Subgroups — C₆₇×#33196 #30712447

Order 17152.

898 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#475×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2868	C₂	1	1	yes	maximal, characteristic
3	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1988	C₂	1	2	yes	maximal
5	C₆₇×#1948	C₂	1	4	yes	maximal
6	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1803	C₂	1	4	yes	maximal
8	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
13	#475×C₆₇	—	2	8	no	—
14	C₂×#106×C₆₇	C₂²	1	4	yes	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
17	C₂×#103×C₆₇	C₂²	1	4	yes	—
18	#384×C₆₇	—	2	2	no	—
19	#384×C₆₇	C₂²	1	4	yes	—
20	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
21	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
22	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₄³×C₆₇	—	2	1	no	—
25	C₄³×C₆₇	C₂²	1	1	yes	characteristic
26	#340×C₆₇	C₂²	1	8	yes	—
27	C₂²×Q₈×C₆₇	—	2	2	no	—
28	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
30	C₄⋊Q₈×C₆₇	—	2	4	no	—
31	C₄⋊Q₈×C₆₇	—	2	4	no	—
32	C₄⋊Q₈×C₆₇	—	4	4	no	—
33	C₄⋊Q₈×C₆₇	—	4	4	no	—
34	C₄×Q₈×C₆₇	—	2	2	no	—
35	C₄×Q₈×C₆₇	—	2	4	no	—
36	C₄×Q₈×C₆₇	D₄	1	4	yes	—
37	C₄×Q₈×C₆₇	Q₈	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
46	C₂×C₄²×C₆₇	—	2	1	no	—
47	C₂×C₄²×C₆₇	—	2	2	no	—
48	C₂×C₄²×C₆₇	—	2	1	no	—
49	C₂×C₄²×C₆₇	—	2	2	no	—
50	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	D₄	1	2	yes	—
52	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
55	#106×C₆₇	—	2	8	no	—
56	C₄⋊C₈×C₆₇	—	2	4	no	—
57	C₄⋊C₈×C₆₇	—	2	8	no	—
58	#103×C₆₇	—	2	8	no	—
59	#95×C₆₇	—	2	1	no	—
60	#95×C₆₇	—	2	2	no	—
61	C₂×Q₈×C₆₇	—	2	1	no	—
62	C₂×Q₈×C₆₇	—	2	2	no	—
63	C₂×Q₈×C₆₇	—	4	4	no	—
64	C₂×Q₈×C₆₇	—	4	4	no	—
65	C₂×Q₈×C₆₇	—	4	4	no	—
66	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
67	C₂×Q₈×C₆₇	C₂×Q₈	1	4	yes	—
68	C₂²×C₄×C₆₇	—	2	1	no	—
69	C₂²×C₄×C₆₇	—	2	1	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	—	2	4	no	—
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₆₇	—	2	4	no	—
78	C₂×C₈×C₆₇	—	2	8	no	—
79	C₄⋊C₄×C₆₇	—	2	4	no	—
80	C₄⋊C₄×C₆₇	—	2	8	no	—
81	C₄⋊C₄×C₆₇	—	2	1	no	—
82	C₄⋊C₄×C₆₇	—	2	2	no	—
83	C₄⋊C₄×C₆₇	—	2	4	no	—
84	C₄⋊C₄×C₆₇	—	2	4	no	—
85	C₄⋊C₄×C₆₇	—	4	8	no	—
86	C₄⋊C₄×C₆₇	—	4	4	no	—
87	C₄⋊C₄×C₆₇	—	4	8	no	—
88	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
89	C₄⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
90	C₄²×C₆₇	—	2	4	no	—
91	C₄²×C₆₇	—	2	8	no	—
92	C₄²×C₆₇	—	2	2	no	—
93	C₄²×C₆₇	—	2	4	no	—
94	C₄²×C₆₇	—	4	2	no	—
95	C₄²×C₆₇	—	4	4	no	—
96	C₄²×C₆₇	SD₁₆	1	4	yes	—
97	C₄²×C₆₇	C₂×D₄	1	2	yes	—
98	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
99	C₄²×C₆₇	#41	1	2	yes	—
100	C₄²×C₆₇	#41	1	4	yes	—
101	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
102	Q₈×C₆₇	—	2	2	no	—
103	Q₈×C₆₇	—	2	4	no	—
104	Q₈×C₆₇	—	8	8	no	—
105	Q₈×C₆₇	C₄⋊Q₈	1	4	yes	complemented
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	—	4	2	no	—
115	C₂×C₄×C₆₇	—	4	2	no	—
116	C₂×C₄×C₆₇	—	4	4	no	—
117	C₂×C₄×C₆₇	—	4	4	no	—
118	C₂×C₄×C₆₇	—	4	4	no	—
119	C₂×C₄×C₆₇	#121	1	2	yes	—
120	C₂×C₄×C₆₇	#121	1	2	yes	—
121	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
122	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
123	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
124	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
125	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₄×C₆₇	#137	1	4	yes	—
127	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
129	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
130	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
131	C₈×C₆₇	—	4	4	no	—
132	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
133	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
134	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
135	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂²×C₆₇	C₂×#137	1	2	yes	—
137	C₄×C₆₇	—	2	4	no	—
138	C₄×C₆₇	—	2	4	no	—
139	C₄×C₆₇	—	2	2	no	—
140	C₄×C₆₇	—	4	4	no	—
141	C₄×C₆₇	—	4	2	no	—
142	C₄×C₆₇	—	8	4	no	—
143	C₄×C₆₇	#460	1	4	yes	—
144	C₄×C₆₇	#475	1	8	yes	complemented
145	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
146	C₄×C₆₇	#532	1	1	yes	characteristic
147	C₄×C₆₇	#533	1	2	yes	—
148	#33196	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂×C₆₇	#2857	1	1	yes	characteristic
150	C₂×C₆₇	C₂×#460	1	1	yes	characteristic
151	C₂×C₆₇	#3039	1	1	yes	characteristic
152	C₂×C₆₇	C₂×#475	1	2	yes	—
153	C₂×C₆₇	#3079	1	2	yes	—
154	C₂×#475	C₂×C₆₇	1	2	yes	—
155	#2868	C₂×C₆₇	1	1	yes	characteristic
156	C₄²×Q₈	C₂×C₆₇	1	1	yes	characteristic
157	#1988	C₂×C₆₇	1	2	yes	—
158	#1948	C₂×C₆₇	1	4	yes	—
159	#1842	C₂×C₆₇	1	1	yes	characteristic
160	#1803	C₂×C₆₇	1	4	yes	—
161	C₆₇	#33196	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×C₄⋊Q₈	—	2	1	no	—
163	C₂×C₄⋊Q₈	—	2	1	no	—
164	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
165	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
166	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
167	#475	—	2	8	no	—
168	C₂×#106	C₂²×C₆₇	1	4	yes	—
169	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
170	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
171	C₂×#103	C₂²×C₆₇	1	4	yes	—
172	#384	—	2	2	no	—
173	#384	C₂²×C₆₇	1	4	yes	—
174	C₄×C₄⋊C₄	—	2	1	no	—
175	C₄×C₄⋊C₄	—	2	2	no	—
176	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
177	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
178	C₄³	—	2	1	no	—
179	C₄³	C₂²×C₆₇	1	1	yes	characteristic
180	#340	C₂²×C₆₇	1	8	yes	—
181	C₂²×Q₈	—	2	2	no	—
182	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
184	C₄⋊Q₈	—	2	4	no	—
185	C₄⋊Q₈	—	2	4	no	—
186	C₄⋊Q₈	—	4	4	no	—
187	C₄⋊Q₈	—	4	4	no	—
188	C₄×Q₈	—	2	2	no	—
189	C₄×Q₈	—	2	4	no	—
190	C₄×Q₈	D₄×C₆₇	1	4	yes	—
191	C₄×Q₈	Q₈×C₆₇	1	8	yes	complemented
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
198	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
199	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
205	C₂×C₄²	D₄×C₆₇	1	2	yes	—
206	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
209	#106	—	2	8	no	—
210	C₄⋊C₈	—	2	4	no	—
211	C₄⋊C₈	—	2	8	no	—
212	#103	—	2	8	no	—
213	#95	—	2	1	no	—
214	#95	—	2	2	no	—
215	C₂×Q₈	—	2	1	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	—	4	4	no	—
218	C₂×Q₈	—	4	4	no	—
219	C₂×Q₈	—	4	4	no	—
220	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
221	C₂×Q₈	C₂×Q₈×C₆₇	1	4	yes	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
228	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
230	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	8	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	4	8	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₄⋊C₄	—	4	8	no	—
242	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
243	C₄⋊C₄	C₂×Q₈×C₆₇	1	4	yes	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	8	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	4	2	no	—
249	C₄²	—	4	4	no	—
250	C₄²	SD₁₆×C₆₇	1	4	yes	—
251	C₄²	C₂×D₄×C₆₇	1	2	yes	—
252	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
253	C₄²	#41×C₆₇	1	2	yes	—
254	C₄²	#41×C₆₇	1	4	yes	—
255	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
256	Q₈	—	2	2	no	—
257	Q₈	—	2	4	no	—
258	Q₈	—	8	8	no	—
259	Q₈	C₄⋊Q₈×C₆₇	1	4	yes	complemented
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	#121×C₆₇	1	2	yes	—
274	C₂×C₄	#121×C₆₇	1	2	yes	—
275	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
276	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
277	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
278	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
279	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
280	C₂×C₄	#137×C₆₇	1	4	yes	—
281	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
282	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
283	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
284	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
285	C₈	—	4	4	no	—
286	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
287	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
288	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
289	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
290	C₂²	C₂×#137×C₆₇	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	4	no	—
293	C₄	—	2	4	no	—
294	C₄	—	4	2	no	—
295	C₄	—	4	4	no	—
296	C₄	—	8	4	no	—
297	C₄	#460×C₆₇	1	4	yes	—
298	C₄	#475×C₆₇	1	8	yes	complemented
299	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
300	C₄	#532×C₆₇	1	1	yes	characteristic
301	C₄	#533×C₆₇	1	2	yes	—
302	C₂	C₆₇×#2857	1	1	yes	characteristic
303	C₂	C₂×#460×C₆₇	1	1	yes	characteristic
304	C₂	C₆₇×#3039	1	1	yes	characteristic
305	C₂	C₂×#475×C₆₇	1	2	yes	—
306	C₂	C₆₇×#3079	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
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
37. C₄×Q₈×C₆₇	Q₈	semidirect
105. Q₈×C₆₇	C₄⋊Q₈	semidirect
144. C₄×C₆₇	#475	semidirect
148. #33196	C₆₇	direct (N × H)
161. C₆₇	#33196	direct (N × H)
191. C₄×Q₈	Q₈×C₆₇	semidirect
259. Q₈	C₄⋊Q₈×C₆₇	semidirect
298. C₄	#475×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#475×C₆₇	2
C₆₇×#2868	2
C₄²×Q₈×C₆₇	2
C₆₇×#1988	2
C₆₇×#1948	2
C₆₇×#1842	2
C₆₇×#1803	2
#33196	67

Subgroups — C67×#33196 #30712447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33196 #30712447