Subgroups — C₆₇×#22702 #30701953

Order 17152.

888 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#435×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2293	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1940	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1851	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1850	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1798	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1729	C₂	1	2	yes	maximal, complemented
10	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
11	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#117×C₆₇	C₂²	1	2	yes	—
14	#435×C₆₇	—	2	8	no	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×#97×C₆₇	C₂²	1	2	yes	—
22	#390×C₆₇	C₄	1	1	yes	characteristic
23	#390×C₆₇	C₄	1	1	yes	characteristic
24	#389×C₆₇	C₄	1	1	yes	characteristic
25	#389×C₆₇	C₄	1	1	yes	characteristic
26	#388×C₆₇	C₄	1	2	yes	—
27	#388×C₆₇	C₄	1	2	yes	—
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₄³×C₆₇	C₂²	1	1	yes	characteristic
31	#336×C₆₇	C₂²	1	1	yes	characteristic
32	#336×C₆₇	C₂²	1	2	yes	—
33	#336×C₆₇	C₂²	1	2	yes	—
34	#336×C₆₇	C₂²	1	2	yes	—
35	#336×C₆₇	C₂²	1	1	yes	characteristic
36	#336×C₆₇	C₂²	1	1	yes	characteristic
37	#336×C₆₇	C₂²	1	1	yes	characteristic
38	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
40	C₂×M₁₆×C₆₇	—	2	4	no	—
41	C₂×M₁₆×C₆₇	—	2	4	no	—
42	C₂×M₁₆×C₆₇	—	4	2	no	—
43	C₂²×C₈×C₆₇	—	2	4	no	—
44	C₂²×C₈×C₆₇	—	2	4	no	—
45	C₂²×C₈×C₆₇	—	4	2	no	—
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₄×D₄×C₆₇	D₄	1	4	yes	—
51	C₄×D₄×C₆₇	Q₈	1	4	yes	—
52	#117×C₆₇	—	4	2	no	—
53	#117×C₆₇	—	4	2	no	—
54	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₆₇	—	2	4	no	—
73	C₄⋊C₈×C₆₇	—	2	4	no	—
74	C₄⋊C₈×C₆₇	—	4	2	no	—
75	C₄⋊C₈×C₆₇	—	4	2	no	—
76	C₂²⋊C₈×C₆₇	—	2	4	no	—
77	C₂²⋊C₈×C₆₇	—	2	4	no	—
78	#97×C₆₇	—	2	4	no	—
79	#95×C₆₇	C₂×C₄	1	2	yes	—
80	#95×C₆₇	C₂×C₄	1	2	yes	—
81	#95×C₆₇	C₂×C₄	1	2	yes	—
82	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂×D₄×C₆₇	—	4	1	no	—
84	C₂×D₄×C₆₇	—	4	1	no	—
85	C₂×D₄×C₆₇	C₄⋊C₄	1	4	yes	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	2	no	—
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
100	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
101	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
102	M₁₆×C₆₇	—	4	8	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	4	8	no	—
115	C₂×C₈×C₆₇	#41	1	4	yes	—
116	C₄⋊C₄×C₆₇	—	4	1	no	—
117	C₄⋊C₄×C₆₇	—	4	1	no	—
118	C₄⋊C₄×C₆₇	—	4	2	no	—
119	C₄⋊C₄×C₆₇	—	4	2	no	—
120	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
121	C₂²⋊C₄×C₆₇	—	2	4	no	—
122	C₂²⋊C₄×C₆₇	—	4	2	no	—
123	C₂²⋊C₄×C₆₇	—	4	2	no	—
124	C₂²⋊C₄×C₆₇	—	4	2	no	—
125	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
126	C₄²×C₆₇	—	2	2	no	—
127	C₄²×C₆₇	—	2	2	no	—
128	C₄²×C₆₇	—	2	2	no	—
129	C₄²×C₆₇	—	2	2	no	—
130	C₄²×C₆₇	—	2	2	no	—
131	C₄²×C₆₇	—	4	1	no	—
132	C₄²×C₆₇	—	4	1	no	—
133	C₄²×C₆₇	—	4	1	no	—
134	C₄²×C₆₇	C₂×D₄	1	2	yes	—
135	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
136	C₂³×C₆₇	—	2	2	no	—
137	C₂³×C₆₇	—	2	2	no	—
138	C₂³×C₆₇	—	2	2	no	—
139	C₂³×C₆₇	—	2	2	no	—
140	C₂³×C₆₇	—	2	2	no	—
141	C₂³×C₆₇	—	2	2	no	—
142	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
143	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
144	D₄×C₆₇	—	4	4	no	—
145	C₂×C₄×C₆₇	—	2	1	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	1	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	1	no	—
151	C₂×C₄×C₆₇	—	2	1	no	—
152	C₂×C₄×C₆₇	—	2	1	no	—
153	C₂×C₄×C₆₇	—	2	4	no	—
154	C₂×C₄×C₆₇	—	2	4	no	—
155	C₂×C₄×C₆₇	—	2	4	no	—
156	C₂×C₄×C₆₇	—	2	4	no	—
157	C₂×C₄×C₆₇	—	2	1	no	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	1	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
173	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
174	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
175	C₂×C₄×C₆₇	#131	1	2	yes	—
176	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
179	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
180	C₈×C₆₇	—	2	4	no	—
181	C₈×C₆₇	—	4	2	no	—
182	C₈×C₆₇	—	4	2	no	—
183	C₈×C₆₇	—	4	2	no	—
184	C₂²×C₆₇	—	2	4	no	—
185	C₂²×C₆₇	—	2	4	no	—
186	C₂²×C₆₇	—	2	4	no	—
187	C₂²×C₆₇	—	4	2	no	—
188	C₂²×C₆₇	—	4	2	no	—
189	C₂²×C₆₇	—	4	2	no	—
190	C₂²×C₆₇	—	4	2	no	—
191	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
192	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
193	C₂²×C₆₇	#518	1	1	yes	characteristic
194	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
195	C₂²×C₆₇	#568	1	1	yes	characteristic
196	C₂²×C₆₇	#568	1	1	yes	characteristic
197	C₂²×C₆₇	#568	1	1	yes	characteristic
198	C₄×C₆₇	—	2	4	no	—
199	C₄×C₆₇	—	2	2	no	—
200	C₄×C₆₇	—	2	2	no	—
201	C₄×C₆₇	—	2	2	no	—
202	C₄×C₆₇	—	4	1	no	—
203	C₄×C₆₇	—	4	1	no	—
204	C₄×C₆₇	—	4	2	no	—
205	C₄×C₆₇	—	4	1	no	—
206	C₄×C₆₇	—	4	2	no	—
207	C₄×C₆₇	#546	1	2	yes	—
208	C₄×C₆₇	#554	1	2	yes	—
209	#22702	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂×C₆₇	—	4	4	no	—
211	C₂×C₆₇	#2349	1	1	yes	characteristic
212	C₂×C₆₇	#2905	1	1	yes	characteristic
213	C₂×C₆₇	#2906	1	1	yes	characteristic
214	C₂×C₆₇	#2967	1	1	yes	characteristic
215	C₂×C₆₇	#2968	1	1	yes	characteristic
216	C₂×C₆₇	#2980	1	1	yes	characteristic
217	C₂×C₆₇	#2981	1	1	yes	characteristic
218	C₂×#435	C₂×C₆₇	1	2	yes	—
219	#2293	C₂×C₆₇	1	1	yes	characteristic
220	#1940	C₂×C₆₇	1	2	yes	complemented
221	#1851	C₂×C₆₇	1	2	yes	complemented
222	#1850	C₂×C₆₇	1	2	yes	complemented
223	#1831	C₂×C₆₇	1	1	yes	characteristic, complemented
224	#1798	C₂×C₆₇	1	2	yes	complemented
225	#1767	C₂×C₆₇	1	1	yes	characteristic, complemented
226	#1729	C₂×C₆₇	1	2	yes	complemented
227	C₆₇	#22702	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
229	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
230	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
231	C₂×#117	C₂²×C₆₇	1	2	yes	—
232	#435	—	2	8	no	—
233	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
234	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
235	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
236	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
237	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
238	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
239	C₂×#97	C₂²×C₆₇	1	2	yes	—
240	#390	C₄×C₆₇	1	1	yes	characteristic
241	#390	C₄×C₆₇	1	1	yes	characteristic
242	#389	C₄×C₆₇	1	1	yes	characteristic
243	#389	C₄×C₆₇	1	1	yes	characteristic
244	#388	C₄×C₆₇	1	2	yes	—
245	#388	C₄×C₆₇	1	2	yes	—
246	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
247	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
248	C₄³	C₂²×C₆₇	1	1	yes	characteristic
249	#336	C₂²×C₆₇	1	1	yes	characteristic
250	#336	C₂²×C₆₇	1	2	yes	—
251	#336	C₂²×C₆₇	1	1	yes	characteristic
252	#336	C₂²×C₆₇	1	1	yes	characteristic
253	#336	C₂²×C₆₇	1	2	yes	—
254	#336	C₂²×C₆₇	1	2	yes	—
255	#336	C₂²×C₆₇	1	1	yes	characteristic
256	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
257	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
258	C₂×M₁₆	—	2	4	no	—
259	C₂×M₁₆	—	2	4	no	—
260	C₂×M₁₆	—	4	2	no	—
261	C₂²×C₈	—	2	4	no	—
262	C₂²×C₈	—	2	4	no	—
263	C₂²×C₈	—	4	2	no	—
264	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
265	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
266	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
267	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
268	C₄×D₄	D₄×C₆₇	1	4	yes	—
269	C₄×D₄	Q₈×C₆₇	1	4	yes	—
270	#117	—	4	2	no	—
271	#117	—	4	2	no	—
272	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
273	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
274	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
275	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
276	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
277	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
278	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
279	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
280	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
281	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
282	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
283	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
284	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
285	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
286	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
287	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
288	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
289	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
290	C₄⋊C₈	—	2	4	no	—
291	C₄⋊C₈	—	2	4	no	—
292	C₄⋊C₈	—	4	2	no	—
293	C₄⋊C₈	—	4	2	no	—
294	C₂²⋊C₈	—	2	4	no	—
295	C₂²⋊C₈	—	2	4	no	—
296	#97	—	2	4	no	—
297	#95	C₂×C₄×C₆₇	1	2	yes	—
298	#95	C₂×C₄×C₆₇	1	2	yes	—
299	#95	C₂×C₄×C₆₇	1	2	yes	—
300	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
301	C₂×D₄	—	4	1	no	—
302	C₂×D₄	—	4	1	no	—
303	C₂×D₄	C₄⋊C₄×C₆₇	1	4	yes	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
309	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
310	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
311	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
312	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
313	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
314	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
315	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
316	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
318	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
319	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
320	M₁₆	—	4	8	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	4	8	no	—
333	C₂×C₈	#41×C₆₇	1	4	yes	—
334	C₄⋊C₄	—	4	2	no	—
335	C₄⋊C₄	—	4	2	no	—
336	C₄⋊C₄	—	4	1	no	—
337	C₄⋊C₄	—	4	1	no	—
338	C₄⋊C₄	C₄⋊C₄×C₆₇	1	4	yes	—
339	C₂²⋊C₄	—	2	4	no	—
340	C₂²⋊C₄	—	4	2	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₂²⋊C₄	—	4	2	no	—
343	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	8	yes	—
344	C₄²	—	2	2	no	—
345	C₄²	—	2	2	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	2	no	—
348	C₄²	—	2	2	no	—
349	C₄²	—	4	1	no	—
350	C₄²	—	4	1	no	—
351	C₄²	—	4	1	no	—
352	C₄²	C₂×D₄×C₆₇	1	2	yes	—
353	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
354	C₂³	—	2	2	no	—
355	C₂³	—	2	2	no	—
356	C₂³	—	2	2	no	—
357	C₂³	—	2	2	no	—
358	C₂³	—	2	2	no	—
359	C₂³	—	2	2	no	—
360	C₂³	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
361	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
362	D₄	—	4	4	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	4	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
391	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
392	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
393	C₂×C₄	#131×C₆₇	1	2	yes	—
394	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
395	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
396	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
397	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
398	C₈	—	2	4	no	—
399	C₈	—	4	2	no	—
400	C₈	—	4	2	no	—
401	C₈	—	4	2	no	—
402	C₂²	—	2	4	no	—
403	C₂²	—	2	4	no	—
404	C₂²	—	2	4	no	—
405	C₂²	—	4	2	no	—
406	C₂²	—	4	2	no	—
407	C₂²	—	4	2	no	—
408	C₂²	—	4	2	no	—
409	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
410	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
411	C₂²	#518×C₆₇	1	1	yes	characteristic
412	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
413	C₂²	#568×C₆₇	1	1	yes	characteristic
414	C₂²	#568×C₆₇	1	1	yes	characteristic
415	C₂²	#568×C₆₇	1	1	yes	characteristic
416	C₄	—	2	4	no	—
417	C₄	—	2	2	no	—
418	C₄	—	2	2	no	—
419	C₄	—	2	2	no	—
420	C₄	—	4	1	no	—
421	C₄	—	4	1	no	—
422	C₄	—	4	2	no	—
423	C₄	—	4	1	no	—
424	C₄	—	4	2	no	—
425	C₄	#546×C₆₇	1	2	yes	—
426	C₄	#554×C₆₇	1	2	yes	—
427	C₂	—	4	4	no	—
428	C₂	C₆₇×#2349	1	1	yes	characteristic
429	C₂	C₆₇×#2905	1	1	yes	characteristic
430	C₂	C₆₇×#2906	1	1	yes	characteristic
431	C₂	C₆₇×#2967	1	1	yes	characteristic
432	C₂	C₆₇×#2968	1	1	yes	characteristic
433	C₂	C₆₇×#2980	1	1	yes	characteristic
434	C₂	C₆₇×#2981	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂²×D₄×C₆₇	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1940	C₂	semidirect
4. C₆₇×#1851	C₂	semidirect
5. C₆₇×#1850	C₂	semidirect
6. C₆₇×#1831	C₂	semidirect
7. C₆₇×#1798	C₂	semidirect
8. C₆₇×#1767	C₂	semidirect
9. C₆₇×#1729	C₂	semidirect
209. #22702	C₆₇	direct (N × H)
220. #1940	C₂×C₆₇	semidirect
221. #1851	C₂×C₆₇	semidirect
222. #1850	C₂×C₆₇	semidirect
223. #1831	C₂×C₆₇	semidirect
224. #1798	C₂×C₆₇	semidirect
225. #1767	C₂×C₆₇	semidirect
226. #1729	C₂×C₆₇	semidirect
227. C₆₇	#22702	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#435×C₆₇	2
C₆₇×#2293	2
C₆₇×#1940	2
C₆₇×#1851	2
C₆₇×#1850	2
C₆₇×#1831	2
C₆₇×#1798	2
C₆₇×#1767	2
C₆₇×#1729	2
#22702	67

Subgroups — C67×#22702 #30701953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22702 #30701953