Subgroups — C₆₇×#33171 #30712422

Order 17152.

1422 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#466×C₆₇	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2856	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1970	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1957	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₆₇	C₂²	1	4	yes	—
9	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
10	C₂×#127×C₆₇	C₂²	1	2	yes	—
11	C₂×#121×C₆₇	—	2	4	no	—
12	C₂×#121×C₆₇	C₂²	1	4	yes	—
13	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
14	#493×C₆₇	—	2	4	no	—
15	#466×C₆₇	—	2	16	no	—
16	C₂×#107×C₆₇	C₂²	1	4	yes	—
17	C₂×#102×C₆₇	C₂²	1	8	yes	—
18	C₂×C₄×C₈×C₆₇	—	2	4	no	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#390×C₆₇	—	2	4	no	—
23	#390×C₆₇	C₂²	1	4	yes	—
24	#384×C₆₇	C₂²	1	2	yes	—
25	C₂²×D₄×C₆₇	—	2	4	no	—
26	C₂²×D₄×C₆₇	—	2	4	no	—
27	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
28	C₂³×C₄×C₆₇	D₄	1	2	yes	—
29	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₂×D₈×C₆₇	—	2	8	no	—
31	C₂×D₈×C₆₇	—	2	8	no	—
32	C₂×D₈×C₆₇	—	2	8	no	—
33	C₂²×C₈×C₆₇	—	2	4	no	—
34	C₂²×C₈×C₆₇	—	2	4	no	—
35	C₂²×C₈×C₆₇	D₄	1	8	yes	complemented
36	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
38	#127×C₆₇	—	2	4	no	—
39	#127×C₆₇	—	2	4	no	—
40	#121×C₆₇	—	2	16	no	—
41	#121×C₆₇	—	4	16	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
44	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
45	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
46	C₂×C₄²×C₆₇	—	2	2	no	—
47	C₂×C₄²×C₆₇	—	2	2	no	—
48	C₂×C₄²×C₆₇	D₄	1	2	yes	—
49	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	#107×C₆₇	—	2	8	no	—
52	#102×C₆₇	—	2	16	no	—
53	C₄×C₈×C₆₇	—	2	4	no	—
54	C₄×C₈×C₆₇	—	2	2	no	—
55	C₄×C₈×C₆₇	D₄	1	4	yes	complemented
56	C₂⁴×C₆₇	—	2	4	no	—
57	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×D₄×C₆₇	—	2	8	no	—
59	C₂×D₄×C₆₇	—	2	4	no	—
60	C₂×D₄×C₆₇	—	2	4	no	—
61	C₂×D₄×C₆₇	—	2	8	no	—
62	C₂×D₄×C₆₇	—	4	8	no	—
63	C₂×D₄×C₆₇	—	4	16	no	—
64	C₂×D₄×C₆₇	—	4	8	no	—
65	C₂×D₄×C₆₇	—	4	8	no	—
66	C₂×D₄×C₆₇	—	4	8	no	—
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	—	2	1	no	—
70	C₂²×C₄×C₆₇	—	2	1	no	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	4	no	—
73	C₂²×C₄×C₆₇	D₈	1	8	yes	complemented
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
79	D₈×C₆₇	—	4	16	no	—
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	—	2	2	no	—
82	C₂×C₈×C₆₇	—	2	16	no	—
83	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
84	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
85	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
86	C₂×C₈×C₆₇	#41	1	4	yes	—
87	C₄⋊C₄×C₆₇	—	2	8	no	—
88	C₄⋊C₄×C₆₇	—	4	8	no	—
89	C₂²⋊C₄×C₆₇	—	4	8	no	—
90	C₂²⋊C₄×C₆₇	—	4	16	no	—
91	C₄²×C₆₇	—	2	4	no	—
92	C₄²×C₆₇	—	2	1	no	—
93	C₄²×C₆₇	—	2	1	no	—
94	C₄²×C₆₇	C₂×D₄	1	2	yes	—
95	C₄²×C₆₇	#41	1	2	yes	—
96	C₂³×C₆₇	—	2	1	no	—
97	C₂³×C₆₇	—	2	1	no	—
98	C₂³×C₆₇	—	2	2	no	—
99	C₂³×C₆₇	—	4	4	no	—
100	C₂³×C₆₇	—	4	4	no	—
101	C₂³×C₆₇	—	4	4	no	—
102	C₂³×C₆₇	—	4	8	no	—
103	C₂³×C₆₇	—	4	8	no	—
104	C₂³×C₆₇	#127	1	2	yes	—
105	C₂³×C₆₇	C₂×D₈	1	4	yes	—
106	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
107	D₄×C₆₇	—	4	8	no	—
108	D₄×C₆₇	—	4	8	no	—
109	D₄×C₆₇	—	8	16	no	—
110	D₄×C₆₇	—	8	16	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	4	no	—
116	C₂×C₄×C₆₇	—	2	4	no	—
117	C₂×C₄×C₆₇	—	2	16	no	—
118	C₂×C₄×C₆₇	—	4	4	no	—
119	C₂×C₄×C₆₇	—	4	8	no	—
120	C₂×C₄×C₆₇	#121	1	4	yes	—
121	C₂×C₄×C₆₇	#127	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
123	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
124	C₂×C₄×C₆₇	#135	1	4	yes	—
125	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
127	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
128	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
129	C₈×C₆₇	—	2	4	no	—
130	C₈×C₆₇	#121	1	8	yes	complemented
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	2	4	no	—
134	C₂²×C₆₇	—	2	4	no	—
135	C₂²×C₆₇	—	8	4	no	—
136	C₂²×C₆₇	—	8	4	no	—
137	C₂²×C₆₇	—	8	4	no	—
138	C₂²×C₆₇	—	8	8	no	—
139	C₂²×C₆₇	—	8	8	no	—
140	C₂²×C₆₇	#493	1	4	yes	complemented
141	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
142	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
143	C₂²×C₆₇	#532	1	1	yes	characteristic
144	C₂²×C₆₇	C₂²×D₈	1	2	yes	—
145	C₂²×C₆₇	C₂×#135	1	2	yes	—
146	C₄×C₆₇	—	2	2	no	—
147	C₄×C₆₇	—	2	4	no	—
148	C₄×C₆₇	—	8	4	no	—
149	C₄×C₆₇	#466	1	8	yes	complemented
150	C₄×C₆₇	C₂×#121	1	2	yes	—
151	C₄×C₆₇	#532	1	2	yes	—
152	#33171	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂×C₆₇	—	2	4	no	—
154	C₂×C₆₇	—	16	4	no	—
155	C₂×C₆₇	#2856	1	1	yes	characteristic
156	C₂×C₆₇	C₂×#466	1	2	yes	—
157	C₂×C₆₇	C₂×#493	1	1	yes	characteristic
158	C₂×C₆₇	#3147	1	1	yes	characteristic
159	C₂×C₆₇	#3169	1	2	yes	—
160	C₂×#493	C₂×C₆₇	1	1	yes	characteristic, complemented
161	C₂×#466	C₂×C₆₇	1	4	yes	complemented
162	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
163	#2856	C₂×C₆₇	1	2	yes	complemented
164	#1970	C₂×C₆₇	1	4	yes	complemented
165	#1957	C₂×C₆₇	1	2	yes	complemented
166	#1845	C₂×C₆₇	1	1	yes	characteristic, complemented
167	C₆₇	#33171	1	1	yes	characteristic, direct product factor, Sylow
168	C₂²×D₈	C₂²×C₆₇	1	4	yes	—
169	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
170	C₂×#127	C₂²×C₆₇	1	2	yes	—
171	C₂×#121	—	2	4	no	—
172	C₂×#121	C₂²×C₆₇	1	4	yes	—
173	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
174	#493	—	2	4	no	—
175	#466	—	2	16	no	—
176	C₂×#107	C₂²×C₆₇	1	4	yes	—
177	C₂×#102	C₂²×C₆₇	1	8	yes	—
178	C₂×C₄×C₈	—	2	4	no	—
179	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
181	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
182	#390	—	2	4	no	—
183	#390	C₂²×C₆₇	1	4	yes	—
184	#384	C₂²×C₆₇	1	2	yes	—
185	C₂²×D₄	—	2	4	no	—
186	C₂²×D₄	—	2	4	no	—
187	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
188	C₂³×C₄	D₄×C₆₇	1	2	yes	—
189	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
190	C₂×D₈	—	2	8	no	—
191	C₂×D₈	—	2	8	no	—
192	C₂×D₈	—	2	8	no	—
193	C₂²×C₈	—	2	4	no	—
194	C₂²×C₈	—	2	4	no	—
195	C₂²×C₈	D₄×C₆₇	1	8	yes	complemented
196	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
197	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
198	#127	—	2	4	no	—
199	#127	—	2	4	no	—
200	#121	—	2	16	no	—
201	#121	—	4	16	no	—
202	C₂×C₄⋊C₄	—	2	4	no	—
203	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
204	C₂×C₂²⋊C₄	—	2	8	no	—
205	C₂×C₂²⋊C₄	—	2	4	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	D₄×C₆₇	1	2	yes	—
209	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
210	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
211	#107	—	2	8	no	—
212	#102	—	2	16	no	—
213	C₄×C₈	—	2	4	no	—
214	C₄×C₈	—	2	2	no	—
215	C₄×C₈	D₄×C₆₇	1	4	yes	complemented
216	C₂⁴	—	2	4	no	—
217	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
218	C₂×D₄	—	2	8	no	—
219	C₂×D₄	—	2	4	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	2	8	no	—
222	C₂×D₄	—	4	8	no	—
223	C₂×D₄	—	4	16	no	—
224	C₂×D₄	—	4	8	no	—
225	C₂×D₄	—	4	8	no	—
226	C₂×D₄	—	4	8	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	D₈×C₆₇	1	8	yes	complemented
234	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
236	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
237	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
238	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
239	D₈	—	4	16	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	16	no	—
243	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
244	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
245	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
246	C₂×C₈	#41×C₆₇	1	4	yes	—
247	C₄⋊C₄	—	2	8	no	—
248	C₄⋊C₄	—	4	8	no	—
249	C₂²⋊C₄	—	4	8	no	—
250	C₂²⋊C₄	—	4	16	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	C₂×D₄×C₆₇	1	2	yes	—
255	C₄²	#41×C₆₇	1	2	yes	—
256	C₂³	—	2	1	no	—
257	C₂³	—	2	1	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	4	4	no	—
260	C₂³	—	4	4	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	4	8	no	—
263	C₂³	—	4	8	no	—
264	C₂³	#127×C₆₇	1	2	yes	—
265	C₂³	C₂×D₈×C₆₇	1	4	yes	—
266	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
267	D₄	—	4	8	no	—
268	D₄	—	4	8	no	—
269	D₄	—	8	16	no	—
270	D₄	—	8	16	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	16	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	8	no	—
280	C₂×C₄	#121×C₆₇	1	4	yes	—
281	C₂×C₄	#127×C₆₇	1	2	yes	—
282	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
283	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
284	C₂×C₄	#135×C₆₇	1	4	yes	—
285	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
286	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
287	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
288	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
289	C₈	—	2	4	no	—
290	C₈	#121×C₆₇	1	8	yes	complemented
291	C₂²	—	2	4	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	8	8	no	—
296	C₂²	—	8	4	no	—
297	C₂²	—	8	4	no	—
298	C₂²	—	8	4	no	—
299	C₂²	—	8	8	no	—
300	C₂²	#493×C₆₇	1	4	yes	complemented
301	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
302	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
303	C₂²	#532×C₆₇	1	1	yes	characteristic
304	C₂²	C₂²×D₈×C₆₇	1	2	yes	—
305	C₂²	C₂×#135×C₆₇	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	2	2	no	—
308	C₄	—	8	4	no	—
309	C₄	#466×C₆₇	1	8	yes	complemented
310	C₄	C₂×#121×C₆₇	1	2	yes	—
311	C₄	#532×C₆₇	1	2	yes	—
312	C₂	—	2	4	no	—
313	C₂	—	16	4	no	—
314	C₂	C₆₇×#2856	1	1	yes	characteristic
315	C₂	C₂×#466×C₆₇	1	2	yes	—
316	C₂	C₂×#493×C₆₇	1	1	yes	characteristic
317	C₂	C₆₇×#3147	1	1	yes	characteristic
318	C₂	C₆₇×#3169	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
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×#493×C₆₇	C₂	semidirect
2. C₂×#466×C₆₇	C₂	semidirect
3. C₂²×C₄×C₈×C₆₇	C₂	semidirect
4. C₆₇×#2856	C₂	semidirect
5. C₆₇×#1970	C₂	semidirect
6. C₆₇×#1957	C₂	semidirect
7. C₆₇×#1845	C₂	semidirect
35. C₂²×C₈×C₆₇	D₄	semidirect
55. C₄×C₈×C₆₇	D₄	semidirect
73. C₂²×C₄×C₆₇	D₈	semidirect
130. C₈×C₆₇	#121	semidirect
140. C₂²×C₆₇	#493	semidirect
149. C₄×C₆₇	#466	semidirect
152. #33171	C₆₇	direct (N × H)
160. C₂×#493	C₂×C₆₇	semidirect
161. C₂×#466	C₂×C₆₇	semidirect
162. C₂²×C₄×C₈	C₂×C₆₇	semidirect
163. #2856	C₂×C₆₇	semidirect
164. #1970	C₂×C₆₇	semidirect
165. #1957	C₂×C₆₇	semidirect
166. #1845	C₂×C₆₇	semidirect
167. C₆₇	#33171	direct (N × H)
195. C₂²×C₈	D₄×C₆₇	semidirect
215. C₄×C₈	D₄×C₆₇	semidirect
233. C₂²×C₄	D₈×C₆₇	semidirect
290. C₈	#121×C₆₇	semidirect
300. C₂²	#493×C₆₇	semidirect
309. C₄	#466×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493×C₆₇	2
C₂×#466×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₆₇×#2856	2
C₆₇×#1970	2
C₆₇×#1957	2
C₆₇×#1845	2
#33171	67

Subgroups — C67×#33171 #30712422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33171 #30712422