Subgroups — C₂×C₆₇×#2500 #30714045

Order 17152.

3222 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#121×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2500	C₂	1	16	yes	maximal, direct product factor
4	C₂×#394×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×#390×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#390×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#388×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#382×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#121×C₆₇	—	2	8	no	—
15	C₂×#121×C₆₇	C₂²	1	8	yes	—
16	C₂×#117×C₆₇	—	2	8	no	—
17	C₂×#117×C₆₇	C₂²	1	8	yes	complemented
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
25	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
27	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
30	#394×C₆₇	C₂²	1	32	yes	complemented
31	#390×C₆₇	C₂²	1	8	yes	complemented
32	#390×C₆₇	C₂²	1	8	yes	complemented
33	#388×C₆₇	C₂²	1	32	yes	complemented
34	#384×C₆₇	C₂²	1	8	yes	complemented
35	#382×C₆₇	C₂²	1	8	yes	complemented
36	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
37	C₂×#95×C₆₇	C₂²	1	2	yes	—
38	C₂×#95×C₆₇	C₂²	1	4	yes	—
39	C₂⁵×C₆₇	C₂³	1	2	yes	—
40	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×D₄×C₆₇	—	2	8	no	—
42	C₂²×D₄×C₆₇	—	2	4	no	—
43	C₂²×D₄×C₆₇	—	2	4	no	—
44	C₂²×D₄×C₆₇	—	2	4	no	—
45	C₂²×D₄×C₆₇	—	4	8	no	—
46	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
47	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
48	C₂³×C₄×C₆₇	—	2	4	no	—
49	C₂³×C₄×C₆₇	—	2	4	no	—
50	C₂³×C₄×C₆₇	D₄	1	8	yes	—
51	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
52	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
53	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
54	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
55	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
56	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
58	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	#121×C₆₇	—	2	32	no	—
60	#117×C₆₇	—	2	32	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
64	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
71	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
74	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
75	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
76	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
77	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
78	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
79	C₂×C₄²×C₆₇	—	2	8	no	—
80	C₂×C₄²×C₆₇	—	2	4	no	—
81	C₂×C₄²×C₆₇	—	2	4	no	—
82	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
83	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
84	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
85	#95×C₆₇	C₂³	1	8	yes	—
86	#95×C₆₇	C₂³	1	16	yes	—
87	C₂⁴×C₆₇	—	2	4	no	—
88	C₂⁴×C₆₇	—	2	2	no	—
89	C₂⁴×C₆₇	—	2	2	no	—
90	C₂⁴×C₆₇	—	2	4	no	—
91	C₂⁴×C₆₇	—	2	4	no	—
92	C₂⁴×C₆₇	—	2	8	no	—
93	C₂⁴×C₆₇	—	2	4	no	—
94	C₂⁴×C₆₇	—	2	4	no	—
95	C₂⁴×C₆₇	—	2	8	no	—
96	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
97	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
98	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
99	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₂×D₄×C₆₇	—	2	16	no	—
101	C₂×D₄×C₆₇	—	2	8	no	—
102	C₂×D₄×C₆₇	—	2	8	no	—
103	C₂×D₄×C₆₇	—	2	8	no	—
104	C₂×D₄×C₆₇	—	4	16	no	—
105	C₂×D₄×C₆₇	—	4	32	no	—
106	C₂×D₄×C₆₇	—	4	16	no	—
107	C₂²×C₄×C₆₇	—	2	4	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	4	no	—
111	C₂²×C₄×C₆₇	—	2	32	no	—
112	C₂²×C₄×C₆₇	—	2	8	no	—
113	C₂²×C₄×C₆₇	—	2	8	no	—
114	C₂²×C₄×C₆₇	—	2	4	no	—
115	C₂²×C₄×C₆₇	—	2	4	no	—
116	C₂²×C₄×C₆₇	—	2	4	no	—
117	C₂²×C₄×C₆₇	—	2	4	no	—
118	C₂²×C₄×C₆₇	—	2	8	no	—
119	C₂²×C₄×C₆₇	—	2	4	no	—
120	C₂²×C₄×C₆₇	—	2	4	no	—
121	C₂²×C₄×C₆₇	—	2	8	no	—
122	C₂²×C₄×C₆₇	—	2	8	no	—
123	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	16	yes	—
125	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
126	C₂²×C₄×C₆₇	#41	1	8	yes	—
127	C₂²×C₄×C₆₇	#41	1	4	yes	—
128	C₂²×C₄×C₆₇	#41	1	4	yes	—
129	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
131	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
132	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
133	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
134	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
135	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
136	C₄⋊C₄×C₆₇	—	2	8	no	—
137	C₄⋊C₄×C₆₇	—	2	8	no	—
138	C₄⋊C₄×C₆₇	—	2	8	no	—
139	C₄⋊C₄×C₆₇	—	4	16	no	—
140	C₂²⋊C₄×C₆₇	—	2	16	no	—
141	C₂²⋊C₄×C₆₇	—	2	16	no	—
142	C₂²⋊C₄×C₆₇	—	2	16	no	—
143	C₂²⋊C₄×C₆₇	—	2	32	no	—
144	C₂²⋊C₄×C₆₇	—	4	16	no	—
145	C₄²×C₆₇	—	2	16	no	—
146	C₄²×C₆₇	—	2	8	no	—
147	C₄²×C₆₇	—	2	8	no	—
148	C₂³×C₆₇	—	2	8	no	—
149	C₂³×C₆₇	—	2	8	no	—
150	C₂³×C₆₇	—	2	8	no	—
151	C₂³×C₆₇	—	2	4	no	—
152	C₂³×C₆₇	—	2	2	no	—
153	C₂³×C₆₇	—	2	2	no	—
154	C₂³×C₆₇	—	2	4	no	—
155	C₂³×C₆₇	—	2	4	no	—
156	C₂³×C₆₇	—	2	8	no	—
157	C₂³×C₆₇	—	2	4	no	—
158	C₂³×C₆₇	—	2	4	no	—
159	C₂³×C₆₇	—	2	8	no	—
160	C₂³×C₆₇	—	4	8	no	—
161	C₂³×C₆₇	—	4	8	no	—
162	C₂³×C₆₇	—	4	8	no	—
163	C₂³×C₆₇	—	4	8	no	—
164	C₂³×C₆₇	—	4	8	no	—
165	C₂³×C₆₇	—	4	16	no	—
166	C₂³×C₆₇	—	4	16	no	—
167	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
168	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
169	C₂³×C₆₇	C₂×#41	1	2	yes	—
170	C₂³×C₆₇	C₂×#41	1	2	yes	—
171	C₂³×C₆₇	C₂×#41	1	4	yes	—
172	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
173	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
174	D₄×C₆₇	—	4	32	no	—
175	D₄×C₆₇	—	4	16	no	—
176	C₂×C₄×C₆₇	—	2	8	no	—
177	C₂×C₄×C₆₇	—	2	16	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	2	4	no	—
182	C₂×C₄×C₆₇	—	2	32	no	—
183	C₂×C₄×C₆₇	—	2	4	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	4	no	—
186	C₂×C₄×C₆₇	—	2	4	no	—
187	C₂×C₄×C₆₇	—	2	8	no	—
188	C₂×C₄×C₆₇	—	2	4	no	—
189	C₂×C₄×C₆₇	—	2	4	no	—
190	C₂×C₄×C₆₇	—	2	8	no	—
191	C₂×C₄×C₆₇	—	2	8	no	—
192	C₂×C₄×C₆₇	—	4	8	no	—
193	C₂×C₄×C₆₇	—	4	8	no	—
194	C₂×C₄×C₆₇	—	4	8	no	—
195	C₂×C₄×C₆₇	—	4	8	no	—
196	C₂×C₄×C₆₇	#123	1	16	yes	—
197	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
198	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
199	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
200	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
201	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
202	C₂²×C₆₇	—	2	4	no	—
203	C₂²×C₆₇	—	2	4	no	—
204	C₂²×C₆₇	—	2	4	no	—
205	C₂²×C₆₇	—	4	4	no	—
206	C₂²×C₆₇	—	4	4	no	—
207	C₂²×C₆₇	—	4	16	no	—
208	C₂²×C₆₇	—	4	4	no	—
209	C₂²×C₆₇	—	4	4	no	—
210	C₂²×C₆₇	—	4	4	no	—
211	C₂²×C₆₇	—	4	8	no	—
212	C₂²×C₆₇	—	4	8	no	—
213	C₂²×C₆₇	—	8	16	no	—
214	C₂²×C₆₇	C₂×#123	1	4	yes	—
215	C₂²×C₆₇	#532	1	4	yes	—
216	C₂²×C₆₇	#535	1	4	yes	—
217	C₂²×C₆₇	#553	1	8	yes	—
218	C₂²×C₆₇	#555	1	8	yes	—
219	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
220	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
221	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
222	C₂²×C₆₇	C₂²×#41	1	2	yes	—
223	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
224	C₄×C₆₇	—	2	4	no	—
225	C₄×C₆₇	—	2	8	no	—
226	C₄×C₆₇	—	4	4	no	—
227	C₄×C₆₇	—	4	4	no	—
228	C₄×C₆₇	—	4	4	no	—
229	C₄×C₆₇	—	4	4	no	—
230	C₄×C₆₇	C₂×#123	1	8	yes	—
231	C₂×#2500	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂×C₆₇	—	4	4	no	—
233	C₂×C₆₇	—	8	4	no	—
234	C₂×C₆₇	#2500	1	8	yes	direct product factor
235	C₂×C₆₇	C₂²×#123	1	1	yes	characteristic
236	C₂×C₆₇	C₂×#532	1	1	yes	characteristic
237	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
238	C₂×C₆₇	C₂×#553	1	2	yes	—
239	C₂×C₆₇	C₂×#555	1	2	yes	—
240	C₂²×#121	C₂×C₆₇	1	1	yes	characteristic
241	C₂²×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
242	#2500	C₂×C₆₇	1	16	yes	direct product factor
243	C₂×#394	C₂×C₆₇	1	4	yes	complemented
244	C₂×#390	C₂×C₆₇	1	1	yes	characteristic, complemented
245	C₂×#390	C₂×C₆₇	1	1	yes	characteristic, complemented
246	C₂×#388	C₂×C₆₇	1	4	yes	complemented
247	C₂×#384	C₂×C₆₇	1	1	yes	characteristic, complemented
248	C₂×#382	C₂×C₆₇	1	1	yes	characteristic, complemented
249	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
250	C₆₇	C₂×#2500	1	1	yes	characteristic, direct product factor, Sylow
251	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
252	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
253	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
254	C₂×#121	—	2	8	no	—
255	C₂×#121	C₂²×C₆₇	1	8	yes	—
256	C₂×#117	—	2	8	no	—
257	C₂×#117	C₂²×C₆₇	1	8	yes	complemented
258	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
259	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
260	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
261	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
262	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
263	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
264	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
265	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
266	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
267	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
268	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
269	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
270	#394	C₂²×C₆₇	1	32	yes	complemented
271	#390	C₂²×C₆₇	1	8	yes	complemented
272	#390	C₂²×C₆₇	1	8	yes	complemented
273	#388	C₂²×C₆₇	1	32	yes	complemented
274	#384	C₂²×C₆₇	1	8	yes	complemented
275	#382	C₂²×C₆₇	1	8	yes	complemented
276	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
277	C₂×#95	C₂²×C₆₇	1	4	yes	—
278	C₂×#95	C₂²×C₆₇	1	2	yes	—
279	C₂⁵	C₂³×C₆₇	1	2	yes	—
280	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
281	C₂²×D₄	—	2	8	no	—
282	C₂²×D₄	—	2	4	no	—
283	C₂²×D₄	—	2	4	no	—
284	C₂²×D₄	—	2	4	no	—
285	C₂²×D₄	—	4	8	no	—
286	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
287	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
288	C₂³×C₄	—	2	4	no	—
289	C₂³×C₄	—	2	4	no	—
290	C₂³×C₄	D₄×C₆₇	1	8	yes	—
291	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
292	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
293	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
294	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
295	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
296	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
297	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
298	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
299	#121	—	2	32	no	—
300	#117	—	2	32	no	—
301	C₂×C₄⋊C₄	—	2	4	no	—
302	C₂×C₄⋊C₄	—	2	4	no	—
303	C₂×C₄⋊C₄	—	2	4	no	—
304	C₂×C₄⋊C₄	—	4	8	no	—
305	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
306	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
307	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
308	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
309	C₂×C₂²⋊C₄	—	2	8	no	—
310	C₂×C₂²⋊C₄	—	2	8	no	—
311	C₂×C₂²⋊C₄	—	2	8	no	—
312	C₂×C₂²⋊C₄	—	2	16	no	—
313	C₂×C₂²⋊C₄	—	4	8	no	—
314	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
315	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
316	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
317	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
318	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
319	C₂×C₄²	—	2	8	no	—
320	C₂×C₄²	—	2	4	no	—
321	C₂×C₄²	—	2	4	no	—
322	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
323	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
324	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
325	#95	C₂³×C₆₇	1	8	yes	—
326	#95	C₂³×C₆₇	1	16	yes	—
327	C₂⁴	—	2	4	no	—
328	C₂⁴	—	2	2	no	—
329	C₂⁴	—	2	2	no	—
330	C₂⁴	—	2	4	no	—
331	C₂⁴	—	2	4	no	—
332	C₂⁴	—	2	8	no	—
333	C₂⁴	—	2	4	no	—
334	C₂⁴	—	2	4	no	—
335	C₂⁴	—	2	8	no	—
336	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
337	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
338	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
339	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
340	C₂×D₄	—	2	16	no	—
341	C₂×D₄	—	2	8	no	—
342	C₂×D₄	—	2	8	no	—
343	C₂×D₄	—	2	8	no	—
344	C₂×D₄	—	4	16	no	—
345	C₂×D₄	—	4	32	no	—
346	C₂×D₄	—	4	16	no	—
347	C₂²×C₄	—	2	4	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	4	no	—
351	C₂²×C₄	—	2	32	no	—
352	C₂²×C₄	—	2	8	no	—
353	C₂²×C₄	—	2	8	no	—
354	C₂²×C₄	—	2	4	no	—
355	C₂²×C₄	—	2	4	no	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	2	8	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	4	no	—
361	C₂²×C₄	—	2	8	no	—
362	C₂²×C₄	—	2	8	no	—
363	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
364	C₂²×C₄	C₂×D₄×C₆₇	1	16	yes	—
365	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
366	C₂²×C₄	#41×C₆₇	1	8	yes	—
367	C₂²×C₄	#41×C₆₇	1	4	yes	—
368	C₂²×C₄	#41×C₆₇	1	4	yes	—
369	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
370	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
371	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
372	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
373	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
374	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
375	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
376	C₄⋊C₄	—	2	8	no	—
377	C₄⋊C₄	—	2	8	no	—
378	C₄⋊C₄	—	2	8	no	—
379	C₄⋊C₄	—	4	16	no	—
380	C₂²⋊C₄	—	2	16	no	—
381	C₂²⋊C₄	—	2	16	no	—
382	C₂²⋊C₄	—	2	16	no	—
383	C₂²⋊C₄	—	2	32	no	—
384	C₂²⋊C₄	—	4	16	no	—
385	C₄²	—	2	16	no	—
386	C₄²	—	2	8	no	—
387	C₄²	—	2	8	no	—
388	C₂³	—	2	8	no	—
389	C₂³	—	2	8	no	—
390	C₂³	—	2	8	no	—
391	C₂³	—	2	4	no	—
392	C₂³	—	2	2	no	—
393	C₂³	—	2	2	no	—
394	C₂³	—	2	4	no	—
395	C₂³	—	2	4	no	—
396	C₂³	—	2	8	no	—
397	C₂³	—	2	4	no	—
398	C₂³	—	2	4	no	—
399	C₂³	—	2	8	no	—
400	C₂³	—	4	8	no	—
401	C₂³	—	4	8	no	—
402	C₂³	—	4	8	no	—
403	C₂³	—	4	8	no	—
404	C₂³	—	4	8	no	—
405	C₂³	—	4	16	no	—
406	C₂³	—	4	16	no	—
407	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
408	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
409	C₂³	C₂×#41×C₆₇	1	2	yes	—
410	C₂³	C₂×#41×C₆₇	1	2	yes	—
411	C₂³	C₂×#41×C₆₇	1	4	yes	—
412	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
413	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
414	D₄	—	4	32	no	—
415	D₄	—	4	16	no	—
416	C₂×C₄	—	2	8	no	—
417	C₂×C₄	—	2	16	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	4	no	—
422	C₂×C₄	—	2	32	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	2	8	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	8	no	—
431	C₂×C₄	—	2	8	no	—
432	C₂×C₄	—	4	8	no	—
433	C₂×C₄	—	4	8	no	—
434	C₂×C₄	—	4	8	no	—
435	C₂×C₄	—	4	8	no	—
436	C₂×C₄	#123×C₆₇	1	16	yes	—
437	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
438	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
439	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
440	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
441	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
442	C₂²	—	2	4	no	—
443	C₂²	—	2	4	no	—
444	C₂²	—	2	4	no	—
445	C₂²	—	4	16	no	—
446	C₂²	—	4	4	no	—
447	C₂²	—	4	4	no	—
448	C₂²	—	4	4	no	—
449	C₂²	—	4	4	no	—
450	C₂²	—	4	4	no	—
451	C₂²	—	4	8	no	—
452	C₂²	—	4	8	no	—
453	C₂²	—	8	16	no	—
454	C₂²	C₂×#123×C₆₇	1	4	yes	—
455	C₂²	#532×C₆₇	1	4	yes	—
456	C₂²	#535×C₆₇	1	4	yes	—
457	C₂²	#553×C₆₇	1	8	yes	—
458	C₂²	#555×C₆₇	1	8	yes	—
459	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
460	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
461	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
462	C₂²	C₂²×#41×C₆₇	1	2	yes	—
463	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
464	C₄	—	2	4	no	—
465	C₄	—	2	8	no	—
466	C₄	—	4	4	no	—
467	C₄	—	4	4	no	—
468	C₄	—	4	4	no	—
469	C₄	—	4	4	no	—
470	C₄	C₂×#123×C₆₇	1	8	yes	—
471	C₂	—	4	4	no	—
472	C₂	—	8	4	no	—
473	C₂	C₆₇×#2500	1	8	yes	direct product factor
474	C₂	C₂²×#123×C₆₇	1	1	yes	characteristic
475	C₂	C₂×#532×C₆₇	1	1	yes	characteristic
476	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
477	C₂	C₂×#553×C₆₇	1	2	yes	—
478	C₂	C₂×#555×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×#121×C₆₇	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#117×C₆₇	C₂	semidirect
3. C₆₇×#2500	C₂	direct (N × H)
4. C₂×#394×C₆₇	C₂	semidirect
5. C₂×#390×C₆₇	C₂	semidirect
6. C₂×#390×C₆₇	C₂	semidirect
7. C₂×#388×C₆₇	C₂	semidirect
8. C₂×#384×C₆₇	C₂	semidirect
9. C₂×#382×C₆₇	C₂	semidirect
10. C₂×C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
17. C₂×#117×C₆₇	C₂²	semidirect
30. #394×C₆₇	C₂²	semidirect
31. #390×C₆₇	C₂²	semidirect
32. #390×C₆₇	C₂²	semidirect
33. #388×C₆₇	C₂²	semidirect
34. #384×C₆₇	C₂²	semidirect
35. #382×C₆₇	C₂²	semidirect
36. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
231. C₂×#2500	C₆₇	direct (N × H)
234. C₂×C₆₇	#2500	direct (N × H)
241. C₂²×#117	C₂×C₆₇	semidirect
242. #2500	C₂×C₆₇	direct (N × H)
243. C₂×#394	C₂×C₆₇	semidirect
244. C₂×#390	C₂×C₆₇	semidirect
245. C₂×#390	C₂×C₆₇	semidirect
246. C₂×#388	C₂×C₆₇	semidirect
247. C₂×#384	C₂×C₆₇	semidirect
248. C₂×#382	C₂×C₆₇	semidirect
249. C₂×C₄×C₂²⋊C₄	C₂×C₆₇	semidirect
250. C₆₇	C₂×#2500	direct (N × H)
257. C₂×#117	C₂²×C₆₇	semidirect
270. #394	C₂²×C₆₇	semidirect
271. #390	C₂²×C₆₇	semidirect
272. #390	C₂²×C₆₇	semidirect
273. #388	C₂²×C₆₇	semidirect
274. #384	C₂²×C₆₇	semidirect
275. #382	C₂²×C₆₇	semidirect
276. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
473. C₂	C₆₇×#2500	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#121×C₆₇	2
C₂²×#117×C₆₇	2
C₆₇×#2500	2
C₂×#394×C₆₇	2
C₂×#390×C₆₇	2
C₂×#388×C₆₇	2
C₂×#384×C₆₇	2
C₂×#382×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₂×#2500	67

Subgroups — C2×C67×#2500 #30714045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2500 #30714045