Subgroups — C₆₇×#22798 #30702049

Order 17152.

808 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#433×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2293	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1955	C₂	1	2	yes	maximal
4	C₆₇×#1924	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1923	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1798	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1771	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#117×C₆₇	C₂²	1	2	yes	—
12	#433×C₆₇	—	2	8	no	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×#97×C₆₇	C₂²	1	2	yes	—
24	C₂×#97×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
26	#390×C₆₇	C₄	1	1	yes	characteristic
27	#390×C₆₇	C₄	1	1	yes	characteristic
28	#389×C₆₇	C₄	1	1	yes	characteristic
29	#389×C₆₇	C₄	1	1	yes	characteristic
30	#388×C₆₇	C₄	1	2	yes	—
31	#388×C₆₇	C₄	1	2	yes	—
32	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
34	C₄³×C₆₇	C₂²	1	1	yes	characteristic
35	#336×C₆₇	C₂²	1	2	yes	—
36	#336×C₆₇	C₂²	1	2	yes	—
37	#336×C₆₇	C₂²	1	2	yes	—
38	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
43	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
44	C₄×D₄×C₆₇	—	2	2	no	—
45	C₄×D₄×C₆₇	—	2	2	no	—
46	#117×C₆₇	—	2	4	no	—
47	#117×C₆₇	—	2	4	no	—
48	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
54	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
57	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₆₇	—	2	4	no	—
67	C₄⋊C₈×C₆₇	—	2	4	no	—
68	C₄⋊C₈×C₆₇	—	2	2	no	—
69	C₄⋊C₈×C₆₇	—	4	1	no	—
70	C₄⋊C₈×C₆₇	—	4	1	no	—
71	C₄⋊C₈×C₆₇	D₄	1	4	yes	complemented
72	C₂²⋊C₈×C₆₇	—	2	4	no	—
73	C₂²⋊C₈×C₆₇	—	2	4	no	—
74	C₂²⋊C₈×C₆₇	—	4	2	no	—
75	C₂²⋊C₈×C₆₇	—	4	2	no	—
76	#97×C₆₇	—	2	4	no	—
77	#97×C₆₇	—	2	4	no	—
78	C₄×C₈×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	—	2	2	no	—
84	C₂×D₄×C₆₇	—	4	1	no	—
85	C₂×D₄×C₆₇	—	4	1	no	—
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	—	2	2	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
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	2	yes	—
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
102	C₂×C₈×C₆₇	—	2	2	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	C₂×D₄	1	4	yes	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	4	no	—
116	C₄⋊C₄×C₆₇	—	2	4	no	—
117	C₄⋊C₄×C₆₇	—	4	1	no	—
118	C₄⋊C₄×C₆₇	—	4	1	no	—
119	C₂²⋊C₄×C₆₇	—	2	4	no	—
120	C₂²⋊C₄×C₆₇	—	2	4	no	—
121	C₂²⋊C₄×C₆₇	—	2	4	no	—
122	C₂²⋊C₄×C₆₇	—	2	4	no	—
123	C₂²⋊C₄×C₆₇	—	4	2	no	—
124	C₄²×C₆₇	—	2	2	no	—
125	C₄²×C₆₇	—	2	2	no	—
126	C₄²×C₆₇	—	2	2	no	—
127	C₄²×C₆₇	—	2	1	no	—
128	C₄²×C₆₇	—	2	1	no	—
129	C₄²×C₆₇	—	4	1	no	—
130	C₄²×C₆₇	—	4	1	no	—
131	C₄²×C₆₇	—	4	1	no	—
132	C₄²×C₆₇	C₂×D₄	1	2	yes	—
133	C₄²×C₆₇	#41	1	2	yes	—
134	C₄²×C₆₇	#41	1	2	yes	—
135	C₄²×C₆₇	#41	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	—	2	2	no	—
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	1	no	—
147	C₂×C₄×C₆₇	—	2	1	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	1	no	—
154	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	2	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	1	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	4	2	no	—
168	C₂×C₄×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	#117	1	2	yes	—
172	C₂×C₄×C₆₇	#117	1	2	yes	—
173	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
174	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
175	C₂×C₄×C₆₇	#131	1	2	yes	—
176	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
179	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
180	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
181	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
182	C₈×C₆₇	—	2	4	no	—
183	C₈×C₆₇	—	4	2	no	—
184	C₈×C₆₇	—	4	2	no	—
185	C₈×C₆₇	—	4	2	no	—
186	C₂²×C₆₇	—	4	2	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	—	4	2	no	—
192	C₂²×C₆₇	—	4	2	no	—
193	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
194	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
195	C₂²×C₆₇	#520	1	1	yes	characteristic
196	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
197	C₂²×C₆₇	#568	1	1	yes	characteristic
198	C₂²×C₆₇	#568	1	1	yes	characteristic
199	C₂²×C₆₇	#568	1	1	yes	characteristic
200	C₄×C₆₇	—	2	2	no	—
201	C₄×C₆₇	—	2	2	no	—
202	C₄×C₆₇	—	2	2	no	—
203	C₄×C₆₇	—	2	2	no	—
204	C₄×C₆₇	—	2	2	no	—
205	C₄×C₆₇	—	4	2	no	—
206	C₄×C₆₇	—	4	2	no	—
207	C₄×C₆₇	—	4	2	no	—
208	C₄×C₆₇	—	4	1	no	—
209	C₄×C₆₇	#547	1	1	yes	characteristic
210	C₄×C₆₇	#550	1	1	yes	characteristic
211	C₄×C₆₇	#555	1	1	yes	characteristic
212	C₄×C₆₇	#556	1	1	yes	characteristic
213	#22798	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
214	C₂×C₆₇	—	8	2	no	—
215	C₂×C₆₇	#2354	1	1	yes	characteristic
216	C₂×C₆₇	#2921	1	1	yes	characteristic
217	C₂×C₆₇	#2925	1	1	yes	characteristic
218	C₂×C₆₇	#2932	1	1	yes	characteristic
219	C₂×C₆₇	#2935	1	1	yes	characteristic
220	C₂×C₆₇	#2992	1	1	yes	characteristic
221	C₂×C₆₇	#2995	1	1	yes	characteristic
222	C₂×#433	C₂×C₆₇	1	2	yes	complemented
223	#2293	C₂×C₆₇	1	1	yes	characteristic
224	#1955	C₂×C₆₇	1	2	yes	—
225	#1924	C₂×C₆₇	1	2	yes	complemented
226	#1923	C₂×C₆₇	1	2	yes	complemented
227	#1842	C₂×C₆₇	1	1	yes	characteristic, complemented
228	#1798	C₂×C₆₇	1	2	yes	complemented
229	#1771	C₂×C₆₇	1	2	yes	complemented
230	#1767	C₂×C₆₇	1	1	yes	characteristic, complemented
231	C₆₇	#22798	1	1	yes	characteristic, direct product factor, Sylow
232	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
233	C₂×#117	C₂²×C₆₇	1	2	yes	—
234	#433	—	2	8	no	—
235	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
237	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
238	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
239	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
240	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
241	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
242	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
243	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
244	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
245	C₂×#97	C₂²×C₆₇	1	2	yes	—
246	C₂×#97	C₂²×C₆₇	1	2	yes	—
247	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
248	#390	C₄×C₆₇	1	1	yes	characteristic
249	#390	C₄×C₆₇	1	1	yes	characteristic
250	#389	C₄×C₆₇	1	1	yes	characteristic
251	#389	C₄×C₆₇	1	1	yes	characteristic
252	#388	C₄×C₆₇	1	2	yes	—
253	#388	C₄×C₆₇	1	2	yes	—
254	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
255	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
256	C₄³	C₂²×C₆₇	1	1	yes	characteristic
257	#336	C₂²×C₆₇	1	2	yes	—
258	#336	C₂²×C₆₇	1	2	yes	—
259	#336	C₂²×C₆₇	1	2	yes	—
260	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
261	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
262	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
263	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
264	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
265	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
266	C₄×D₄	—	2	2	no	—
267	C₄×D₄	—	2	2	no	—
268	#117	—	2	4	no	—
269	#117	—	2	4	no	—
270	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
271	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
272	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
273	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
274	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
275	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
276	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
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×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
281	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
282	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
283	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
284	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
285	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
286	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
287	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
288	C₄⋊C₈	—	2	4	no	—
289	C₄⋊C₈	—	2	4	no	—
290	C₄⋊C₈	—	2	2	no	—
291	C₄⋊C₈	—	4	1	no	—
292	C₄⋊C₈	—	4	1	no	—
293	C₄⋊C₈	D₄×C₆₇	1	4	yes	complemented
294	C₂²⋊C₈	—	2	4	no	—
295	C₂²⋊C₈	—	2	4	no	—
296	C₂²⋊C₈	—	4	2	no	—
297	C₂²⋊C₈	—	4	2	no	—
298	#97	—	2	4	no	—
299	#97	—	2	4	no	—
300	C₄×C₈	—	2	4	no	—
301	#95	C₂×C₄×C₆₇	1	2	yes	—
302	#95	C₂×C₄×C₆₇	1	2	yes	—
303	#95	C₂×C₄×C₆₇	1	2	yes	—
304	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	4	1	no	—
307	C₂×D₄	—	4	1	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
315	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
316	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
317	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
318	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
319	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
320	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
321	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
322	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
323	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
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	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	4	no	—
338	C₄⋊C₄	—	2	4	no	—
339	C₄⋊C₄	—	4	1	no	—
340	C₄⋊C₄	—	4	1	no	—
341	C₂²⋊C₄	—	2	4	no	—
342	C₂²⋊C₄	—	2	4	no	—
343	C₂²⋊C₄	—	2	4	no	—
344	C₂²⋊C₄	—	2	4	no	—
345	C₂²⋊C₄	—	4	2	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	2	no	—
348	C₄²	—	2	2	no	—
349	C₄²	—	2	1	no	—
350	C₄²	—	2	1	no	—
351	C₄²	—	4	1	no	—
352	C₄²	—	4	1	no	—
353	C₄²	—	4	1	no	—
354	C₄²	C₂×D₄×C₆₇	1	2	yes	—
355	C₄²	#41×C₆₇	1	2	yes	—
356	C₄²	#41×C₆₇	1	2	yes	—
357	C₄²	#41×C₆₇	1	2	yes	—
358	C₂³	—	2	2	no	—
359	C₂³	—	2	2	no	—
360	C₂³	—	2	2	no	—
361	C₂³	—	2	2	no	—
362	C₂³	—	2	2	no	—
363	C₂³	—	2	2	no	—
364	C₂³	—	2	2	no	—
365	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
366	D₄	—	4	4	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	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	#117×C₆₇	1	2	yes	—
394	C₂×C₄	#117×C₆₇	1	2	yes	—
395	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
396	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
397	C₂×C₄	#131×C₆₇	1	2	yes	—
398	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
399	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
400	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
401	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
402	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
403	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
404	C₈	—	2	4	no	—
405	C₈	—	4	2	no	—
406	C₈	—	4	2	no	—
407	C₈	—	4	2	no	—
408	C₂²	—	4	2	no	—
409	C₂²	—	4	2	no	—
410	C₂²	—	4	2	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	4	2	no	—
413	C₂²	—	4	2	no	—
414	C₂²	—	4	2	no	—
415	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
416	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
417	C₂²	#520×C₆₇	1	1	yes	characteristic
418	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
419	C₂²	#568×C₆₇	1	1	yes	characteristic
420	C₂²	#568×C₆₇	1	1	yes	characteristic
421	C₂²	#568×C₆₇	1	1	yes	characteristic
422	C₄	—	2	2	no	—
423	C₄	—	2	2	no	—
424	C₄	—	2	2	no	—
425	C₄	—	2	2	no	—
426	C₄	—	2	2	no	—
427	C₄	—	4	2	no	—
428	C₄	—	4	2	no	—
429	C₄	—	4	2	no	—
430	C₄	—	4	1	no	—
431	C₄	#547×C₆₇	1	1	yes	characteristic
432	C₄	#550×C₆₇	1	1	yes	characteristic
433	C₄	#555×C₆₇	1	1	yes	characteristic
434	C₄	#556×C₆₇	1	1	yes	characteristic
435	C₂	—	8	2	no	—
436	C₂	C₆₇×#2354	1	1	yes	characteristic
437	C₂	C₆₇×#2921	1	1	yes	characteristic
438	C₂	C₆₇×#2925	1	1	yes	characteristic
439	C₂	C₆₇×#2932	1	1	yes	characteristic
440	C₂	C₆₇×#2935	1	1	yes	characteristic
441	C₂	C₆₇×#2992	1	1	yes	characteristic
442	C₂	C₆₇×#2995	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
1. C₂×#433×C₆₇	C₂	semidirect
4. C₆₇×#1924	C₂	semidirect
5. C₆₇×#1923	C₂	semidirect
6. C₆₇×#1842	C₂	semidirect
7. C₆₇×#1798	C₂	semidirect
8. C₆₇×#1771	C₂	semidirect
9. C₆₇×#1767	C₂	semidirect
71. C₄⋊C₈×C₆₇	D₄	semidirect
213. #22798	C₆₇	direct (N × H)
222. C₂×#433	C₂×C₆₇	semidirect
225. #1924	C₂×C₆₇	semidirect
226. #1923	C₂×C₆₇	semidirect
227. #1842	C₂×C₆₇	semidirect
228. #1798	C₂×C₆₇	semidirect
229. #1771	C₂×C₆₇	semidirect
230. #1767	C₂×C₆₇	semidirect
231. C₆₇	#22798	direct (N × H)
293. C₄⋊C₈	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#433×C₆₇	2
C₆₇×#2293	2
C₆₇×#1955	2
C₆₇×#1924	2
C₆₇×#1923	2
C₆₇×#1842	2
C₆₇×#1798	2
C₆₇×#1771	2
C₆₇×#1767	2
#22798	67

Subgroups — C67×#22798 #30702049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22798 #30702049