Subgroups — C₆₇×#33176 #30712427

Order 17152.

1090 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#470×C₆₇	C₂	1	4	yes	maximal
3	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2857	C₂	1	2	yes	maximal
5	C₆₇×#1971	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1956	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
9	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
11	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
12	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
13	#494×C₆₇	—	2	4	no	—
14	#470×C₆₇	—	2	16	no	—
15	C₂×#106×C₆₇	C₂²	1	4	yes	—
16	C₂×#103×C₆₇	C₂²	1	8	yes	—
17	C₄×M₁₆×C₆₇	—	2	4	no	—
18	C₂×#97×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	#391×C₆₇	C₂²	1	4	yes	—
22	#389×C₆₇	—	2	4	no	—
23	#384×C₆₇	C₂²	1	2	yes	—
24	#380×C₆₇	—	2	4	no	—
25	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
26	C₂³×C₄×C₆₇	D₄	1	2	yes	—
27	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×Q₁₆×C₆₇	—	2	8	no	—
29	C₂×Q₁₆×C₆₇	—	2	8	no	—
30	C₂×Q₁₆×C₆₇	—	2	8	no	—
31	C₂×M₁₆×C₆₇	—	2	4	no	—
32	C₂×M₁₆×C₆₇	—	2	4	no	—
33	C₂×M₁₆×C₆₇	D₄	1	8	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
36	C₄⋊Q₈×C₆₇	—	2	4	no	—
37	C₄⋊Q₈×C₆₇	—	2	4	no	—
38	C₂²⋊Q₈×C₆₇	—	2	16	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
43	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
44	C₂×C₄²×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	—	2	2	no	—
46	C₂×C₄²×C₆₇	D₄	1	2	yes	—
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	#106×C₆₇	—	2	8	no	—
50	#103×C₆₇	—	2	16	no	—
51	#97×C₆₇	—	2	4	no	—
52	C₄×C₈×C₆₇	—	2	2	no	—
53	C₄×C₈×C₆₇	D₄	1	4	yes	—
54	#95×C₆₇	—	2	8	no	—
55	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×Q₈×C₆₇	—	2	8	no	—
57	C₂×Q₈×C₆₇	—	2	4	no	—
58	C₂×Q₈×C₆₇	—	2	4	no	—
59	C₂×Q₈×C₆₇	—	2	8	no	—
60	C₂²×C₄×C₆₇	—	2	1	no	—
61	C₂²×C₄×C₆₇	—	2	1	no	—
62	C₂²×C₄×C₆₇	—	2	2	no	—
63	C₂²×C₄×C₆₇	—	2	4	no	—
64	C₂²×C₄×C₆₇	—	2	8	no	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
72	Q₁₆×C₆₇	—	4	16	no	—
73	M₁₆×C₆₇	—	2	16	no	—
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₂×C₈×C₆₇	—	2	2	no	—
76	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
77	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₆₇	#41	1	4	yes	—
80	C₄⋊C₄×C₆₇	—	2	8	no	—
81	C₄⋊C₄×C₆₇	—	4	8	no	—
82	C₄⋊C₄×C₆₇	—	4	8	no	—
83	C₄⋊C₄×C₆₇	—	4	8	no	—
84	C₂²⋊C₄×C₆₇	—	4	8	no	—
85	C₄²×C₆₇	—	2	4	no	—
86	C₄²×C₆₇	—	2	1	no	—
87	C₄²×C₆₇	—	2	1	no	—
88	C₄²×C₆₇	C₂×D₄	1	2	yes	—
89	C₄²×C₆₇	#41	1	2	yes	—
90	C₂³×C₆₇	—	2	1	no	—
91	C₂³×C₆₇	—	2	1	no	—
92	C₂³×C₆₇	—	2	4	no	—
93	C₂³×C₆₇	#127	1	2	yes	—
94	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
95	Q₈×C₆₇	—	4	8	no	—
96	Q₈×C₆₇	—	4	8	no	—
97	C₂×C₄×C₆₇	—	2	1	no	—
98	C₂×C₄×C₆₇	—	2	1	no	—
99	C₂×C₄×C₆₇	—	2	4	no	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	4	no	—
103	C₂×C₄×C₆₇	—	4	8	no	—
104	C₂×C₄×C₆₇	—	4	4	no	—
105	C₂×C₄×C₆₇	—	4	4	no	—
106	C₂×C₄×C₆₇	—	4	4	no	—
107	C₂×C₄×C₆₇	—	4	4	no	—
108	C₂×C₄×C₆₇	—	4	8	no	—
109	C₂×C₄×C₆₇	#121	1	4	yes	—
110	C₂×C₄×C₆₇	#127	1	2	yes	—
111	C₂×C₄×C₆₇	#137	1	8	yes	—
112	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
114	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
115	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
116	C₈×C₆₇	—	2	4	no	—
117	C₈×C₆₇	#121	1	8	yes	—
118	C₂²×C₆₇	—	2	2	no	—
119	C₂²×C₆₇	—	2	4	no	—
120	C₂²×C₆₇	—	4	2	no	—
121	C₂²×C₆₇	—	4	2	no	—
122	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
123	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
124	C₂²×C₆₇	#532	1	1	yes	characteristic
125	C₂²×C₆₇	C₂×#137	1	4	yes	—
126	C₄×C₆₇	—	2	2	no	—
127	C₄×C₆₇	—	4	2	no	—
128	C₄×C₆₇	—	8	4	no	—
129	C₄×C₆₇	—	8	4	no	—
130	C₄×C₆₇	#470	1	8	yes	complemented
131	C₄×C₆₇	C₂×#121	1	2	yes	—
132	C₄×C₆₇	#532	1	2	yes	—
133	#33176	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂×C₆₇	—	4	2	no	—
135	C₂×C₆₇	#2856	1	1	yes	characteristic
136	C₂×C₆₇	C₂×#470	1	2	yes	—
137	C₂×C₆₇	#3152	1	2	yes	—
138	C₂×C₆₇	#3185	1	2	yes	—
139	C₂×#494	C₂×C₆₇	1	1	yes	characteristic, complemented
140	C₂×#470	C₂×C₆₇	1	4	yes	—
141	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic
142	#2857	C₂×C₆₇	1	2	yes	—
143	#1971	C₂×C₆₇	1	4	yes	complemented
144	#1956	C₂×C₆₇	1	2	yes	complemented
145	#1844	C₂×C₆₇	1	1	yes	characteristic, complemented
146	C₆₇	#33176	1	1	yes	characteristic, direct product factor, Sylow
147	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	—
148	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
149	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
150	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
151	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
152	#494	—	2	4	no	—
153	#470	—	2	16	no	—
154	C₂×#106	C₂²×C₆₇	1	4	yes	—
155	C₂×#103	C₂²×C₆₇	1	8	yes	—
156	C₄×M₁₆	—	2	4	no	—
157	C₂×#97	C₂²×C₆₇	1	2	yes	—
158	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
160	#391	C₂²×C₆₇	1	4	yes	—
161	#389	—	2	4	no	—
162	#384	C₂²×C₆₇	1	2	yes	—
163	#380	—	2	4	no	—
164	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
165	C₂³×C₄	D₄×C₆₇	1	2	yes	—
166	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
167	C₂×Q₁₆	—	2	8	no	—
168	C₂×Q₁₆	—	2	8	no	—
169	C₂×Q₁₆	—	2	8	no	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	2	4	no	—
172	C₂×M₁₆	D₄×C₆₇	1	8	yes	—
173	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
174	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
175	C₄⋊Q₈	—	2	4	no	—
176	C₄⋊Q₈	—	2	4	no	—
177	C₂²⋊Q₈	—	2	16	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	4	no	—
181	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
182	C₂×C₂²⋊C₄	—	2	4	no	—
183	C₂×C₄²	—	2	2	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	D₄×C₆₇	1	2	yes	—
186	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
187	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
188	#106	—	2	8	no	—
189	#103	—	2	16	no	—
190	#97	—	2	4	no	—
191	C₄×C₈	—	2	2	no	—
192	C₄×C₈	D₄×C₆₇	1	4	yes	—
193	#95	—	2	8	no	—
194	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
195	C₂×Q₈	—	2	8	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	—	2	8	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	8	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
209	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
210	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
211	Q₁₆	—	4	16	no	—
212	M₁₆	—	2	16	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
216	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
217	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
218	C₂×C₈	#41×C₆₇	1	4	yes	—
219	C₄⋊C₄	—	2	8	no	—
220	C₄⋊C₄	—	4	8	no	—
221	C₄⋊C₄	—	4	8	no	—
222	C₄⋊C₄	—	4	8	no	—
223	C₂²⋊C₄	—	4	8	no	—
224	C₄²	—	2	4	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	1	no	—
227	C₄²	C₂×D₄×C₆₇	1	2	yes	—
228	C₄²	#41×C₆₇	1	2	yes	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	4	no	—
232	C₂³	#127×C₆₇	1	2	yes	—
233	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
234	Q₈	—	4	8	no	—
235	Q₈	—	4	8	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	#121×C₆₇	1	4	yes	—
249	C₂×C₄	#127×C₆₇	1	2	yes	—
250	C₂×C₄	#137×C₆₇	1	8	yes	—
251	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
253	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
254	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
255	C₈	—	2	4	no	—
256	C₈	#121×C₆₇	1	8	yes	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
262	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
263	C₂²	#532×C₆₇	1	1	yes	characteristic
264	C₂²	C₂×#137×C₆₇	1	4	yes	—
265	C₄	—	2	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	8	4	no	—
268	C₄	—	8	4	no	—
269	C₄	#470×C₆₇	1	8	yes	complemented
270	C₄	C₂×#121×C₆₇	1	2	yes	—
271	C₄	#532×C₆₇	1	2	yes	—
272	C₂	—	4	2	no	—
273	C₂	C₆₇×#2856	1	1	yes	characteristic
274	C₂	C₂×#470×C₆₇	1	2	yes	—
275	C₂	C₆₇×#3152	1	2	yes	—
276	C₂	C₆₇×#3185	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
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×#494×C₆₇	C₂	semidirect
5. C₆₇×#1971	C₂	semidirect
6. C₆₇×#1956	C₂	semidirect
7. C₆₇×#1844	C₂	semidirect
130. C₄×C₆₇	#470	semidirect
133. #33176	C₆₇	direct (N × H)
139. C₂×#494	C₂×C₆₇	semidirect
143. #1971	C₂×C₆₇	semidirect
144. #1956	C₂×C₆₇	semidirect
145. #1844	C₂×C₆₇	semidirect
146. C₆₇	#33176	direct (N × H)
269. C₄	#470×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#494×C₆₇	2
C₂×#470×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₆₇×#2857	2
C₆₇×#1971	2
C₆₇×#1956	2
C₆₇×#1844	2
#33176	67

Subgroups — C67×#33176 #30712427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33176 #30712427