Subgroups — C₆₇×#33197 #30712448

Order 17152.

1018 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#476×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1987	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1948	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1802	C₂	1	4	yes	maximal
8	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
12	C₂²×C₄²×C₆₇	—	2	1	no	—
13	#476×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×#102×C₆₇	C₂²	1	4	yes	—
18	#384×C₆₇	—	2	1	no	—
19	#384×C₆₇	—	2	1	no	—
20	#384×C₆₇	C₂²	1	4	yes	—
21	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
24	C₄×C₂²⋊C₄×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×D₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	—	2	1	no	—
30	C₂³×C₄×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
32	C₄⋊Q₈×C₆₇	—	2	4	no	—
33	C₄⋊Q₈×C₆₇	—	2	4	no	—
34	C₄⋊Q₈×C₆₇	—	4	8	no	—
35	C₄×D₄×C₆₇	—	2	2	no	—
36	C₄×D₄×C₆₇	—	2	4	no	—
37	C₄×D₄×C₆₇	D₄	1	4	yes	—
38	C₄×D₄×C₆₇	Q₈	1	8	yes	complemented
39	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
45	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
46	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₄²×C₆₇	—	2	1	no	—
48	C₂×C₄²×C₆₇	—	2	2	no	—
49	C₂×C₄²×C₆₇	—	4	4	no	—
50	C₂×C₄²×C₆₇	D₄	1	2	yes	—
51	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
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	#102×C₆₇	—	2	8	no	—
59	#95×C₆₇	—	2	1	no	—
60	#95×C₆₇	—	2	2	no	—
61	C₂⁴×C₆₇	—	2	1	no	—
62	C₂×Q₈×C₆₇	—	4	8	no	—
63	C₂×Q₈×C₆₇	—	4	4	no	—
64	C₂×D₄×C₆₇	—	2	2	no	—
65	C₂×D₄×C₆₇	—	2	1	no	—
66	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
67	C₂×D₄×C₆₇	C₂×Q₈	1	4	yes	—
68	C₂²×C₄×C₆₇	—	2	1	no	—
69	C₂²×C₄×C₆₇	—	2	2	no	—
70	C₂²×C₄×C₆₇	—	2	4	no	—
71	C₂²×C₄×C₆₇	—	2	8	no	—
72	C₂²×C₄×C₆₇	—	2	4	no	—
73	C₂²×C₄×C₆₇	—	4	2	no	—
74	C₂²×C₄×C₆₇	—	4	4	no	—
75	C₂²×C₄×C₆₇	—	4	4	no	—
76	C₂²×C₄×C₆₇	—	4	2	no	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₆₇	—	2	4	no	—
83	C₂×C₈×C₆₇	—	2	8	no	—
84	C₄⋊C₄×C₆₇	—	2	1	no	—
85	C₄⋊C₄×C₆₇	—	2	2	no	—
86	C₄⋊C₄×C₆₇	—	2	4	no	—
87	C₄⋊C₄×C₆₇	—	2	4	no	—
88	C₄⋊C₄×C₆₇	—	4	8	no	—
89	C₄⋊C₄×C₆₇	—	4	4	no	—
90	C₄⋊C₄×C₆₇	—	4	8	no	—
91	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
92	C₄⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
93	C₂²⋊C₄×C₆₇	—	2	4	no	—
94	C₂²⋊C₄×C₆₇	—	2	8	no	—
95	C₄²×C₆₇	—	2	4	no	—
96	C₄²×C₆₇	—	2	2	no	—
97	C₄²×C₆₇	—	4	2	no	—
98	C₄²×C₆₇	—	4	4	no	—
99	C₄²×C₆₇	SD₁₆	1	4	yes	—
100	C₄²×C₆₇	C₂×D₄	1	2	yes	—
101	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
102	C₄²×C₆₇	#41	1	2	yes	—
103	C₄²×C₆₇	#41	1	4	yes	—
104	C₂³×C₆₇	—	2	2	no	—
105	C₂³×C₆₇	—	2	4	no	—
106	C₂³×C₆₇	—	4	1	no	—
107	C₂³×C₆₇	—	4	2	no	—
108	C₂³×C₆₇	—	4	1	no	—
109	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
110	Q₈×C₆₇	—	8	8	no	—
111	D₄×C₆₇	—	2	2	no	—
112	D₄×C₆₇	—	2	4	no	—
113	D₄×C₆₇	C₄⋊Q₈	1	4	yes	complemented
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	4	no	—
119	C₂×C₄×C₆₇	—	4	1	no	—
120	C₂×C₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	4	1	no	—
123	C₂×C₄×C₆₇	—	4	8	no	—
124	C₂×C₄×C₆₇	—	4	16	no	—
125	C₂×C₄×C₆₇	—	4	4	no	—
126	C₂×C₄×C₆₇	—	4	8	no	—
127	C₂×C₄×C₆₇	#121	1	2	yes	—
128	C₂×C₄×C₆₇	#121	1	2	yes	—
129	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
130	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
131	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
132	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
133	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
134	C₂×C₄×C₆₇	#137	1	2	yes	—
135	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
137	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
138	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
139	C₈×C₆₇	—	4	4	no	—
140	C₂²×C₆₇	—	2	4	no	—
141	C₂²×C₆₇	—	4	4	no	—
142	C₂²×C₆₇	—	4	2	no	—
143	C₂²×C₆₇	—	4	2	no	—
144	C₂²×C₆₇	—	4	4	no	—
145	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
146	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
147	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
148	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
149	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
150	C₄×C₆₇	—	2	4	no	—
151	C₄×C₆₇	—	2	2	no	—
152	C₄×C₆₇	—	4	2	no	—
153	C₄×C₆₇	—	4	4	no	—
154	C₄×C₆₇	—	8	4	no	—
155	C₄×C₆₇	#461	1	4	yes	—
156	C₄×C₆₇	#476	1	8	yes	complemented
157	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
158	C₄×C₆₇	#532	1	1	yes	characteristic
159	C₄×C₆₇	#533	1	2	yes	—
160	#33197	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂×C₆₇	—	4	4	no	—
162	C₂×C₆₇	#2857	1	1	yes	characteristic
163	C₂×C₆₇	C₂×#461	1	1	yes	characteristic
164	C₂×C₆₇	#3038	1	1	yes	characteristic
165	C₂×C₆₇	C₂×#476	1	2	yes	—
166	C₂×C₆₇	#3079	1	2	yes	—
167	C₂×#476	C₂×C₆₇	1	2	yes	—
168	#2868	C₂×C₆₇	1	1	yes	characteristic, complemented
169	C₄²×D₄	C₂×C₆₇	1	1	yes	characteristic
170	#1987	C₂×C₆₇	1	2	yes	complemented
171	#1948	C₂×C₆₇	1	4	yes	complemented
172	#1842	C₂×C₆₇	1	1	yes	characteristic, complemented
173	#1802	C₂×C₆₇	1	4	yes	—
174	C₆₇	#33197	1	1	yes	characteristic, direct product factor, Sylow
175	C₂×C₄⋊Q₈	—	2	2	no	—
176	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
177	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
178	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
179	C₂²×C₄²	—	2	1	no	—
180	#476	—	2	8	no	—
181	C₂×#106	C₂²×C₆₇	1	4	yes	—
182	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
183	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
184	C₂×#102	C₂²×C₆₇	1	4	yes	—
185	#384	—	2	1	no	—
186	#384	—	2	1	no	—
187	#384	C₂²×C₆₇	1	4	yes	—
188	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
189	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
190	C₄×C₂²⋊C₄	—	2	2	no	—
191	C₄×C₂²⋊C₄	—	2	1	no	—
192	C₄³	C₂²×C₆₇	1	1	yes	characteristic
193	#340	C₂²×C₆₇	1	8	yes	—
194	C₂²×Q₈	—	2	2	no	—
195	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
196	C₂³×C₄	—	2	1	no	—
197	C₂³×C₄	—	2	2	no	—
198	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
199	C₄⋊Q₈	—	2	4	no	—
200	C₄⋊Q₈	—	2	4	no	—
201	C₄⋊Q₈	—	4	8	no	—
202	C₄×D₄	—	2	2	no	—
203	C₄×D₄	—	2	4	no	—
204	C₄×D₄	D₄×C₆₇	1	4	yes	—
205	C₄×D₄	Q₈×C₆₇	1	8	yes	complemented
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
210	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
211	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
212	C₂×C₂²⋊C₄	—	2	1	no	—
213	C₂×C₂²⋊C₄	—	2	2	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	—	4	4	no	—
217	C₂×C₄²	D₄×C₆₇	1	2	yes	—
218	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
219	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
220	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
222	#106	—	2	8	no	—
223	C₄⋊C₈	—	2	4	no	—
224	C₄⋊C₈	—	2	8	no	—
225	#102	—	2	8	no	—
226	#95	—	2	1	no	—
227	#95	—	2	2	no	—
228	C₂⁴	—	2	1	no	—
229	C₂×Q₈	—	4	8	no	—
230	C₂×Q₈	—	4	4	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
234	C₂×D₄	C₂×Q₈×C₆₇	1	4	yes	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	8	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	4	no	—
242	C₂²×C₄	—	4	4	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
248	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	8	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	4	8	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	4	8	no	—
258	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
259	C₄⋊C₄	C₂×Q₈×C₆₇	1	4	yes	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₂²⋊C₄	—	2	8	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	4	2	no	—
265	C₄²	—	4	4	no	—
266	C₄²	SD₁₆×C₆₇	1	4	yes	—
267	C₄²	C₂×D₄×C₆₇	1	2	yes	—
268	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
269	C₄²	#41×C₆₇	1	2	yes	—
270	C₄²	#41×C₆₇	1	4	yes	—
271	C₂³	—	2	2	no	—
272	C₂³	—	2	4	no	—
273	C₂³	—	4	1	no	—
274	C₂³	—	4	2	no	—
275	C₂³	—	4	1	no	—
276	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
277	Q₈	—	8	8	no	—
278	D₄	—	2	2	no	—
279	D₄	—	2	4	no	—
280	D₄	C₄⋊Q₈×C₆₇	1	4	yes	complemented
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	8	no	—
291	C₂×C₄	—	4	16	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	8	no	—
294	C₂×C₄	#121×C₆₇	1	2	yes	—
295	C₂×C₄	#121×C₆₇	1	2	yes	—
296	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
297	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
298	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
299	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
300	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
301	C₂×C₄	#137×C₆₇	1	2	yes	—
302	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
303	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
304	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
305	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
306	C₈	—	4	4	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
313	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
314	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
315	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
316	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
317	C₄	—	2	2	no	—
318	C₄	—	2	4	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	4	no	—
321	C₄	—	8	4	no	—
322	C₄	#461×C₆₇	1	4	yes	—
323	C₄	#476×C₆₇	1	8	yes	complemented
324	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
325	C₄	#532×C₆₇	1	1	yes	characteristic
326	C₄	#533×C₆₇	1	2	yes	—
327	C₂	—	4	4	no	—
328	C₂	C₆₇×#2857	1	1	yes	characteristic
329	C₂	C₂×#461×C₆₇	1	1	yes	characteristic
330	C₂	C₆₇×#3038	1	1	yes	characteristic
331	C₂	C₂×#476×C₆₇	1	2	yes	—
332	C₂	C₆₇×#3079	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
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
2. C₆₇×#2868	C₂	semidirect
4. C₆₇×#1987	C₂	semidirect
5. C₆₇×#1948	C₂	semidirect
6. C₆₇×#1842	C₂	semidirect
38. C₄×D₄×C₆₇	Q₈	semidirect
113. D₄×C₆₇	C₄⋊Q₈	semidirect
156. C₄×C₆₇	#476	semidirect
160. #33197	C₆₇	direct (N × H)
168. #2868	C₂×C₆₇	semidirect
170. #1987	C₂×C₆₇	semidirect
171. #1948	C₂×C₆₇	semidirect
172. #1842	C₂×C₆₇	semidirect
174. C₆₇	#33197	direct (N × H)
205. C₄×D₄	Q₈×C₆₇	semidirect
280. D₄	C₄⋊Q₈×C₆₇	semidirect
323. C₄	#476×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#476×C₆₇	2
C₆₇×#2868	2
C₄²×D₄×C₆₇	2
C₆₇×#1987	2
C₆₇×#1948	2
C₆₇×#1842	2
C₆₇×#1802	2
#33197	67

Subgroups — C67×#33197 #30712448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33197 #30712448