	no	—
200	C₂×C₂²⋊C₄	—	2	3	no	—
201	C₂×C₂²⋊C₄	—	2	3	no	—
202	#106	—	2	6	no	—
203	#103	—	2	6	no	—
204	#102	—	2	6	no	—
205	#95	—	2	3	no	—
206	#95	—	2	3	no	—
207	C₂⁴	—	2	1	no	—
208	C₂⁴	C₂×D₄×C₆₇	1	3	yes	—
209	#41	—	2	8	no	—
210	#41	—	2	24	no	—
211	C₂×Q₈	—	2	6	no	—
212	C₂×Q₈	—	2	3	no	—
213	C₂×Q₈	—	2	3	no	—
214	C₂×D₄	—	2	12	no	—
215	C₂×D₄	—	2	6	no	—
216	C₂×D₄	—	2	3	no	—
217	C₂×D₄	—	2	3	no	—
218	C₂×D₄	—	4	6	no	—
219	C₂×D₄	—	8	6	no	—
220	C₂²×C₄	—	2	6	no	—
221	C₂²×C₄	—	2	6	no	—
222	C₂²×C₄	—	2	3	no	—
223	C₂²×C₄	—	2	3	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
226	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
227	C₂²×C₄	#41×C₆₇	1	6	yes	—
228	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
229	SD₁₆	—	4	4	no	—
230	M₁₆	—	2	24	no	—
231	C₂×C₈	—	2	6	no	—
232	C₂×C₈	—	2	24	no	—
233	C₂×C₈	C₂×D₄×C₆₇	1	6	yes	—
234	C₂×C₈	C₂×D₄×C₆₇	1	6	yes	—
235	C₄⋊C₄	—	2	3	no	—
236	C₄⋊C₄	—	2	3	no	—
237	C₄⋊C₄	—	2	3	no	—
238	C₄⋊C₄	—	2	3	no	—
239	C₄⋊C₄	—	4	6	no	—
240	C₂²⋊C₄	—	4	6	no	—
241	C₂²⋊C₄	—	4	6	no	—
242	C₂²⋊C₄	—	8	3	no	—
243	C₂²⋊C₄	—	8	3	no	—
244	C₂³	—	2	6	no	—
245	C₂³	—	2	3	no	—
246	C₂³	—	2	3	no	—
247	C₂³	—	2	6	no	—
248	C₂³	—	4	3	no	—
249	C₂³	—	4	3	no	—
250	C₂³	—	4	1	no	—
251	C₂³	#121×C₆₇	1	6	yes	—
252	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
253	Q₈	—	2	8	no	—
254	Q₈	—	4	1	no	—
255	Q₈	—	4	3	no	—
256	D₄	—	2	24	no	—
257	D₄	—	4	1	no	—
258	D₄	—	4	3	no	—
259	D₄	—	8	12	no	—
260	C₂×C₄	—	2	6	no	—
261	C₂×C₄	—	2	6	no	—
262	C₂×C₄	—	2	6	no	—
263	C₂×C₄	—	2	12	no	—
264	C₂×C₄	—	4	3	no	—
265	C₂×C₄	—	4	3	no	—
266	C₂×C₄	—	4	3	no	—
267	C₂×C₄	—	4	3	no	—
268	C₂×C₄	—	4	6	no	—
269	C₂×C₄	—	4	3	no	—
270	C₂×C₄	#121×C₆₇	1	6	yes	—
271	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
272	C₂×C₄	C₂×#41×C₆₇	1	3	yes	—
273	C₈	—	2	6	no	—
274	C₈	#120×C₆₇	1	4	yes	—
275	C₂²	—	2	6	no	—
276	C₂²	—	2	6	no	—
277	C₂²	—	2	6	no	—
278	C₂²	—	4	6	no	—
279	C₂²	—	4	6	no	—
280	C₂²	—	8	3	no	—
281	C₂²	—	8	3	no	—
282	C₂²	—	8	1	no	—
283	C₂²	C₂×#121×C₆₇	1	3	yes	—
284	C₂²	#577×C₆₇	1	3	yes	—
285	C₂²	#577×C₆₇	1	1	yes	characteristic
286	C₄	—	2	6	no	—
287	C₄	—	8	3	no	—
288	C₄	—	8	3	no	—
289	C₄	—	8	1	no	—
290	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
291	C₄	#525×C₆₇	1	3	yes	—
292	C₂	—	4	6	no	—
293	C₂	—	16	1	no	—
294	C₂	C₆₇×#2847	1	1	yes	characteristic
295	C₂	C₆₇×#3057	1	3	yes	—
296	C₂	C₆₇×#3060	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2615	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×#41×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#131×C₆₇	C₂	semidirect
2. C₂×#465×C₆₇	C₂	semidirect
3. C₆₇×#2616	C₂	semidirect
5. C₆₇×#1895	C₂	semidirect
6. C₆₇×#1894	C₂	semidirect
7. C₆₇×#1807	C₂	semidirect
39. C₂²×C₈×C₆₇	D₄	semidirect
143. #32932	C₆₇	direct (N × H)
149. C₂²×#131	C₂×C₆₇	semidirect
150. C₂×#465	C₂×C₆₇	semidirect
151. #2616	C₂×C₆₇	semidirect
153. #1895	C₂×C₆₇	semidirect
154. #1894	C₂×C₆₇	semidirect
155. #1807	C₂×C₆₇	semidirect
156. C₆₇	#32932	direct (N × H)
188. C₂²×C₈	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#131×C₆₇	2
C₂×#465×C₆₇	2
C₆₇×#2616	2
C₆₇×#2615	2
C₆₇×#1895	2
C₆₇×#1894	2
C₆₇×#1807	2
#32932	67

Subgroups — C67×#32932 #30712183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32932 #30712183