Subgroups — C₆₇×#9894 #30689145

Order 17152.

424 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2859	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1745	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1295	C₂	1	4	yes	maximal
5	C₂×#125×C₆₇	C₄	1	2	yes	—
6	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
7	C₂×#97×C₆₇	C₂²	1	2	yes	—
8	C₂×#97×C₆₇	C₂²	1	2	yes	—
9	#389×C₆₇	—	2	1	no	—
10	#389×C₆₇	—	2	1	no	—
11	#389×C₆₇	—	2	1	no	—
12	#389×C₆₇	—	2	1	no	—
13	#384×C₆₇	C₄	1	1	yes	characteristic
14	#384×C₆₇	C₄	1	1	yes	characteristic
15	#382×C₆₇	C₂²	1	2	yes	—
16	C₄³×C₆₇	C₂²	1	1	yes	characteristic
17	#336×C₆₇	—	2	1	no	—
18	#336×C₆₇	—	2	1	no	—
19	#336×C₆₇	—	2	1	no	—
20	C₂²×C₈×C₆₇	—	2	2	no	—
21	C₂²×C₈×C₆₇	—	2	2	no	—
22	#125×C₆₇	—	4	4	no	—
23	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
24	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
25	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
26	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
27	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
28	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₆₇	—	2	4	no	—
39	#97×C₆₇	—	2	2	no	—
40	#97×C₆₇	—	2	4	no	—
41	#97×C₆₇	Q₈	1	4	yes	—
42	#95×C₆₇	—	2	2	no	—
43	#95×C₆₇	—	2	2	no	—
44	C₂²×C₄×C₆₇	—	2	2	no	—
45	C₂²×C₄×C₆₇	—	2	2	no	—
46	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₆₇	—	2	4	no	—
54	C₂×C₈×C₆₇	—	2	4	no	—
55	C₂×C₈×C₆₇	—	4	2	no	—
56	C₂×C₈×C₆₇	—	4	2	no	—
57	C₂×C₈×C₆₇	—	4	2	no	—
58	C₂×C₈×C₆₇	—	4	2	no	—
59	C₄⋊C₄×C₆₇	—	4	2	no	—
60	C₄⋊C₄×C₆₇	—	4	2	no	—
61	C₄⋊C₄×C₆₇	—	4	2	no	—
62	C₄⋊C₄×C₆₇	—	4	2	no	—
63	C₄⋊C₄×C₆₇	—	4	2	no	—
64	C₄⋊C₄×C₆₇	—	4	4	no	—
65	C₄⋊C₄×C₆₇	—	4	4	no	—
66	C₄⋊C₄×C₆₇	—	4	2	no	—
67	C₄²×C₆₇	—	2	2	no	—
68	C₄²×C₆₇	—	2	2	no	—
69	C₄²×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	—	4	1	no	—
71	C₄²×C₆₇	—	4	1	no	—
72	C₄²×C₆₇	—	4	1	no	—
73	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
74	C₄²×C₆₇	#41	1	2	yes	—
75	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	—	2	1	no	—
77	C₂×C₄×C₆₇	—	2	1	no	—
78	C₂×C₄×C₆₇	—	2	1	no	—
79	C₂×C₄×C₆₇	—	2	1	no	—
80	C₂×C₄×C₆₇	—	2	1	no	—
81	C₂×C₄×C₆₇	—	2	1	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	1	no	—
87	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	4	2	no	—
92	C₂×C₄×C₆₇	—	4	2	no	—
93	C₂×C₄×C₆₇	—	4	2	no	—
94	C₂×C₄×C₆₇	—	4	2	no	—
95	C₂×C₄×C₆₇	—	4	2	no	—
96	C₂×C₄×C₆₇	—	4	2	no	—
97	C₂×C₄×C₆₇	#100	1	2	yes	—
98	C₂×C₄×C₆₇	#101	1	2	yes	—
99	C₂×C₄×C₆₇	#104	1	2	yes	—
100	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
101	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
104	C₈×C₆₇	—	4	4	no	—
105	C₈×C₆₇	—	4	4	no	—
106	C₂²×C₆₇	#391	1	1	yes	characteristic
107	C₂²×C₆₇	C₂×#100	1	1	yes	characteristic
108	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
109	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
110	C₂²×C₆₇	#421	1	1	yes	characteristic
111	C₂²×C₆₇	#421	1	1	yes	characteristic
112	C₂²×C₆₇	#421	1	1	yes	characteristic
113	C₄×C₆₇	—	2	1	no	—
114	C₄×C₆₇	—	2	1	no	—
115	C₄×C₆₇	—	2	2	no	—
116	C₄×C₆₇	—	2	2	no	—
117	C₄×C₆₇	—	2	2	no	—
118	C₄×C₆₇	—	2	2	no	—
119	C₄×C₆₇	—	2	2	no	—
120	C₄×C₆₇	—	4	1	no	—
121	C₄×C₆₇	—	8	2	no	—
122	C₄×C₆₇	—	8	2	no	—
123	#9894	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×C₆₇	#1528	1	1	yes	characteristic
125	C₂×C₆₇	#1530	1	1	yes	characteristic
126	C₂×C₆₇	#1534	1	1	yes	characteristic
127	C₂×C₆₇	#1536	1	1	yes	characteristic
128	C₂×C₆₇	#1986	1	1	yes	characteristic
129	C₂×C₆₇	#1995	1	1	yes	characteristic
130	C₂×C₆₇	#1997	1	1	yes	characteristic
131	#2859	C₂×C₆₇	1	1	yes	characteristic
132	#1767	C₂×C₆₇	1	1	yes	characteristic
133	#1745	C₂×C₆₇	1	1	yes	characteristic
134	#1295	C₂×C₆₇	1	4	yes	—
135	C₆₇	#9894	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×#125	C₄×C₆₇	1	2	yes	—
137	C₂×C₄⋊C₈	—	2	1	no	—
138	C₂×#97	C₂²×C₆₇	1	2	yes	—
139	C₂×#97	C₂²×C₆₇	1	2	yes	—
140	#389	—	2	1	no	—
141	#389	—	2	1	no	—
142	#389	—	2	1	no	—
143	#389	—	2	1	no	—
144	#384	C₄×C₆₇	1	1	yes	characteristic
145	#384	C₄×C₆₇	1	1	yes	characteristic
146	#382	C₂²×C₆₇	1	2	yes	—
147	C₄³	C₂²×C₆₇	1	1	yes	characteristic
148	#336	—	2	1	no	—
149	#336	—	2	1	no	—
150	#336	—	2	1	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	#125	—	4	4	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
161	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
162	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
163	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
164	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
165	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
166	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
167	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
168	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
169	C₄⋊C₈	—	2	4	no	—
170	#97	—	2	2	no	—
171	#97	—	2	4	no	—
172	#97	Q₈×C₆₇	1	4	yes	—
173	#95	—	2	2	no	—
174	#95	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
178	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
179	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
181	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	4	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	4	1	no	—
202	C₄²	—	4	1	no	—
203	C₄²	—	4	1	no	—
204	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
205	C₄²	#41×C₆₇	1	2	yes	—
206	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	#100×C₆₇	1	2	yes	—
229	C₂×C₄	#101×C₆₇	1	2	yes	—
230	C₂×C₄	#104×C₆₇	1	2	yes	—
231	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
232	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
233	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
234	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
235	C₈	—	4	4	no	—
236	C₈	—	4	4	no	—
237	C₂²	#391×C₆₇	1	1	yes	characteristic
238	C₂²	C₂×#100×C₆₇	1	1	yes	characteristic
239	C₂²	C₂×#101×C₆₇	1	1	yes	characteristic
240	C₂²	C₂×#104×C₆₇	1	1	yes	characteristic
241	C₂²	#421×C₆₇	1	1	yes	characteristic
242	C₂²	#421×C₆₇	1	1	yes	characteristic
243	C₂²	#421×C₆₇	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	4	1	no	—
252	C₄	—	8	2	no	—
253	C₄	—	8	2	no	—
254	C₂	C₆₇×#1528	1	1	yes	characteristic
255	C₂	C₆₇×#1530	1	1	yes	characteristic
256	C₂	C₆₇×#1534	1	1	yes	characteristic
257	C₂	C₆₇×#1536	1	1	yes	characteristic
258	C₂	C₆₇×#1986	1	1	yes	characteristic
259	C₂	C₆₇×#1995	1	1	yes	characteristic
260	C₂	C₆₇×#1997	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
123. #9894	C₆₇	direct (N × H)
135. C₆₇	#9894	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2859	2
C₆₇×#1767	2
C₆₇×#1745	2
C₆₇×#1295	2
#9894	67

Subgroups — C67×#9894 #30689145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#9894 #30689145