Subgroups — C₂×C₆₇×#1797 #30699875

Order 17152.

2222 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊C₈×C₆₇	C₂	1	2	yes	maximal
4	C₆₇×#1797	C₂	1	8	yes	maximal, direct product factor
5	C₂×#336×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₆₇	C₄	1	1	yes	characteristic
7	C₂⁴×C₄×C₆₇	C₄	1	1	yes	characteristic
8	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₆₇	C₂²	1	4	yes	—
10	C₂²×C₄²×C₆₇	—	2	8	no	—
11	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
12	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
13	C₂²×C₄²×C₆₇	C₄	1	4	yes	—
14	C₂²×C₄²×C₆₇	C₂²	1	4	yes	—
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₆₇	—	2	8	no	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	8	yes	complemented
19	C₂×C₂²⋊C₈×C₆₇	—	2	8	no	—
20	C₂×C₂²⋊C₈×C₆₇	C₂²	1	8	yes	—
21	#336×C₆₇	C₂²	1	8	yes	complemented
22	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₆₇	—	2	4	no	—
24	C₂³×C₄×C₆₇	—	2	4	no	—
25	C₂³×C₄×C₆₇	—	2	4	no	—
26	C₂³×C₄×C₆₇	—	2	4	no	—
27	C₂³×C₄×C₆₇	—	2	4	no	—
28	C₂³×C₄×C₆₇	—	2	4	no	—
29	C₂³×C₄×C₆₇	—	2	4	no	—
30	C₂³×C₄×C₆₇	C₈	1	8	yes	complemented
31	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
32	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
37	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	D₄	1	4	yes	—
40	C₂³×C₄×C₆₇	Q₈	1	4	yes	—
41	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
42	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	—	2	8	no	—
44	C₂²×C₈×C₆₇	—	2	8	no	—
45	C₂²×C₈×C₆₇	—	2	8	no	—
46	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
47	C₂×C₄²×C₆₇	—	2	4	no	—
48	C₂×C₄²×C₆₇	—	2	4	no	—
49	C₂×C₄²×C₆₇	—	2	4	no	—
50	C₂×C₄²×C₆₇	—	2	16	no	—
51	C₂×C₄²×C₆₇	—	2	16	no	—
52	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₆₇	D₄	1	8	yes	—
57	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
58	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
59	C₄⋊C₈×C₆₇	—	2	16	no	—
60	C₂²⋊C₈×C₆₇	—	2	16	no	—
61	C₂⁴×C₆₇	—	2	2	no	—
62	C₂⁴×C₆₇	—	2	2	no	—
63	C₂⁴×C₆₇	—	2	2	no	—
64	C₂⁴×C₆₇	—	2	2	no	—
65	C₂⁴×C₆₇	C₄⋊C₄	1	4	yes	—
66	C₂⁴×C₆₇	C₂×C₈	1	4	yes	—
67	C₂⁴×C₆₇	M₁₆	1	4	yes	—
68	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	—	2	32	no	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	—	2	8	no	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	—	2	2	no	—
78	C₂²×C₄×C₆₇	—	2	2	no	—
79	C₂²×C₄×C₆₇	—	2	2	no	—
80	C₂²×C₄×C₆₇	—	2	32	no	—
81	C₂²×C₄×C₆₇	—	2	8	no	—
82	C₂²×C₄×C₆₇	—	2	4	no	—
83	C₂²×C₄×C₆₇	—	2	8	no	—
84	C₂²×C₄×C₆₇	—	2	4	no	—
85	C₂²×C₄×C₆₇	—	2	16	no	—
86	C₂²×C₄×C₆₇	—	2	16	no	—
87	C₂²×C₄×C₆₇	—	2	8	no	—
88	C₂²×C₄×C₆₇	—	2	8	no	—
89	C₂²×C₄×C₆₇	—	2	4	no	—
90	C₂²×C₄×C₆₇	—	2	4	no	—
91	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
92	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
93	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
94	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
95	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
96	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
97	C₂²×C₄×C₆₇	C₂×C₈	1	16	yes	complemented
98	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
99	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
107	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
110	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
111	C₂²×C₄×C₆₇	#41	1	4	yes	—
112	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
113	C₂×C₈×C₆₇	—	2	8	no	—
114	C₂×C₈×C₆₇	—	2	8	no	—
115	C₂×C₈×C₆₇	—	2	8	no	—
116	C₂×C₈×C₆₇	—	4	16	no	—
117	C₄²×C₆₇	—	2	8	no	—
118	C₄²×C₆₇	—	2	4	no	—
119	C₄²×C₆₇	—	2	8	no	—
120	C₄²×C₆₇	—	2	4	no	—
121	C₄²×C₆₇	C₂²⋊C₄	1	8	yes	—
122	C₄²×C₆₇	C₂×D₄	1	8	yes	—
123	C₂³×C₆₇	—	2	2	no	—
124	C₂³×C₆₇	—	2	2	no	—
125	C₂³×C₆₇	—	2	2	no	—
126	C₂³×C₆₇	—	2	2	no	—
127	C₂³×C₆₇	—	2	8	no	—
128	C₂³×C₆₇	—	2	8	no	—
129	C₂³×C₆₇	—	2	8	no	—
130	C₂³×C₆₇	—	2	8	no	—
131	C₂³×C₆₇	—	2	8	no	—
132	C₂³×C₆₇	—	2	8	no	—
133	C₂³×C₆₇	C₄⋊C₈	1	16	yes	complemented
134	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
135	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
136	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
137	C₂³×C₆₇	#117	1	2	yes	—
138	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
139	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
140	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
141	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
142	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
143	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
144	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	—	2	32	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	2	8	no	—
155	C₂×C₄×C₆₇	—	2	8	no	—
156	C₂×C₄×C₆₇	—	2	8	no	—
157	C₂×C₄×C₆₇	—	2	8	no	—
158	C₂×C₄×C₆₇	—	2	8	no	—
159	C₂×C₄×C₆₇	—	2	8	no	—
160	C₂×C₄×C₆₇	—	2	16	no	—
161	C₂×C₄×C₆₇	—	2	32	no	—
162	C₂×C₄×C₆₇	C₂²⋊C₈	1	16	yes	complemented
163	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
164	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
165	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
166	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
167	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
168	C₂×C₄×C₆₇	#117	1	2	yes	—
169	C₂×C₄×C₆₇	#121	1	8	yes	—
170	C₂×C₄×C₆₇	C₂²⋊Q₈	1	8	yes	—
171	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
172	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
173	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
174	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
175	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
177	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
179	C₈×C₆₇	—	4	8	no	—
180	C₂²×C₆₇	—	2	32	no	—
181	C₂²×C₆₇	—	2	4	no	—
182	C₂²×C₆₇	—	2	4	no	—
183	C₂²×C₆₇	—	2	4	no	—
184	C₂²×C₆₇	—	2	4	no	—
185	C₂²×C₆₇	—	2	4	no	—
186	C₂²×C₆₇	—	2	4	no	—
187	C₂²×C₆₇	#380	1	4	yes	—
188	C₂²×C₆₇	C₂×C₂²⋊C₈	1	4	yes	—
189	C₂²×C₆₇	#407	1	4	yes	—
190	C₂²×C₆₇	C₂×C₄⋊C₈	1	4	yes	—
191	C₂²×C₆₇	C₂×C₄⋊C₈	1	8	yes	complemented
192	C₂²×C₆₇	#423	1	4	yes	—
193	C₂²×C₆₇	#431	1	4	yes	—
194	C₂²×C₆₇	#432	1	4	yes	—
195	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
196	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
197	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
198	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
199	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
200	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
201	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
202	C₄×C₆₇	—	2	4	no	—
203	C₄×C₆₇	—	2	4	no	—
204	C₄×C₆₇	—	2	4	no	—
205	C₄×C₆₇	—	2	4	no	—
206	C₄×C₆₇	#380	1	4	yes	—
207	C₄×C₆₇	C₂×C₂²⋊C₈	1	8	yes	complemented
208	C₄×C₆₇	C₂×#121	1	2	yes	—
209	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
210	C₂×#1797	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂×C₆₇	—	2	8	no	—
212	C₂×C₆₇	#1797	1	8	yes	direct product factor
213	C₂×C₆₇	C₂×#380	1	1	yes	characteristic
214	C₂×C₆₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
215	C₂×C₆₇	C₂×#407	1	1	yes	characteristic
216	C₂×C₆₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
217	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
218	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
219	C₂×C₆₇	C₂×#432	1	1	yes	characteristic
220	C₂³×C₄²	C₂×C₆₇	1	1	yes	characteristic
221	C₂²×C₄⋊C₈	C₂×C₆₇	1	2	yes	complemented
222	C₂²×C₂²⋊C₈	C₂×C₆₇	1	2	yes	—
223	#1797	C₂×C₆₇	1	8	yes	direct product factor
224	C₂×#336	C₂×C₆₇	1	2	yes	complemented
225	C₆₇	C₂×#1797	1	1	yes	characteristic, direct product factor, Sylow
226	C₂⁴×C₄	C₄×C₆₇	1	1	yes	characteristic
227	C₂⁴×C₄	C₄×C₆₇	1	1	yes	characteristic
228	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
229	C₂³×C₈	C₂²×C₆₇	1	4	yes	—
230	C₂²×C₄²	—	2	8	no	—
231	C₂²×C₄²	C₄×C₆₇	1	1	yes	characteristic
232	C₂²×C₄²	C₄×C₆₇	1	1	yes	characteristic
233	C₂²×C₄²	C₄×C₆₇	1	4	yes	—
234	C₂²×C₄²	C₂²×C₆₇	1	4	yes	—
235	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
236	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
237	C₂×C₄⋊C₈	—	2	8	no	—
238	C₂×C₄⋊C₈	C₂²×C₆₇	1	8	yes	complemented
239	C₂×C₂²⋊C₈	—	2	8	no	—
240	C₂×C₂²⋊C₈	C₂²×C₆₇	1	8	yes	—
241	#336	C₂²×C₆₇	1	8	yes	complemented
242	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
243	C₂³×C₄	—	2	4	no	—
244	C₂³×C₄	—	2	4	no	—
245	C₂³×C₄	—	2	4	no	—
246	C₂³×C₄	—	2	4	no	—
247	C₂³×C₄	—	2	4	no	—
248	C₂³×C₄	—	2	4	no	—
249	C₂³×C₄	—	2	4	no	—
250	C₂³×C₄	C₈×C₆₇	1	8	yes	complemented
251	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
252	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
253	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
254	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
255	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
256	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
257	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
258	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
259	C₂³×C₄	D₄×C₆₇	1	4	yes	—
260	C₂³×C₄	Q₈×C₆₇	1	4	yes	—
261	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
262	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
263	C₂²×C₈	—	2	8	no	—
264	C₂²×C₈	—	2	8	no	—
265	C₂²×C₈	—	2	8	no	—
266	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
267	C₂×C₄²	—	2	4	no	—
268	C₂×C₄²	—	2	4	no	—
269	C₂×C₄²	—	2	4	no	—
270	C₂×C₄²	—	2	16	no	—
271	C₂×C₄²	—	2	16	no	—
272	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
273	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
274	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
275	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
276	C₂×C₄²	D₄×C₆₇	1	8	yes	—
277	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
278	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
279	C₄⋊C₈	—	2	16	no	—
280	C₂²⋊C₈	—	2	16	no	—
281	C₂⁴	—	2	2	no	—
282	C₂⁴	—	2	2	no	—
283	C₂⁴	—	2	2	no	—
284	C₂⁴	—	2	2	no	—
285	C₂⁴	C₄⋊C₄×C₆₇	1	4	yes	—
286	C₂⁴	C₂×C₈×C₆₇	1	4	yes	—
287	C₂⁴	M₁₆×C₆₇	1	4	yes	—
288	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
289	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
290	C₂²×C₄	—	2	32	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	2	8	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	8	no	—
301	C₂²×C₄	—	2	4	no	—
302	C₂²×C₄	—	2	8	no	—
303	C₂²×C₄	—	2	4	no	—
304	C₂²×C₄	—	2	16	no	—
305	C₂²×C₄	—	2	16	no	—
306	C₂²×C₄	—	2	8	no	—
307	C₂²×C₄	—	2	8	no	—
308	C₂²×C₄	—	2	4	no	—
309	C₂²×C₄	—	2	4	no	—
310	C₂²×C₄	—	2	32	no	—
311	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
312	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
313	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
314	C₂²×C₄	C₄⋊C₄×C₆₇	1	8	yes	—
315	C₂²×C₄	C₂×C₈×C₆₇	1	4	yes	—
316	C₂²×C₄	C₂×C₈×C₆₇	1	4	yes	—
317	C₂²×C₄	C₂×C₈×C₆₇	1	16	yes	complemented
318	C₂²×C₄	M₁₆×C₆₇	1	4	yes	—
319	C₂²×C₄	M₁₆×C₆₇	1	4	yes	—
320	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
321	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
322	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
323	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
324	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
325	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
326	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
327	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
328	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
329	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
330	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
331	C₂²×C₄	#41×C₆₇	1	4	yes	—
332	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
333	C₂×C₈	—	2	8	no	—
334	C₂×C₈	—	2	8	no	—
335	C₂×C₈	—	2	8	no	—
336	C₂×C₈	—	4	16	no	—
337	C₄²	—	2	8	no	—
338	C₄²	—	2	4	no	—
339	C₄²	—	2	8	no	—
340	C₄²	—	2	4	no	—
341	C₄²	C₂²⋊C₄×C₆₇	1	8	yes	—
342	C₄²	C₂×D₄×C₆₇	1	8	yes	—
343	C₂³	—	2	2	no	—
344	C₂³	—	2	2	no	—
345	C₂³	—	2	2	no	—
346	C₂³	—	2	2	no	—
347	C₂³	—	2	8	no	—
348	C₂³	—	2	8	no	—
349	C₂³	—	2	8	no	—
350	C₂³	—	2	8	no	—
351	C₂³	—	2	8	no	—
352	C₂³	—	2	8	no	—
353	C₂³	C₄⋊C₈×C₆₇	1	16	yes	complemented
354	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
355	C₂³	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
356	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
357	C₂³	#117×C₆₇	1	2	yes	—
358	C₂³	C₂²×C₈×C₆₇	1	4	yes	—
359	C₂³	C₂²×C₈×C₆₇	1	2	yes	—
360	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
361	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
362	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
363	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
364	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
365	C₂×C₄	—	2	32	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	8	no	—
375	C₂×C₄	—	2	8	no	—
376	C₂×C₄	—	2	8	no	—
377	C₂×C₄	—	2	8	no	—
378	C₂×C₄	—	2	8	no	—
379	C₂×C₄	—	2	8	no	—
380	C₂×C₄	—	2	16	no	—
381	C₂×C₄	—	2	32	no	—
382	C₂×C₄	C₂²⋊C₈×C₆₇	1	16	yes	complemented
383	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
384	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
385	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
386	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
387	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
388	C₂×C₄	#117×C₆₇	1	2	yes	—
389	C₂×C₄	#121×C₆₇	1	8	yes	—
390	C₂×C₄	C₂²⋊Q₈×C₆₇	1	8	yes	—
391	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
392	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
393	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
394	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
395	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
396	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
397	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
398	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
399	C₈	—	4	8	no	—
400	C₂²	—	2	4	no	—
401	C₂²	—	2	32	no	—
402	C₂²	—	2	4	no	—
403	C₂²	—	2	4	no	—
404	C₂²	—	2	4	no	—
405	C₂²	—	2	4	no	—
406	C₂²	—	2	4	no	—
407	C₂²	#380×C₆₇	1	4	yes	—
408	C₂²	C₂×C₂²⋊C₈×C₆₇	1	4	yes	—
409	C₂²	#407×C₆₇	1	4	yes	—
410	C₂²	C₂×C₄⋊C₈×C₆₇	1	4	yes	—
411	C₂²	C₂×C₄⋊C₈×C₆₇	1	8	yes	complemented
412	C₂²	#423×C₆₇	1	4	yes	—
413	C₂²	#431×C₆₇	1	4	yes	—
414	C₂²	#432×C₆₇	1	4	yes	—
415	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
416	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
417	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
418	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
419	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
420	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
421	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
422	C₄	—	2	4	no	—
423	C₄	—	2	4	no	—
424	C₄	—	2	4	no	—
425	C₄	—	2	4	no	—
426	C₄	#380×C₆₇	1	4	yes	—
427	C₄	C₂×C₂²⋊C₈×C₆₇	1	8	yes	complemented
428	C₄	C₂×#121×C₆₇	1	2	yes	—
429	C₄	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
430	C₂	—	2	8	no	—
431	C₂	C₆₇×#1797	1	8	yes	direct product factor
432	C₂	C₂×#380×C₆₇	1	1	yes	characteristic
433	C₂	C₂²×C₂²⋊C₈×C₆₇	1	1	yes	characteristic
434	C₂	C₂×#407×C₆₇	1	1	yes	characteristic
435	C₂	C₂²×C₄⋊C₈×C₆₇	1	1	yes	characteristic
436	C₂	C₂×#423×C₆₇	1	1	yes	characteristic
437	C₂	C₂×#431×C₆₇	1	1	yes	characteristic
438	C₂	C₂×#432×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₆₇	C₂³
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂⁴×C₆₇	C₂²×C₄
prime radical	C₂⁵×C₆₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊C₈×C₆₇	C₂	semidirect
4. C₆₇×#1797	C₂	direct (N × H)
5. C₂×#336×C₆₇	C₂	semidirect
18. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
21. #336×C₆₇	C₂²	semidirect
30. C₂³×C₄×C₆₇	C₈	semidirect
97. C₂²×C₄×C₆₇	C₂×C₈	semidirect
133. C₂³×C₆₇	C₄⋊C₈	semidirect
162. C₂×C₄×C₆₇	C₂²⋊C₈	semidirect
191. C₂²×C₆₇	C₂×C₄⋊C₈	semidirect
207. C₄×C₆₇	C₂×C₂²⋊C₈	semidirect
210. C₂×#1797	C₆₇	direct (N × H)
212. C₂×C₆₇	#1797	direct (N × H)
221. C₂²×C₄⋊C₈	C₂×C₆₇	semidirect
223. #1797	C₂×C₆₇	direct (N × H)
224. C₂×#336	C₂×C₆₇	semidirect
225. C₆₇	C₂×#1797	direct (N × H)
238. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
241. #336	C₂²×C₆₇	semidirect
250. C₂³×C₄	C₈×C₆₇	semidirect
317. C₂²×C₄	C₂×C₈×C₆₇	semidirect
353. C₂³	C₄⋊C₈×C₆₇	semidirect
382. C₂×C₄	C₂²⋊C₈×C₆₇	semidirect
411. C₂²	C₂×C₄⋊C₈×C₆₇	semidirect
427. C₄	C₂×C₂²⋊C₈×C₆₇	semidirect
431. C₂	C₆₇×#1797	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄²×C₆₇	2
C₂²×C₄⋊C₈×C₆₇	2
C₂²×C₂²⋊C₈×C₆₇	2
C₆₇×#1797	2
C₂×#336×C₆₇	2
C₂×#1797	67

Subgroups — C2×C67×#1797 #30699875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1797 #30699875