Subgroups — C₆₇×#21322 #30700573

Order 17152.

1270 conjugacy classes of subgroups.

The subgroup lattice has 302 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×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1992	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1990	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1736	C₂	1	4	yes	maximal
7	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
8	#533×C₆₇	—	2	4	no	—
9	#533×C₆₇	C₄	1	8	yes	complemented
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×C₄×Q₈×C₆₇	—	2	2	no	—
14	C₂×#117×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	C₄×M₁₆×C₆₇	—	2	4	no	—
19	C₂×#97×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
21	#384×C₆₇	—	2	2	no	—
22	#384×C₆₇	C₂²	1	2	yes	—
23	#384×C₆₇	C₂²	1	2	yes	—
24	#340×C₆₇	C₂²	1	16	yes	—
25	C₂²×Q₈×C₆₇	—	2	2	no	—
26	C₂³×C₄×C₆₇	—	2	2	no	—
27	C₂³×C₄×C₆₇	D₄	1	2	yes	—
28	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₆₇	—	2	4	no	—
30	C₂×M₁₆×C₆₇	—	2	4	no	—
31	C₂×M₁₆×C₆₇	Q₈	1	8	yes	complemented
32	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
33	C₄⋊Q₈×C₆₇	—	2	4	no	—
34	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
35	#125×C₆₇	—	2	4	no	—
36	#125×C₆₇	C₂×C₄	1	8	yes	complemented
37	C₂²⋊Q₈×C₆₇	—	2	16	no	—
38	C₄×Q₈×C₆₇	—	2	16	no	—
39	#117×C₆₇	—	2	4	no	—
40	#117×C₆₇	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
48	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
50	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄²×C₆₇	—	2	2	no	—
52	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₆₇	D₄	1	2	yes	—
54	C₂×C₄²×C₆₇	D₄	1	4	yes	—
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	#97×C₆₇	—	2	4	no	—
58	C₄×C₈×C₆₇	—	2	4	no	—
59	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂×Q₈×C₆₇	—	2	8	no	—
61	C₂×Q₈×C₆₇	—	4	4	no	—
62	C₂×Q₈×C₆₇	—	4	4	no	—
63	C₂²×C₄×C₆₇	—	2	4	no	—
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	4	8	no	—
68	C₂²×C₄×C₆₇	—	4	4	no	—
69	C₂²×C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
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	2	yes	—
75	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₆₇	#41	1	4	yes	—
77	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
78	M₁₆×C₆₇	—	2	16	no	—
79	C₂×C₈×C₆₇	—	2	4	no	—
80	C₂×C₈×C₆₇	—	2	4	no	—
81	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
82	C₄⋊C₄×C₆₇	—	2	16	no	—
83	C₄⋊C₄×C₆₇	—	2	8	no	—
84	C₄⋊C₄×C₆₇	—	2	4	no	—
85	C₄⋊C₄×C₆₇	—	2	8	no	—
86	C₄⋊C₄×C₆₇	—	4	8	no	—
87	C₄⋊C₄×C₆₇	—	4	4	no	—
88	C₄⋊C₄×C₆₇	—	4	8	no	—
89	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
90	C₂²⋊C₄×C₆₇	—	2	8	no	—
91	C₄²×C₆₇	—	2	8	no	—
92	C₄²×C₆₇	C₂²⋊C₄	1	8	yes	—
93	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
95	C₄²×C₆₇	C₂×D₄	1	2	yes	—
96	C₄²×C₆₇	C₂×D₄	1	2	yes	—
97	C₂³×C₆₇	—	2	2	no	—
98	C₂³×C₆₇	—	2	2	no	—
99	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂³×C₆₇	#124	1	2	yes	—
101	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
102	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
103	Q₈×C₆₇	—	4	16	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	16	no	—
107	C₂×C₄×C₆₇	—	2	8	no	—
108	C₂×C₄×C₆₇	—	4	4	no	—
109	C₂×C₄×C₆₇	—	4	4	no	—
110	C₂×C₄×C₆₇	—	4	4	no	—
111	C₂×C₄×C₆₇	—	4	8	no	—
112	C₂×C₄×C₆₇	—	8	8	no	—
113	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
114	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
116	C₂×C₄×C₆₇	C₄×Q₈	1	8	yes	—
117	C₂×C₄×C₆₇	C₂²⋊Q₈	1	8	yes	—
118	C₂×C₄×C₆₇	#124	1	2	yes	—
119	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
120	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
121	C₂×C₄×C₆₇	#137	1	4	yes	—
122	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
125	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
126	C₈×C₆₇	—	2	8	no	—
127	C₂²×C₆₇	—	2	2	no	—
128	C₂²×C₆₇	—	2	2	no	—
129	C₂²×C₆₇	—	2	4	no	—
130	C₂²×C₆₇	—	2	4	no	—
131	C₂²×C₆₇	#391	1	4	yes	—
132	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
134	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
135	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
136	C₂²×C₆₇	C₂×#137	1	2	yes	—
137	C₄×C₆₇	—	2	4	no	—
138	C₄×C₆₇	—	4	8	no	—
139	C₄×C₆₇	—	8	4	no	—
140	C₄×C₆₇	#391	1	4	yes	—
141	C₄×C₆₇	#417	1	8	yes	—
142	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
143	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
144	#21322	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×C₆₇	—	2	4	no	—
146	C₂×C₆₇	C₂×#391	1	1	yes	characteristic
147	C₂×C₆₇	C₂×#417	1	2	yes	—
148	C₂×C₆₇	#3138	1	1	yes	characteristic
149	C₂×C₆₇	#3139	1	1	yes	characteristic
150	C₂×C₆₇	#3165	1	2	yes	—
151	C₂×#533	C₂×C₆₇	1	1	yes	characteristic
152	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic
153	C₂×#384	C₂×C₆₇	1	1	yes	characteristic
154	#1992	C₂×C₆₇	1	4	yes	complemented
155	#1990	C₂×C₆₇	1	4	yes	complemented
156	#1736	C₂×C₆₇	1	4	yes	—
157	C₆₇	#21322	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
159	#533	—	2	4	no	—
160	#533	C₄×C₆₇	1	8	yes	complemented
161	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
162	C₂×#125	C₂²×C₆₇	1	2	yes	—
163	C₂×C₂²⋊Q₈	—	2	2	no	—
164	C₂×C₄×Q₈	—	2	2	no	—
165	C₂×#117	C₂²×C₆₇	1	2	yes	—
166	C₂²×C₄⋊C₄	—	2	2	no	—
167	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
168	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
169	C₄×M₁₆	—	2	4	no	—
170	C₂×#97	C₂²×C₆₇	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
172	#384	—	2	2	no	—
173	#384	C₂²×C₆₇	1	2	yes	—
174	#384	C₂²×C₆₇	1	2	yes	—
175	#340	C₂²×C₆₇	1	16	yes	—
176	C₂²×Q₈	—	2	2	no	—
177	C₂³×C₄	—	2	2	no	—
178	C₂³×C₄	D₄×C₆₇	1	2	yes	—
179	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	Q₈×C₆₇	1	8	yes	complemented
183	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
184	C₄⋊Q₈	—	2	4	no	—
185	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
186	#125	—	2	4	no	—
187	#125	C₂×C₄×C₆₇	1	8	yes	complemented
188	C₂²⋊Q₈	—	2	16	no	—
189	C₄×Q₈	—	2	16	no	—
190	#117	—	2	4	no	—
191	#117	C₂×C₄×C₆₇	1	8	yes	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	8	no	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄⋊C₄	—	2	4	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	4	8	no	—
199	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
200	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
201	C₂×C₂²⋊C₄	—	2	2	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
204	C₂×C₄²	D₄×C₆₇	1	2	yes	—
205	C₂×C₄²	D₄×C₆₇	1	4	yes	—
206	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
208	#97	—	2	4	no	—
209	C₄×C₈	—	2	4	no	—
210	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
211	C₂×Q₈	—	2	8	no	—
212	C₂×Q₈	—	4	4	no	—
213	C₂×Q₈	—	4	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	4	8	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
221	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
226	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
227	C₂²×C₄	#41×C₆₇	1	4	yes	—
228	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
229	M₁₆	—	2	16	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	C₂×Q₈×C₆₇	1	8	yes	—
233	C₄⋊C₄	—	2	16	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	4	8	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	8	no	—
240	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₄²	—	2	8	no	—
243	C₄²	C₂²⋊C₄×C₆₇	1	8	yes	—
244	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
245	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
246	C₄²	C₂×D₄×C₆₇	1	2	yes	—
247	C₄²	C₂×D₄×C₆₇	1	2	yes	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
251	C₂³	#124×C₆₇	1	2	yes	—
252	C₂³	C₄⋊Q₈×C₆₇	1	2	yes	—
253	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
254	Q₈	—	4	16	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	8	no	—
263	C₂×C₄	—	8	8	no	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
265	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
266	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
267	C₂×C₄	C₄×Q₈×C₆₇	1	8	yes	—
268	C₂×C₄	C₂²⋊Q₈×C₆₇	1	8	yes	—
269	C₂×C₄	#124×C₆₇	1	2	yes	—
270	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
271	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
272	C₂×C₄	#137×C₆₇	1	4	yes	—
273	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
275	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
276	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
277	C₈	—	2	8	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	#391×C₆₇	1	4	yes	—
283	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
284	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
285	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
286	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
287	C₂²	C₂×#137×C₆₇	1	2	yes	—
288	C₄	—	2	4	no	—
289	C₄	—	4	8	no	—
290	C₄	—	8	4	no	—
291	C₄	#391×C₆₇	1	4	yes	—
292	C₄	#417×C₆₇	1	8	yes	—
293	C₄	C₂×C₄×Q₈×C₆₇	1	2	yes	—
294	C₄	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
295	C₂	—	2	4	no	—
296	C₂	C₂×#391×C₆₇	1	1	yes	characteristic
297	C₂	C₂×#417×C₆₇	1	2	yes	—
298	C₂	C₆₇×#3138	1	1	yes	characteristic
299	C₂	C₆₇×#3139	1	1	yes	characteristic
300	C₂	C₆₇×#3165	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂×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
4. C₆₇×#1992	C₂	semidirect
5. C₆₇×#1990	C₂	semidirect
9. #533×C₆₇	C₄	semidirect
31. C₂×M₁₆×C₆₇	Q₈	semidirect
34. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
36. #125×C₆₇	C₂×C₄	semidirect
144. #21322	C₆₇	direct (N × H)
154. #1992	C₂×C₆₇	semidirect
155. #1990	C₂×C₆₇	semidirect
157. C₆₇	#21322	direct (N × H)
160. #533	C₄×C₆₇	semidirect
182. C₂×M₁₆	Q₈×C₆₇	semidirect
185. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
187. #125	C₂×C₄×C₆₇	semidirect

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×C₄×M₁₆×C₆₇	2
C₂×#384×C₆₇	2
C₆₇×#1992	2
C₆₇×#1990	2
C₆₇×#1736	2
#21322	67

Subgroups — C67×#21322 #30700573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21322 #30700573